Single Best Answer MCQs for Sub-Topics of Topic 16: Statistics & Probability

Question 1. The collection of facts, figures, and other information in a definite form is called:

(A) Information

(B) Data

(C) Statistics

(D) Raw data

Question 2. Data collected from a survey regarding the number of children in 50 households in a village, before any processing, is an example of:

(A) Organized data

(B) Secondary data

(C) Raw data

(D) Summarized data

Question 3. A single measurement or observation of a variable is known as:

(A) Data

(B) Variable

(C) Raw data

(D) Observation

Question 4. A characteristic that can vary from one individual to another is called:

(A) Constant

(B) Attribute

(C) Variable

(D) Data

Question 5. Which of the following is a quantitative variable?

(A) Eye colour

(B) Marital status

(C) Number of students in a class

(D) Religion

Question 6. Which of the following is a qualitative variable?

(A) Height

(B) Temperature

(C) Favourite colour

(D) Number of siblings

Question 7. Data that can only take specific values, often integers, is called:

(A) Continuous data

(B) Qualitative data

(C) Discrete data

(D) Primary data

Question 8. Data that can take any value within a given range is called:

(A) Discrete data

(B) Qualitative data

(C) Categorical data

(D) Continuous data

Question 9. The branch of mathematics concerned with collecting, organizing, analyzing, interpreting, and presenting data is called:

(A) Algebra

(B) Calculus

(C) Statistics

(D) Geometry

Question 10. The main purpose of statistics is to:

(A) Provide exact predictions about future events.

(B) Make inferences about a population based on a sample.

(C) Replace detailed qualitative analysis with numbers.

(D) Only describe numerical data without interpretation.

Question 11. Data collected by the investigator himself for a specific purpose is called:

(A) Secondary data

(B) Primary data

(C) External data

(D) Published data

Question 12. Data collected from published reports, like a government census report, is an example of:

(A) Primary data

(B) Internal data

(C) Secondary data

(D) Raw data

Question 13. Which of the following is the correct sequence of stages in data handling?

(A) Collection, Organization, Analysis, Presentation, Interpretation

(B) Organization, Collection, Presentation, Analysis, Interpretation

(C) Collection, Organization, Presentation, Analysis, Interpretation

(D) Collection, Presentation, Organization, Analysis, Interpretation

Question 14. Arranging data in a systematic order or grouping it into classes is part of which stage of data handling?

(A) Collection

(B) Presentation

(C) Organization

(D) Analysis

Question 15. Summarizing data using tables, charts, or graphs is part of which stage of data handling?

(A) Organization

(B) Presentation

(C) Analysis

(D) Interpretation

Question 16. Applying statistical methods to extract meaningful insights from data is part of which stage of data handling?

(A) Presentation

(B) Analysis

(C) Interpretation

(D) Collection

Question 17. Drawing conclusions and making decisions based on the results of data analysis is part of which stage of data handling?

(A) Analysis

(B) Presentation

(C) Interpretation

(D) Organization

Question 18. Grouping data into classes with specified ranges is called:

(A) Data collection

(B) Tabulation

(C) Data interpretation

(D) Data grouping

Question 19. A value that remains constant throughout a study is called a:

(A) Variable

(B) Parameter

(C) Statistic

(D) Constant

Question 20. A survey conducted by a researcher collecting data directly from respondents for their thesis is an example of collecting:

(A) Secondary data

(B) Tertiary data

(C) Primary data

(D) Published data

Question 21. The process of extracting conclusions from the data presented in tables and graphs is called:

(A) Data collection

(B) Data organization

(C) Data presentation

(D) Data interpretation

Question 22. Which of the following is NOT typically considered a stage of data handling?

(A) Data Destruction

(B) Data Collection

(C) Data Analysis

(D) Data Organization

Question 23. When data is presented in an array (ordered list), it is a form of:

(A) Raw data

(B) Grouped data

(C) Organized data

(D) Classified data

Question 24. The unit on which an observation is made is called a:

(A) Variable unit

(B) Observation unit

(C) Statistical unit

(D) Data unit

Question 25. Data related to attributes like honesty or beauty are examples of:

(A) Quantitative data

(B) Numerical data

(C) Qualitative data

(D) Discrete data

Question 1. The number of times a particular observation occurs in a data set is called its:

(A) Class limit

(B) Frequency

(C) Class interval

(D) Cumulative frequency

Question 2. A table that shows the frequency of each observation or group of observations is called a:

(A) Data table

(B) Frequency distribution table

(C) Tally table

(D) Class table

Question 3. In a tally chart, the symbol $\bcancel{||||}$ represents a frequency of:

(A) 4

(B) 5

(C) 6

(D) 10

Question 4. An ungrouped frequency distribution table is typically used for:

(A) Large data sets with a wide range of values

(B) Continuous data

(C) Small data sets or data with a limited number of distinct values

(D) Qualitative data only

Question 5. In the class interval 10-20, the lower class limit is:

(A) 10

(B) 20

(C) 15

(D) 30

Question 6. In the class interval 10-20 (exclusive), the upper class limit is:

(A) 10

(B) 19

(C) 20

(D) 20.5

Question 7. The difference between the upper class limit and the lower class limit of a class interval is called:

(A) Class mark

(B) Class boundary

(C) Class frequency

(D) Class size/width

Question 8. The mid-point of a class interval is called the:

(A) Class limit

(B) Class boundary

(C) Class mark

(D) Class frequency

Question 9. For the class interval 20-30, the class mark is:

(A) 20

(B) 30

(C) 25

(D) 10

Question 10. In an inclusive class interval like 10-19, an observation with value 19 is included in:

(A) The class 0-9

(B) The class 10-19

(C) The class 20-29

(D) Both 10-19 and 20-29

Question 11. In an exclusive class interval like 10-20, an observation with value 20 is included in:

(A) The class 10-20

(B) The class 20-30

(C) The class 0-10

(D) Neither 10-20 nor 20-30

Question 12. The sum of frequencies of all observations up to a certain value or class interval is called:

(A) Relative frequency

(B) Class frequency

(C) Cumulative frequency

(D) Total frequency

Question 13. In a 'less than' cumulative frequency distribution, the cumulative frequency for a class interval represents the number of observations:

(A) Greater than the lower limit

(B) Less than the upper limit

(C) Exactly equal to the class mark

(D) Within that class interval

Question 14. In a 'more than' cumulative frequency distribution, the cumulative frequency for a class interval represents the number of observations:

(A) Less than the upper limit

(B) Greater than or equal to the lower limit

(C) Exactly equal to the class mark

(D) Within that class interval

Question 15. The sum of all frequencies in a frequency distribution table is equal to:

(A) The number of classes

(B) The range of the data

(C) The total number of observations

(D) The sum of class marks

Question 16. If the cumulative frequency 'less than' 30 is 15 and 'less than' 40 is 35, the frequency of the class interval 30-40 is:

(A) 15

(B) 20

(C) 35

(D) 50

Question 17. Open-ended class intervals are used when:

(A) All observations have the same value

(B) There are extreme values at the ends of the data range

(C) The data is continuous

(D) The total number of observations is small

Question 18. Converting exclusive class intervals (e.g., 10-20, 20-30) to inclusive class intervals (e.g., 10-19, 20-29) is sometimes done for:

(A) Calculating class marks easily

(B) Avoiding confusion with boundary values

(C) Graphical representation like histograms

(D) Manual tabulation clarity

Question 19. The advantage of grouping data is to:

(A) Preserve all the original information

(B) Make the data easier to understand and analyze

(C) Always make calculations more precise

(D) Reduce the total number of observations

Question 20. The disadvantage of grouping data is that it:

(A) Increases the number of classes

(B) Leads to some loss of information about individual observations

(C) Makes it difficult to calculate frequencies

(D) Requires more space for presentation

Question 21. If the upper boundary of a class is 50 and its class size is 10, the lower boundary is:

(A) 40

(B) 45

(C) 50

(D) 60

Question 22. The class mark of the class interval 15-25 is:

(A) 10

(B) 20

(C) 25

(D) 40

Question 23. If the class marks of a distribution are 12, 18, 24, 30, the class size is:

(A) 4

(B) 6

(C) 8

(D) 10

Question 24. If the lower boundary of a class is 60 and its class size is 8, the class mark is:

(A) 64

(B) 66

(C) 68

(D) 70

Question 25. The total number of observations is 100. The 'less than' cumulative frequency for the last class interval is:

(A) Less than 100

(B) 100

(C) Greater than 100

(D) Cannot be determined

Question 1. Graphical representation of data helps in:

(A) Only precise calculation

(B) Understanding trends and patterns easily

(C) Eliminating the need for data analysis

(D) Storing raw data efficiently

Question 2. A graph that represents data using pictures or symbols is called a:

(A) Bar graph

(B) Histogram

(C) Pictograph

(D) Pie chart

Question 3. In a pictograph, if one symbol of a car represents 100 cars, how many symbols are needed to represent 450 cars?

(A) 4

(B) 4.5

(C) 5

(D) 45

Question 4. A graph using bars of uniform width, where the height of each bar is proportional to the value it represents, is called a:

(A) Pictograph

(B) Histogram

(C) Bar graph

(D) Ogive

Question 5. In a bar graph, the bars are separated by equal spaces. This is because it is used to represent:

(A) Continuous data

(B) Grouped data

(C) Discrete or categorical data

(D) Cumulative frequency

Question 6. A double bar graph is useful for:

(A) Showing proportion of parts to a whole

(B) Displaying a continuous frequency distribution

(C) Comparing two sets of related data simultaneously

(D) Representing data using symbols

Question 7. A circle graph divided into sectors, where the area of each sector is proportional to the part of the whole it represents, is called a:

(A) Bar graph

(B) Pie chart

(C) Histogram

(D) Frequency polygon

Question 8. The total angle at the centre of a pie chart is:

(A) $90^\circ$

(B) $180^\circ$

(C) $270^\circ$

(D) $360^\circ$

Question 9. If a category represents 25% of the total in a pie chart, the angle of the corresponding sector is:

(A) $25^\circ$

(B) $90^\circ$

(C) $180^\circ$

(D) $360^\circ$

Question 10. Which type of graph is best suited for showing the expenditure on different items in a household?

(A) Bar graph

(B) Double bar graph

(C) Pie chart

(D) Histogram

Question 11. Which type of graph is best suited for comparing the marks obtained by students in two different subjects?

(A) Single bar graph

(B) Pie chart

(C) Double bar graph

(D) Pictograph

Question 12. In a bar graph, the height of a bar represents:

(A) Class interval

(B) Frequency or value

(C) Class mark

(D) Cumulative frequency

Question 13. If a pie chart shows favourite sports of 200 students, and cricket is favoured by 100 students, the angle for cricket sector is:

(A) $90^\circ$

(B) $120^\circ$

(C) $180^\circ$

(D) $200^\circ$

Question 14. Pictographs are often used for:

(A) Representing very precise data values

(B) Presenting data in a simple and visually appealing way, especially for a general audience

(C) Analyzing complex statistical relationships

(D) Handling continuous data

Question 15. The bars in a single bar graph have:

(A) Varying widths and varying spaces

(B) Uniform widths and varying spaces

(C) Uniform widths and uniform spaces

(D) Varying widths and uniform spaces

Question 16. If a double bar graph compares sales of two products over 5 months, how many pairs of bars will be shown?

(A) 2

(B) 5

(C) 10

(D) 7

Question 17. Which of the following graphs is NOT suitable for representing qualitative data like colours or regions?

(A) Bar graph

(B) Pictograph

(C) Pie chart

(D) Histogram

Question 18. To compare the population of four different cities in India using a single bar graph, the cities would be plotted on the:

(A) Vertical axis

(B) Horizontal axis

(C) Diagonal axis

(D) Not possible with a single bar graph

Question 19. A pie chart is also known as a:

(A) Line graph

(B) Circle graph

(C) Scatter plot

(D) Stem-and-leaf plot

Question 20. If a survey found that 40% of people prefer tea, 30% coffee, 20% milk, and 10% juice, how would you represent this distribution using basic charts?

(A) Histogram

(B) Frequency polygon

(C) Ogive

(D) Pie chart or Bar graph

Question 21. In a pie chart, the proportion of each category is represented by the:

(A) Radius of the circle

(B) Area of the sector

(C) Circumference of the circle

(D) Length of the arc only

Question 22. Which graphical representation would you use to show the production of wheat in Punjab over the last 5 years?

(A) Pie Chart

(B) Histogram

(C) Single Bar Graph (or line graph, though not listed as option)

(D) Double Bar Graph

Question 23. To compare the performance of two students in multiple subjects, the most suitable graph is:

(A) Pictograph

(B) Single Bar Graph

(C) Double Bar Graph

(D) Pie Chart

Question 24. When constructing a pie chart, the first step is usually to:

(A) Draw a circle

(B) Calculate the total value of the data

(C) Calculate the angle for each sector

(D) Divide the circle into equal parts

Question 25. In a pictograph, if half a symbol represents 5 units, a full symbol represents:

(A) 2.5 units

(B) 5 units

(C) 10 units

(D) 15 units

Question 1. A histogram is a graphical representation of a frequency distribution for:

(A) Ungrouped discrete data

(B) Categorical data

(C) Grouped continuous data

(D) Time series data

Question 2. In a histogram, the bars are adjacent to each other because:

(A) The data is discrete

(B) The class intervals are continuous

(C) It represents frequencies

(D) It is used for comparison

Question 3. The height of a bar in a histogram with equal class widths represents the:

(A) Class mark

(B) Class limit

(C) Frequency of the class

(D) Cumulative frequency

Question 4. If a histogram has unequal class widths, the height of the bars is adjusted. This adjustment is made based on the:

(A) Frequency of the class

(B) Class width

(C) Frequency density (Frequency / Class Width)

(D) Class mark

Question 5. A frequency polygon is a graphical representation of a frequency distribution formed by joining the mid-points of the tops of adjacent bars in a:

(A) Bar graph

(B) Histogram

(C) Pie chart

(D) Ogive

Question 6. To plot a frequency polygon directly from a frequency table, we plot points by taking the class marks on the x-axis and the corresponding on the y-axis.

(A) Class limits

(B) Class boundaries

(C) Frequencies

(D) Cumulative frequencies

Question 7. To make the area under the frequency polygon equal to the area under the corresponding histogram, the frequency polygon is closed by joining the mid-points of the extreme classes to the:

(A) Origin (0,0)

(B) X-axis at the upper boundary of the last class

(C) X-axis at the lower boundary of the first class and upper boundary of the last class (assuming zero frequency classes before first and after last)

(D) Highest frequency point

Question 8. Histograms are generally used for:

(A) Comparing proportions of a whole

(B) Showing the frequency of discrete items

(C) Representing the distribution of continuous data

(D) Showing cumulative frequencies

Question 9. A frequency polygon helps in visualizing the __________ of the distribution.

(A) Central tendency

(B) Spread

(C) Shape

(D) Cumulative total

Question 10. Which graph is suitable for comparing the distribution of marks of two different sections of students on the same test?

(A) Double bar graph

(B) Histogram (if grouped data)

(C) Frequency polygon (two overlaid)

(D) Pie chart

Question 11. If the class intervals are 10-20, 20-30, 30-40 with frequencies 5, 10, 8, what are the class marks?

(A) 10, 20, 30

(B) 15, 25, 35

(C) 5, 10, 8

(D) 20, 30, 40

Question 12. When drawing a histogram, the variable (e.g., marks, heights) is represented on the:

(A) Vertical axis

(B) Horizontal axis

(C) Both axes simultaneously

(D) Not represented on either axis

Question 13. If the class boundaries are used for constructing a histogram, the lower boundary of the first class is plotted at:

(A) The origin

(B) The starting point on the x-axis representing the first class boundary

(C) The midpoint of the first class

(D) Any arbitrary point

Question 14. The area of each rectangle in a histogram is proportional to the _________ of the corresponding class.

(A) Class width

(B) Class mark

(C) Cumulative frequency

(D) Frequency

Question 15. For a frequency distribution with class intervals 0-10, 10-20, 20-30, etc., a histogram would have bars placed:

(A) With gaps between them

(B) Adjacent to each other

(C) Over the class marks

(D) Randomly placed

Question 16. If the class intervals are 10-19, 20-29, 30-39, etc. (inclusive), what must be done before constructing a histogram?

(A) Calculate cumulative frequencies

(B) Find class marks

(C) Convert them to exclusive intervals/find class boundaries

(D) Calculate relative frequencies

Question 17. The graphical representation showing frequencies plotted against class marks, joined by line segments, is a:

(A) Histogram

(B) Bar graph

(C) Frequency polygon

(D) Ogive

Question 18. The sum of the areas of all rectangles in a histogram is proportional to the:

(A) Number of classes

(B) Total frequency

(C) Range of data

(D) Mean of the data

Question 19. When comparing two frequency distributions using frequency polygons, it is important that:

(A) They have the same total frequency

(B) They are plotted on the same axes and scale

(C) They have the same class intervals

(D) All of the above are helpful but (B) is essential for direct comparison

Question 20. The point on a frequency polygon corresponding to a class interval represents the frequency of that class at its:

(A) Lower limit

(B) Upper limit

(C) Class mark

(D) Class boundary

Question 1. A graphical representation of a cumulative frequency distribution is called:

(A) Histogram

(B) Frequency polygon

(C) Ogive

(D) Bar graph

Question 2. In a 'less than' ogive, the cumulative frequency is plotted against the:

(A) Lower limits of the class intervals

(B) Upper limits of the class intervals

(C) Class marks of the class intervals

(D) Frequencies of the class intervals

Question 3. In a 'more than' ogive, the cumulative frequency is plotted against the:

(A) Lower limits of the class intervals

(B) Upper limits of the class intervals

(C) Class marks of the class intervals

(D) Frequencies of the class intervals

Question 4. A 'less than' ogive is generally:

(A) Decreasing

(B) Increasing

(C) Bell-shaped

(D) Horizontal

Question 5. A 'more than' ogive is generally:

(A) Decreasing

(B) Increasing

(C) Bell-shaped

(D) Horizontal

Question 6. The Median of a distribution can be estimated graphically using:

(A) Histogram

(B) Frequency polygon

(C) Ogives

(D) Bar graph

Question 7. To estimate the Median from a 'less than' ogive with total frequency N, you locate N/2 on the y-axis, draw a horizontal line to the ogive, and then drop a vertical line to the x-axis. The value on the x-axis is the estimated Median. What should be plotted on the x-axis?

(A) Class marks

(B) Frequencies

(C) Upper class boundaries

(D) Cumulative frequencies

Question 8. The point of intersection of the 'less than' ogive and the 'more than' ogive gives the:

(A) Mean

(B) Median

(C) Mode

(D) Quartiles

Question 9. The x-coordinate of the intersection point of the two ogives represents the:

(A) Total frequency

(B) Frequency of the modal class

(C) Median

(D) Mean

Question 10. The y-coordinate of the intersection point of the two ogives represents the:

(A) Total frequency

(B) Half of the total frequency (N/2)

(C) Median class frequency

(D) Cumulative frequency of the first class

Question 11. Ogives are primarily used to determine:

(A) Frequencies of classes

(B) Measures of central tendency like Median and other positional values

(C) The shape of the frequency distribution curve

(D) The range of the data

Question 12. To find the number of observations below a certain value from a 'less than' ogive, you would:

(A) Locate the value on the y-axis and draw a horizontal line

(B) Locate the value on the x-axis and draw a vertical line to the ogive

(C) Look at the total frequency

(D) Find the frequency of the class containing that value

Question 13. To find the number of observations above a certain value from a 'more than' ogive, you would:

(A) Locate the value on the y-axis and draw a horizontal line

(B) Locate the value on the x-axis and draw a vertical line to the ogive

(C) Look at the total frequency

(D) Find the frequency of the class containing that value

Question 14. Can quartiles (Q1, Q3) be estimated from ogives?

(A) No, only Median can be estimated.

(B) Yes, using the same method as Median but with N/4 and 3N/4 on the y-axis.

(C) Yes, but only from the point of intersection of the two ogives.

(D) Yes, but you need a frequency polygon as well.

Question 15. For constructing an ogive, which values are typically plotted on the x-axis?

(A) Class marks

(B) Frequencies

(C) Class boundaries

(D) Cumulative frequencies

Question 16. The first point plotted for a 'less than' ogive is usually:

(A) (lower boundary of first class, 0)

(B) (upper boundary of first class, frequency of first class)

(C) (class mark of first class, cumulative frequency of first class)

(D) (0, 0)

Question 17. The last point plotted for a 'less than' ogive is at the cumulative frequency equal to:

(A) The frequency of the last class

(B) Half of the total frequency

(C) The total frequency

(D) The frequency density of the last class

Question 18. Which type of ogive starts from a cumulative frequency of 0 on the y-axis?

(A) More than ogive

(B) Less than ogive

(C) Both less than and more than ogives

(D) Neither type of ogive starts at 0

Question 19. The total number of students in a class is 50. To find the median marks graphically, you would look for the mark corresponding to a cumulative frequency of:

(A) 100

(B) 50

(C) 25

(D) Cannot be determined without the data

Question 20. Which graph is most suitable for showing the number of students who scored marks less than 60 in a test?

(A) Histogram

(B) Frequency polygon

(C) Less than ogive

(D) More than ogive

Question 21. The steepness of an ogive indicates the concentration of observations. A steeper section means:

(A) Lower frequency in that range

(B) Higher frequency in that range

(C) The median is in that range

(D) The data is less spread out

Question 22. When constructing an ogive from a frequency distribution with class intervals, it is crucial to use:

(A) Class marks

(B) Class frequencies

(C) Class boundaries

(D) Class limits (exclusive format)

Question 23. An ogive helps in determining __________ values visually.

(A) Extreme values

(B) Positional values like median, quartiles, percentiles

(C) Mean and Mode

(D) Range and standard deviation

Question 24. If a point on a 'less than' ogive is (70, 45), it means:

(A) 45 observations have a value of exactly 70.

(B) 70 observations have a value less than 45.

(C) 45 observations have a value less than 70.

(D) The frequency of the class with upper boundary 70 is 45.

Question 25. If a point on a 'more than' ogive is (30, 60), it means:

(A) 60 observations have a value of exactly 30.

(B) 30 observations have a value greater than or equal to 60.

(C) 60 observations have a value greater than or equal to 30.

(D) The cumulative frequency for values less than 30 is 60.

Question 1. A single value that attempts to describe a set of data by identifying the central position within that set is called a:

(A) Measure of dispersion

(B) Measure of central tendency

(C) Measure of correlation

(D) Measure of skewness

Question 2. Measures of central tendency are also known as:

(A) Measures of variation

(B) Averages

(C) Positional values

(D) Measures of symmetry

Question 3. The most commonly used measure of central tendency is the:

(A) Median

(B) Mode

(C) Arithmetic Mean

(D) Range

Question 4. The sum of all observations divided by the number of observations is the:

(A) Median

(B) Mode

(C) Mean

(D) Range

Question 5. For the data set {10, 15, 20, 25, 30}, the arithmetic mean is:

(A) 15

(B) 20

(C) 25

(D) 100

Question 6. If a constant 'c' is added to every observation in a data set, the mean of the new data set will be:

(A) Original mean + c

(B) Original mean - c

(C) Original mean $\times$ c

(D) Original mean / c

Question 7. If every observation in a data set is multiplied by a constant 'k' (k $\neq$ 0), the mean of the new data set will be:

(A) Original mean + k

(B) Original mean - k

(C) Original mean $\times$ k

(D) Original mean / k

Question 8. The arithmetic mean is significantly affected by:

(A) The number of observations

(B) The order of observations

(C) Extreme values (outliers)

(D) The scale used for measurement

Question 9. Which method of calculating mean for grouped data is generally simplest for small, simple numbers?

(A) Assumed Mean Method

(B) Step-Deviation Method

(C) Direct Method

(D) Median Method

Question 10. The formula for calculating the mean of grouped data using the Direct Method is: ($\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$). Here $x_i$ represents:

(A) Lower limit of the class

(B) Upper limit of the class

(C) Class mark (mid-point) of the class

(D) Frequency of the class

Question 11. In the Assumed Mean Method, the deviation $d_i$ is calculated as:

(A) $d_i = x_i - \bar{x}$

(B) $d_i = x_i - A$ (where A is assumed mean)

(C) $d_i = f_i x_i$

(D) $d_i = x_i / h$ (where h is class size)

Question 12. The Step-Deviation Method is a simplification of the Assumed Mean Method, particularly useful when:

(A) Class widths are unequal

(B) Class marks are large and have a common factor (class size)

(C) The mean is expected to be very small

(D) The data is ungrouped

Question 13. If the mean of 5 observations $x, x+2, x+4, x+6, x+8$ is 18, the value of x is:

(A) 10

(B) 12

(C) 14

(D) 16

Question 14. The mean of the first 10 natural numbers (1, 2, ..., 10) is:

(A) 5

(B) 5.5

(C) 6

(D) 10

Question 15. The sum of the deviations of individual observations from their mean is always:

(A) Positive

(B) Negative

(C) Zero

(D) Equal to the mean itself

Question 16. If the mean of a data set is 50 and a calculation error is found where a value 20 was incorrectly read as 30 for one observation, what is the corrected mean if there are 10 observations?

(A) 49

(B) 50

(C) 51

(D) 52

Question 17. Two sections of Class X have 30 and 35 students. The average marks in a test for the first section was 60, and for the second section was 55. The average marks for the entire class is approximately:

(A) 57.5

(B) 57.3

(C) 58.1

(D) 57.7

Question 18. If all the observations in a data set are the same (e.g., {5, 5, 5, 5}), the mean is:

(A) 0

(B) 5

(C) Undefined

(D) Depends on the number of observations

Question 19. The mean is most appropriate for data that is:

(A) Qualitative

(B) Highly skewed

(C) Symmetric and interval/ratio scale

(D) Ordinal

Question 20. The mean of the first 5 even numbers (2, 4, 6, 8, 10) is:

(A) 5

(B) 6

(C) 7

(D) 10

Question 21. If the mean of 20 observations is 40, and one observation 60 is removed, the mean of the remaining 19 observations is:

(A) 38

(B) 39

(C) 40

(D) 41

Question 22. Which of the following statements about the arithmetic mean is true?

(A) It is not affected by extreme values.

(B) It can be calculated for qualitative data.

(C) It is the balance point of the data set.

(D) It must be one of the observations in the data set.

Question 23. If the mean of the data 6, 8, x, 10, 12 is 9, the value of x is:

(A) 7

(B) 9

(C) 11

(D) 13

Question 24. The mean is a measure of:

(A) Dispersion

(B) Central tendency

(C) Association

(D) Variability

Question 25. The sum of the deviations of observations from an assumed mean is 150, and the total number of observations is 30. If the assumed mean is 40, the actual mean is:

(A) 45

(B) 35

(C) 50

(D) 40

Question 1. The middle value of a data set when arranged in ascending or descending order is called the:

(A) Mean

(B) Median

(C) Mode

(D) Range

Question 2. For the ungrouped data set {7, 12, 5, 9, 15}, the Median is:

(A) 5

(B) 9

(C) 12

(D) 15

Question 3. For the ungrouped data set {10, 20, 15, 30, 25, 5}, the Median is:

(A) 15

(B) 17.5

(C) 20

(D) 22.5

Question 4. The Median is a better measure of central tendency than the Mean when the data contains:

(A) Only positive values

(B) A large number of observations

(C) Extreme values or outliers

(D) Values close to each other

Question 5. The Median is a positional average because it is determined by its position in the data set, not by the value of every observation. (True/False)

(A) True

(B) False

(C) Depends on the data type

(D) Only true for ungrouped data

Question 6. To find the Median for grouped data, the first step is to calculate:

(A) Class marks

(B) Relative frequencies

(C) Cumulative frequencies

(D) Frequency densities

Question 7. The Median class for grouped data is the class interval where the (N/2)th observation falls, where N is the total frequency. Which cumulative frequency table is used to find this class?

(A) Any cumulative frequency table

(B) 'Less than' cumulative frequency table

(C) 'More than' cumulative frequency table

(D) Both 'less than' and 'more than' tables must be used together

Question 8. In the formula for Median of grouped data, $M = L + \frac{(N/2 - cf)}{f} \times h$, 'cf' represents:

(A) Cumulative frequency of the median class

(B) Cumulative frequency of the class succeeding the median class

(C) Cumulative frequency of the class preceding the median class

(D) Frequency of the median class

Question 9. In the formula for Median of grouped data, 'L' represents:

(A) Lower limit of the median class

(B) Upper limit of the median class

(C) Class mark of the median class

(D) Lower boundary of the median class

Question 10. The Median can be estimated graphically from:

(A) Histogram

(B) Frequency polygon

(C) Ogives (Cumulative frequency graphs)

(D) Bar graph

Question 11. For symmetric distributions, the Median is equal to the:

(A) Mode only

(B) Mean only

(C) Both Mean and Mode

(D) Range

Question 12. If the Median of a data set is 45, it means:

(A) Half of the observations are exactly 45.

(B) Half of the observations are less than or equal to 45, and half are greater than or equal to 45.

(C) The average value is 45.

(D) 45 is the most frequent value.

Question 13. The Median is not defined for data that is:

(A) Discrete numerical

(B) Continuous numerical

(C) Ordinal categorical

(D) Nominal categorical (like colours)

Question 14. If a data set has 11 observations, the Median is the value of the observation at position:

(A) 5th

(B) 6th

(C) 5.5th

(D) Depends on the values

Question 15. If a data set has 12 observations, the Median is the average of the values at positions:

(A) 6th and 7th

(B) 5th and 6th

(C) 6th only

(D) 6.5th (which means averaging the 6th and 7th)

Question 16. The Median of the first 5 odd numbers (1, 3, 5, 7, 9) is:

(A) 3

(B) 5

(C) 7

(D) 9

Question 17. The Median of the first 6 natural numbers (1, 2, 3, 4, 5, 6) is:

(A) 3

(B) 3.5

(C) 4

(D) 4.5

Question 18. Which measure of central tendency is least affected by extreme values?

(A) Mean

(B) Median

(C) Mode

(D) Geometric Mean

Question 19. For grouped data, if the cumulative frequency less than 50 is 20 and less than 60 is 35, and the total frequency is 80, what is the median class if N/2 is 40?

(A) 40-50

(B) 50-60

(C) 60-70

(D) Cannot be determined

Question 1. The value that appears most frequently in a data set is called the:

(A) Mean

(B) Median

(C) Mode

(D) Range

Question 2. For the ungrouped data set {2, 5, 7, 5, 2, 5, 9, 2, 5, 1}, the Mode is:

(A) 2

(B) 5

(C) 2 and 5

(D) 9

Question 3. A data set with two modes is called:

(A) Unimodal

(B) Bimodal

(C) Multimodal

(D) Non-modal

Question 4. If all observations in a data set occur with the same frequency, the data set has:

(A) One mode

(B) Multiple modes

(C) No mode

(D) Depends on the number of observations

Question 5. The Mode is most useful for which type of data?

(A) Continuous numerical data

(B) Qualitative (Nominal) data

(C) Highly skewed numerical data

(D) Symmetric numerical data

Question 6. For grouped data, the Modal class is the class interval with the:

(A) Highest class mark

(B) Smallest frequency

(C) Highest frequency

(D) Highest cumulative frequency

Question 7. The empirical formula relating Mean, Median, and Mode for moderately skewed distributions is approximately:

(A) Mean = 3 Median - 2 Mode

(B) Mode = 3 Mean - 2 Median

(C) Mode = 3 Median - 2 Mean

(D) Mean = 3 Mode - 2 Median

Question 8. If the Mean is 50 and the Median is 52 for a distribution, the estimated Mode using the empirical formula is:

(A) 54

(B) 56

(C) 50

(D) 51

Question 9. For a symmetric distribution, which of the following is true?

(A) Mean < Median < Mode

(B) Mean > Median > Mode

(C) Mean = Median = Mode

(D) Mean $\neq$ Median $\neq$ Mode

Question 10. For a positively skewed distribution, which of the following is generally true?

(A) Mean < Median < Mode

(B) Mean > Median > Mode

(C) Mean = Median = Mode

(D) Mean $\approx$ Median $\approx$ Mode

Question 11. For a negatively skewed distribution, which of the following is generally true?

(A) Mean < Median < Mode

(B) Mean > Median > Mode

(C) Mean = Median = Mode

(D) Median > Mean > Mode

Question 12. Which measure of central tendency is the only one applicable to nominal data?

(A) Mean

(B) Median

(C) Mode

(D) All three

Question 13. In the formula for Mode of grouped data, $Mode = L + \frac{(f_1 - f_0)}{(2f_1 - f_0 - f_2)} \times h$, $f_1$ represents:

(A) Frequency of the class preceding the modal class

(B) Frequency of the modal class

(C) Frequency of the class succeeding the modal class

(D) Cumulative frequency of the modal class

Question 14. If the Modal class is 30-40 with frequency 15, and the preceding and succeeding class frequencies are 10 and 12 respectively, the value of $f_0$ in the mode formula is:

(A) 10

(B) 12

(C) 15

(D) 30

Question 15. Which measure of central tendency can be estimated from a histogram?

(A) Mean

(B) Median

(C) Mode

(D) All three

Question 16. The mode is the peak of the distribution. In a histogram, the mode can be estimated by drawing lines from the top corners of the modal class bar to the opposite top corners of the adjacent bars and dropping a perpendicular from the intersection to the x-axis. The value on the x-axis is the estimated mode. (True/False)

(A) True

(B) False

(C) Only true for symmetric distributions

(D) Only true for ungrouped data

Question 17. In a distribution where Mean=50, Median=50, and Mode=50, the distribution is likely:

(A) Positively skewed

(B) Negatively skewed

(C) Symmetric

(D) Bimodal

Question 18. The empirical relationship between Mean, Median, and Mode is strictly true only for:

(A) Any distribution

(B) Moderately skewed distributions

(C) Perfectly symmetric distributions

(D) Binomial distributions

Question 19. If the Mode of a grouped data with class size 10 is 65, and the modal class is 60-70, which frequencies among $f_0, f_1, f_2$ is the highest?

(A) $f_0$

(B) $f_1$

(C) $f_2$

(D) Cannot be determined from Mode value alone

Question 20. The Mean, Median, and Mode are all equal in value for a:

(A) Uniform distribution

(B) Exponential distribution

(C) Normal distribution

(D) Chi-square distribution

Question 1. Measures of dispersion indicate the extent to which data values are:

(A) Concentrated around the center

(B) Symmetric or asymmetric

(C) Spread out or scattered

(D) Related to another variable

Question 2. Which of the following is NOT a measure of dispersion?

(A) Range

(B) Median

(C) Standard Deviation

(D) Variance

Question 3. The difference between the highest and lowest value in a data set is called the:

(A) Mean Deviation

(B) Interquartile Range

(C) Range

(D) Standard Deviation

Question 4. For the data set {15, 20, 10, 25, 30}, the Range is:

(A) 10

(B) 20

(C) 25

(D) 30

Question 5. The Range is a simple measure of dispersion, but its main limitation is that it:

(A) Is difficult to calculate

(B) Is affected by all observations

(C) Only considers the two extreme values

(D) Cannot be calculated for discrete data

Question 6. The average of the absolute deviations of individual observations from a measure of central tendency (Mean or Median) is called:

(A) Range

(B) Variance

(C) Standard Deviation

(D) Mean Deviation

Question 7. When calculating Mean Deviation, absolute values of deviations are used to:

(A) Make the calculation simpler

(B) Ensure the sum of deviations is zero

(C) Avoid the deviations canceling each other out

(D) Give more weight to larger deviations

Question 8. For the data set {2, 4, 6, 8}, the Mean is 5. The Mean Deviation from the Mean is:

(A) 1.5

(B) 2

(C) 2.5

(D) 3

Question 9. Mean Deviation is minimum when calculated from the:

(A) Mean

(B) Median

(C) Mode

(D) Range

Question 10. If a constant 'c' is added to every observation in a data set, the Range of the new data set will:

(A) Increase by c

(B) Decrease by c

(C) Remain unchanged

(D) Be multiplied by c

Question 11. If every observation in a data set is multiplied by a positive constant 'k', the Mean Deviation of the new data set will be:

(A) Original Mean Deviation + k

(B) Original Mean Deviation $\times$ k

(C) Original Mean Deviation - k

(D) Original Mean Deviation / k

Question 12. Mean Deviation is less affected by extreme values compared to:

(A) Median

(B) Quartile Deviation

(C) Standard Deviation

(D) Range

Question 13. The coefficient of Range is given by the formula:

(A) (Max - Min) / 2

(B) (Max - Min) / (Max + Min)

(C) (Max + Min) / (Max - Min)

(D) Max - Min

Question 14. Mean Deviation calculated from the Mean is always greater than or equal to 0. (True/False)

(A) True

(B) False

(C) Depends on the data

(D) Only true for positive data

Question 15. If the Mean Deviation of a data set is 5 units, the units of the original data must be:

(A) Square of the data units

(B) The same as the data units

(C) Dimensionless

(D) Cannot be determined

Question 16. Which measure of dispersion is based on the absolute deviations from an average?

(A) Range

(B) Variance

(C) Mean Deviation

(D) Standard Deviation

Question 17. For grouped data, the Mean Deviation is calculated using the formula $\frac{\sum f_i |x_i - A|}{\sum f_i}$, where A is either the Mean or Median. Here $x_i$ represents the:

(A) Lower limit of the class

(B) Upper limit of the class

(C) Class mark

(D) Frequency

Question 18. If the range of a data set is 50 and the minimum value is 10, the maximum value is:

(A) 40

(B) 50

(C) 60

(D) 70

Question 19. Which method of calculating mean deviation is usually preferred in practice?

(A) Mean Deviation from Mean

(B) Mean Deviation from Median

(C) Both are equally preferred

(D) Neither is preferred over Standard Deviation

Question 20. The measure of dispersion that only considers the spread between the two extreme values is the:

(A) Mean Deviation

(B) Variance

(C) Standard Deviation

(D) Range

Question 21. The coefficient of Mean Deviation is a measure of relative dispersion. It is calculated as Mean Deviation divided by the chosen average (Mean or Median). (True/False)

(A) True

(B) False

(C) Only true for Mean

(D) Only true for Median

Question 22. If the Range of a data set is 0, it means:

(A) The mean deviation is very large

(B) All observations have the same value

(C) The data set is empty

(D) The data is highly dispersed

Question 23. A limitation of Mean Deviation is that it ignores the algebraic signs of deviations, which makes it mathematically less rigorous than variance or standard deviation. (True/False)

(A) True

(B) False

(C) Depends on the context

(D) It is never a limitation

Question 24. Which measure of dispersion is useful for open-ended distributions where calculating Mean or Standard Deviation might be difficult?

(A) Mean Deviation

(B) Standard Deviation

(C) Range (if not open-ended)

(D) Quartile Deviation (Interquartile Range)

Question 1. The average of the squared deviations of individual observations from their mean is called the:

(A) Mean Deviation

(B) Standard Deviation

(C) Variance

(D) Range

Question 2. The positive square root of the Variance is called the:

(A) Mean Deviation

(B) Standard Deviation

(C) Range

(D) Coefficient of Variation

Question 3. For the ungrouped data set {1, 2, 3, 4, 5}, the mean is 3. The Variance is:

(A) 1

(B) 2

(C) 2.5

(D) 3

Question 4. For the ungrouped data set {1, 2, 3, 4, 5}, the Standard Deviation is:

(A) $\sqrt{1} = 1$

(B) $\sqrt{2} \approx 1.414$

(C) $\sqrt{2.5} \approx 1.581$

(D) $\sqrt{3} \approx 1.732$

Question 5. If a constant 'c' is added to every observation in a data set, the Variance of the new data set will:

(A) Increase by c

(B) Remain unchanged

(C) Increase by c$^2$

(D) Be multiplied by c

Question 6. If every observation in a data set is multiplied by a constant 'k', the Standard Deviation of the new data set will be:

(A) Original Standard Deviation + k

(B) Original Standard Deviation - k

(C) Original Standard Deviation $\times$ k

(D) Original Standard Deviation $\times |k|$

Question 7. The units of Variance are:

(A) The same as the data units

(B) Square of the data units

(C) Square root of the data units

(D) Dimensionless

Question 8. The units of Standard Deviation are:

(A) The same as the data units

(B) Square of the data units

(C) Square root of the data units

(D) Dimensionless

Question 9. When is the Standard Deviation equal to 0?

(A) When all observations are 0.

(B) When the mean is 0.

(C) When all observations have the same value.

(D) When the range is 0.

Question 10. Standard Deviation is considered a more appropriate measure of dispersion than Mean Deviation because it:

(A) Is easier to calculate

(B) Is based on squares of deviations, making it amenable to further mathematical treatment

(C) Is not affected by extreme values

(D) Is always smaller than Mean Deviation

Question 11. The sum of squared deviations from the Mean, $\sum (x_i - \bar{x})^2$, is minimum. (True/False)

(A) True

(B) False

(C) Only true for symmetric distributions

(D) Only true for positive data

Question 12. Variance and Standard Deviation are primarily used for:

(A) Nominal data

(B) Ordinal data

(C) Interval or Ratio scale data

(D) Qualitative data

Question 13. For grouped data, the formula for Variance is $\frac{\sum f_i (x_i - \bar{x})^2}{\sum f_i}$. Here $x_i$ is the:

(A) Lower limit

(B) Upper limit

(C) Class mark

(D) Frequency

Question 14. The standard deviation is always:

(A) Positive

(B) Negative

(C) Non-negative

(D) An integer

Question 15. If the variance of a dataset is 36, its standard deviation is:

(A) 6

(B) 18

(C) $\sqrt{36}$ which is 6 (or -6, but SD is positive)

(D) 36

Question 16. The standard deviation measures the typical distance of observations from the:

(A) Median

(B) Mode

(C) Mean

(D) Range

Question 17. Which measure of dispersion is considered the most comprehensive and widely used?

(A) Range

(B) Mean Deviation

(C) Standard Deviation

(D) Quartile Deviation

Question 18. If all the data points in a set are identical, then its variance and standard deviation are:

(A) Maximum

(B) Equal to the mean

(C) Zero

(D) Undefined

Question 19. The variance calculated from a sample is often divided by $(n-1)$ instead of $n$ to get an unbiased estimate of the population variance. What is $(n-1)$ called?

(A) Number of observations

(B) Degrees of freedom

(C) Sample size

(D) Population size

Question 20. The square of the standard deviation is equal to the:

(A) Mean

(B) Variance

(C) Median

(D) Range

Question 1. Measures of relative dispersion are used to:

(A) Compare the spread of two or more data sets with different units or means

(B) Measure the central location of a data set

(C) Determine the frequency of observations

(D) Analyze the relationship between two variables

Question 2. The Coefficient of Variation (CV) is a measure of relative dispersion. Its formula is:

(A) $(\text{Standard Deviation} / \text{Mean}) \times 100$

(B) $(\text{Mean} / \text{Standard Deviation}) \times 100$

(C) Range / Mean

(D) Mean Deviation / Median

Question 3. The Coefficient of Variation is usually expressed as a:

(A) Ratio

(B) Decimal

(C) Percentage

(D) Frequency

Question 4. A higher Coefficient of Variation indicates:

(A) Less variability relative to the mean

(B) More variability relative to the mean

(C) The data is symmetric

(D) The data has no outliers

Question 5. Two share prices had the following statistics: Stock A (Mean price = $\textsf{₹}500$, SD = $\textsf{₹}50$), Stock B (Mean price = $\textsf{₹}100$, SD = $\textsf{₹}20$). Which stock is more volatile (variable)?

(A) Stock A

(B) Stock B

(C) Both are equally volatile

(D) Cannot be determined from these values

Question 6. The Coefficient of Variation is meaningful only when the Mean is:

(A) Positive

(B) Negative

(C) Zero

(D) Non-zero and preferably positive

Question 7. If the Mean of a distribution is 20 and its Standard Deviation is 5, the Coefficient of Variation is:

(A) 25%

(B) 20%

(C) 400%

(D) 4%

Question 8. Comparing the consistency of two cricket players' scores, you would typically use the:

(A) Mean score

(B) Standard Deviation of scores

(C) Coefficient of Variation of scores

(D) Highest score

Question 9. A lower Coefficient of Variation indicates:

(A) More consistency

(B) Less consistency

(C) Higher variability

(D) Lower mean

Question 10. If $\bar{x}_1 = 100$, $\sigma_1 = 10$ and $\bar{x}_2 = 20$, $\sigma_2 = 5$, which data set has more variability (relative)?

(A) Data set 1

(B) Data set 2

(C) Both have equal relative variability

(D) Cannot be determined

Question 11. The concept of moments is related to the average of powers of deviations. The first moment about the origin is the:

(A) Variance

(B) Standard Deviation

(C) Mean

(D) Median

Question 12. The second central moment (moment about the mean) is equal to the:

(A) Mean

(B) Variance

(C) Skewness

(D) Kurtosis

Question 13. Moments about the mean are called:

(A) Raw moments

(B) Central moments

(C) Absolute moments

(D) Factorial moments

Question 14. Moments about an arbitrary value 'A' are called:

(A) Central moments

(B) Standardized moments

(C) Raw moments

(D) Absolute moments

Question 15. Coefficient of Variation is useful when comparing variability between two groups, for example, comparing the variability of wages in two different industries. (True/False)

(A) True

(B) False

(C) Only if the means are identical

(D) Only if the standard deviations are identical

Question 16. If the mean is 0, the Coefficient of Variation is:

(A) 0

(B) 100

(C) Infinite or Undefined

(D) Equal to the standard deviation

Question 17. Given Mean = 80, CV = 20%. The Standard Deviation is:

(A) 16

(B) 4

(C) 400

(D) 0.25

Question 18. The first raw moment (moment about the origin) is denoted by $\mu_1'$ and is calculated as $E(X^1) = E(X)$. This is equal to the Mean. (True/False)

(A) True

(B) False

(C) Only for discrete data

(D) Only for continuous data

Question 1. Skewness measures the __________ of a distribution.

(A) Central location

(B) Spread

(C) Symmetry or asymmetry

(D) Peakedness

Question 2. A distribution is said to be positively skewed if the tail is longer on the:

(A) Left side

(B) Right side

(C) Both sides equally

(D) Center

Question 3. For a negatively skewed distribution, the relationship between Mean, Median, and Mode is typically:

(A) Mean = Median = Mode

(B) Mean > Median > Mode

(C) Mean < Median < Mode

(D) Median > Mode > Mean

Question 4. For a symmetric distribution, the coefficient of skewness is:

(A) Positive

(B) Negative

(C) Zero

(D) 1

Question 5. Karl Pearson's coefficient of skewness is calculated using the formula:

(A) (Median - Mode) / Standard Deviation

(B) (Mean - Mode) / Standard Deviation

(C) (Q3 - Q1) / Median

(D) 3 * (Mean - Median) / Standard Deviation

Question 6. If Karl Pearson's coefficient of skewness is +0.8, the distribution is:

(A) Symmetric

(B) Moderately positively skewed

(C) Moderately negatively skewed

(D) Highly symmetric

Question 7. Bowley's coefficient of skewness is based on the relationship between:

(A) Mean and Standard Deviation

(B) Median and Mode

(C) Quartiles and Median

(D) Mean and Median

Question 8. Kurtosis measures the __________ of a distribution.

(A) Asymmetry

(B) Dispersion

(C) Central tendency

(D) Peakedness and tail heaviness

Question 9. A distribution that is more peaked and has heavier tails than the normal distribution is called:

(A) Platykurtic

(B) Mesokurtic

(C) Leptokurtic

(D) Symmetric

Question 10. The normal distribution is considered to be:

(A) Platykurtic

(B) Mesokurtic

(C) Leptokurtic

(D) Positively skewed

Question 11. A distribution that is flatter and has lighter tails than the normal distribution is called:

(A) Platykurtic

(B) Mesokurtic

(C) Leptokurtic

(D) Negatively skewed

Question 12. Skewness tells us about the direction of the asymmetry, while kurtosis tells us about the shape of the distribution's central peak and tails. (True/False)

(A) True

(B) False

(C) They measure the same thing

(D) Only applies to non-normal distributions

Question 13. For a distribution, if Mean > Median > Mode, it is likely:

(A) Symmetric

(B) Positively skewed

(C) Negatively skewed

(D) Bimodal

Question 14. Bowley's coefficient of skewness ranges between:

(A) -1 and +1

(B) 0 and 1

(C) -3 and +3

(D) $-\infty$ and $+\infty$

Question 15. If for a distribution, Q1 = 20, Median = 30, and Q3 = 45, the distribution is:

(A) Symmetric

(B) Positively skewed

(C) Negatively skewed

(D) Bimodal

Question 16. Measures of skewness and kurtosis are based on:

(A) Only the central tendency

(B) The shape of the distribution

(C) Only the spread of the data

(D) The total frequency

Question 17. A distribution with zero skewness is also called:

(A) Mesokurtic

(B) Leptokurtic

(C) Platykurtic

(D) Symmetric

Question 18. If Karl Pearson's coefficient of skewness is calculated as 0, it implies that:

(A) Mean = Mode

(B) Median = Mode

(C) Mean = Median = Mode

(D) The distribution is bimodal

Question 19. Kurtosis is primarily used to understand the behaviour of the data in the:

(A) Center of the distribution only

(B) Tails of the distribution only

(C) Center and tails of the distribution

(D) Skewness of the distribution

Question 20. A leptokurtic distribution is characterized by a peak that is __________ than the normal distribution.

(A) Flatter

(B) Taller and sharper

(C) Wider

(D) Lower

Question 1. Quartiles divide a data set into __________ equal parts.

(A) 2

(B) 3

(C) 4

(D) 100

Question 2. The first quartile (Q1) is also known as the:

(A) Median

(B) Lower Quartile

(C) Upper Quartile

(D) 10th Percentile

Question 3. The second quartile (Q2) is always equal to the:

(A) Mean

(B) Mode

(C) Median

(D) Average of Q1 and Q3

Question 4. Percentiles divide a data set into __________ equal parts.

(A) 4

(B) 10

(C) 20

(D) 100

Question 5. The 50th percentile (P50) is equal to the:

(A) Mean

(B) Mode

(C) Median (Q2)

(D) Third Quartile (Q3)

Question 6. For the ungrouped data set {10, 15, 20, 25, 30, 35, 40}, the Median (Q2) is 25. The first quartile (Q1) is:

(A) 15

(B) 17.5

(C) 20

(D) 22.5

Question 7. For the ungrouped data set {10, 15, 20, 25, 30, 35, 40}, the third quartile (Q3) is:

(A) 30

(B) 32.5

(C) 35

(D) 37.5

Question 8. The Interquartile Range (IQR) is the difference between the third quartile (Q3) and the first quartile (Q1). What does IQR measure?

(A) Central tendency

(B) Total spread

(C) Spread of the middle 50% of the data

(D) Skewness

Question 9. The Quartile Deviation (QD) or Semi-Interquartile Range is calculated as:

(A) Q3 - Q1

(B) (Q3 + Q1) / 2

(C) (Q3 - Q1) / 2

(D) (Q3 + Q1) / (Q3 - Q1)

Question 10. If Q1 = 20 and Q3 = 40 for a data set, the Interquartile Range (IQR) is:

(A) 10

(B) 20

(C) 30

(D) 60

Question 11. Quartile Deviation is a measure of:

(A) Central tendency

(B) Relative dispersion

(C) Absolute dispersion

(D) Skewness

Question 12. Percentiles are useful for determining the value below which a certain percentage of observations fall. (True/False)

(A) True

(B) False

(C) Only for symmetric data

(D) Only for grouped data

Question 13. If a student scores in the 80th percentile on a test, it means they scored better than approximately what percentage of the students who took the test?

(A) 20%

(B) 50%

(C) 80%

(D) 100%

Question 14. Quartiles and Percentiles can be estimated graphically from:

(A) Histograms

(B) Frequency polygons

(C) Ogives

(D) Bar graphs

Question 15. To find the 75th percentile (P75) from a 'less than' ogive with total frequency N, you would locate the cumulative frequency corresponding to:

(A) N/4

(B) N/2

(C) 3N/4

(D) N

Question 16. Quartile Deviation is useful for distributions with extreme values or open-ended classes because it does not depend on all observations. (True/False)

(A) True

(B) False

(C) It depends on the class size

(D) It is less reliable than Range in such cases

Question 17. The percentile rank of a value is the percentage of values in the data set that are _________ or equal to that value.

(A) Less than

(B) Greater than

(C) Exactly equal to

(D) Not equal to

Question 18. For a symmetric distribution, the distance between Q1 and Median is equal to the distance between Median and Q3. (True/False)

(A) True

(B) False

(C) Only if the Mean is also equal

(D) Only for continuous data

Question 19. The 25th percentile (P25) is equal to the:

(A) Median (Q2)

(B) First Quartile (Q1)

(C) Third Quartile (Q3)

(D) Mean

Question 20. The 75th percentile (P75) is equal to the:

(A) Median (Q2)

(B) First Quartile (Q1)

(C) Third Quartile (Q3)

(D) Mean

Question 1. Correlation measures the __________ between two variables.

(A) Cause and effect relationship

(B) Difference

(C) Linear relationship or association

(D) Variability

Question 2. If two variables tend to increase or decrease together, they have:

(A) Zero correlation

(B) Negative correlation

(C) Positive correlation

(D) Perfect correlation

Question 3. If one variable tends to increase as the other decreases, they have:

(A) Zero correlation

(B) Negative correlation

(C) Positive correlation

(D) No relationship

Question 4. A scatter diagram is a graphical representation used to:

(A) Show the frequency distribution of a single variable

(B) Illustrate the relationship between two variables

(C) Show the proportion of parts to a whole

(D) Display cumulative frequencies

Question 5. In a scatter diagram, if the points cluster around a straight line sloping upwards from left to right, it indicates:

(A) Negative correlation

(B) Positive correlation

(C) Zero correlation

(D) Non-linear relationship

Question 6. If the points in a scatter diagram are randomly scattered without any apparent pattern, it suggests:

(A) Strong positive correlation

(B) Strong negative correlation

(C) Weak or zero correlation

(D) Perfect correlation

Question 7. Karl Pearson's Coefficient of Correlation (r) measures the strength and direction of the __________ relationship between two variables.

(A) Non-linear

(B) Exponential

(C) Linear

(D) Quadratic

Question 8. The value of Karl Pearson's coefficient of correlation (r) ranges from:

(A) 0 to 1

(B) -1 to +1

(C) $-\infty$ to $+\infty$

(D) -100 to +100

Question 9. If r = +1, it indicates:

(A) Perfect negative linear correlation

(B) Perfect positive linear correlation

(C) No linear correlation

(D) Perfect non-linear relationship

Question 10. If r = -1, it indicates:

(A) Perfect negative linear correlation

(B) Perfect positive linear correlation

(C) No linear correlation

(D) Perfect non-linear relationship

Question 11. If r = 0, it indicates:

(A) Perfect linear correlation

(B) No linear correlation

(C) A perfect non-linear relationship

(D) The variables are independent

Question 12. A strong positive correlation (e.g., r = 0.9) means:

(A) As one variable increases, the other tends to decrease significantly

(B) As one variable increases, the other tends to increase significantly

(C) There is no relationship between the variables

(D) There is a perfect cause and effect relationship

Question 13. Correlation does NOT imply:

(A) Association

(B) Relationship

(C) Cause and effect

(D) Covariation

Question 14. Spearman's Rank Correlation Coefficient is used when:

(A) The relationship is linear only

(B) The data is ordinal or the relationship is monotonic (increasing or decreasing but not necessarily linear)

(C) The data is continuous and normally distributed

(D) We want to measure the average of the two variables

Question 15. The range of Spearman's Rank Correlation Coefficient is:

(A) 0 to 1

(B) -1 to +1

(C) $-\infty$ to $+\infty$

(D) Depends on the number of observations

Question 16. If the heights of fathers and sons show a positive correlation, it means:

(A) Taller fathers tend to have taller sons

(B) Taller fathers tend to have shorter sons

(C) Father's height has no effect on son's height

(D) The average height of fathers is greater than sons

Question 17. Which method of correlation measurement is less sensitive to extreme values?

(A) Karl Pearson's coefficient

(B) Spearman's rank correlation

(C) Both are equally sensitive

(D) Scatter diagram

Question 18. If the consumption of ice cream increases as the temperature rises, this is an example of:

(A) Negative correlation

(B) Positive correlation

(C) Zero correlation

(D) Perfect correlation

Question 19. The strength of a linear relationship is indicated by the __________ of the correlation coefficient.

(A) Sign

(B) Magnitude (absolute value)

(C) Unit

(D) Square

Question 20. A perfect linear relationship would appear as a straight line on a:

(A) Histogram

(B) Pie chart

(C) Scatter diagram

(D) Box plot

Question 21. Which of the following correlation coefficients indicates the strongest relationship?

(A) +0.5

(B) -0.8

(C) +0.2

(D) -0.1

Question 1. Probability is a measure of the __________ of an event occurring.

(A) Certainty

(B) Impossibility

(C) Likelihood

(D) Outcome

Question 2. An experiment where the outcomes cannot be predicted with certainty, but all possible outcomes are known, is called a:

(A) Certain experiment

(B) Deterministic experiment

(C) Random experiment

(D) Simple experiment

Question 3. The set of all possible outcomes of a random experiment is called the:

(A) Event set

(B) Outcome set

(C) Sample space

(D) Universal set

Question 4. For the experiment of tossing a coin, the sample space is:

(A) {H}

(B) {T}

(C) {H, T}

(D) {Coin}

Question 5. For the experiment of rolling a standard six-sided die, the sample space is:

(A) {6}

(B) {1, 2, 3, 4, 5, 6}

(C) {Even, Odd}

(D) {Number}

Question 6. A subset of the sample space is called an:

(A) Experiment

(B) Outcome

(C) Event

(D) Probability

Question 7. An event consisting of a single outcome is called a(n):

(A) Compound event

(B) Simple event

(C) Impossible event

(D) Sure event

Question 8. An event that is sure to occur is represented by the:

(A) Impossible event

(B) Sample space itself

(C) Empty set

(D) Single outcome event

Question 9. An event that cannot occur is represented by the:

(A) Sample space

(B) Impossible event (Empty set $\emptyset$)

(C) Simple event

(D) Sure event

Question 10. Two events A and B are mutually exclusive if:

(A) $P(A \cap B) = P(A) P(B)$

(B) $A \cap B = \emptyset$ (They cannot occur simultaneously)

(C) $A \cup B = \Omega$ (Their union is the sample space)

(D) $P(A|B) = P(A)$

Question 11. According to the classical (theoretical) definition of probability, the probability of an event E is given by:

(A) (Number of trials) / (Number of times E occurred)

(B) (Number of favourable outcomes) / (Total number of outcomes)

(C) Sum of probabilities of all outcomes in E

(D) Cannot be determined theoretically

Question 12. The range of probability of any event E is:

(A) $0 < P(E) < 1$

(B) $0 \le P(E) \le 1$

(C) $-1 \le P(E) \le 1$

(D) $0 \le P(E)$

Question 13. The probability of a sure event is:

(A) 0

(B) 0.5

(C) 1

(D) Depends on the experiment

Question 14. The probability of an impossible event is:

(A) 0

(B) 0.5

(C) 1

(D) Undefined

Question 15. If a bag contains 5 red and 3 blue balls, and one ball is drawn randomly, the total number of outcomes in the sample space is:

(A) 2

(B) 8

(C) 15

(D) Depends on the colour

Question 16. If a bag contains 5 red and 3 blue balls, and one ball is drawn randomly, the probability of drawing a red ball is:

(A) $3/8$

(B) $5/8$

(C) $5/3$

(D) $1/8$

Question 17. Experimental (Empirical) Probability is based on:

(A) Theoretical calculations of outcomes

(B) The subjective belief of the observer

(C) The actual results of conducting an experiment multiple times

(D) Axioms of probability

Question 18. If a coin is tossed 100 times and heads appears 55 times, the experimental probability of getting a head is:

(A) $0.5$

(B) $0.55$

(C) $0.45$

(D) $1$

Question 19. As the number of trials in an experiment increases, the experimental probability of an event tends to get closer to its __________ probability.

(A) Subjective

(B) Theoretical (Classical)

(C) Impossible

(D) Conditional

Question 20. In a well-shuffled deck of 52 playing cards, the probability of drawing a King is:

(A) $1/52$

(B) $4/52 = 1/13$

(C) $13/52 = 1/4$

(D) $1/4$

Question 21. Which of the following values CANNOT be a probability of an event?

(A) 0.7

(B) -0.2

(C) 0

(D) 1

Question 22. If event A is 'getting an even number' when rolling a die, the outcomes in event A are:

(A) {1, 3, 5}

(B) {2, 4, 6}

(C) {1, 2, 3, 4, 5, 6}

(D) {Even}

Question 23. If event A is 'getting a head' and event B is 'getting a tail' in a single coin toss, then A and B are:

(A) Independent events

(B) Mutually exclusive events

(C) Sure events

(D) Simple events

Question 24. A compound event is an event that:

(A) Contains only one outcome

(B) Contains more than one outcome

(C) Is impossible

(D) Is certain to happen

Question 25. The empirical probability of an event is found by dividing the number of times the event occurred by the total number of __________.

(A) Outcomes

(B) Favourable outcomes

(C) Trials

(D) Events

Question 1. In the axiomatic approach to probability, the probability of any event A, denoted by P(A), must satisfy the condition:

(A) $P(A) < 0$

(B) $P(A) > 1$

(C) $0 \le P(A) \le 1$

(D) $P(A) = 1$

Question 2. According to the axioms of probability, the probability of the sample space $\Omega$ is:

(A) 0

(B) 0.5

(C) 1

(D) Undefined

Question 3. For any event A, the probability of its complement A' (not A) is given by:

(A) $P(A')$ = $P(A)$

(B) $P(A')$ = $1 + P(A)$

(C) $P(A')$ = $1 - P(A)$

(D) $P(A')$ = $1 / P(A)$

Question 4. If P(A) = 0.6, then P(not A) is:

(A) 0.6

(B) 0.4

(C) 1

(D) 0

Question 5. For any two mutually exclusive events A and B, the probability that A or B occurs is given by the Addition Law:

(A) $P(A \cup B) = P(A) \times P(B)$

(B) $P(A \cup B) = P(A) + P(B)$

(C) $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

(D) $P(A \cup B) = P(A \cap B)$

Question 6. If A and B are NOT mutually exclusive events, the probability that A or B occurs is given by:

(A) $P(A \cup B) = P(A) + P(B)$

(B) $P(A \cup B) = P(A) + P(B) - P(A \cap B)$

(C) $P(A \cup B) = P(A \cap B)$

(D) $P(A \cup B) = 1 - P(A \cap B)$

Question 7. If $P(A) = 0.4$, $P(B) = 0.5$, and A and B are mutually exclusive, then $P(A \cup B)$ is:

(A) 0.1

(B) 0.9

(C) 0.2

(D) 0

Question 8. If $P(A) = 0.4$, $P(B) = 0.5$, and $P(A \cap B) = 0.2$, then $P(A \cup B)$ is:

(A) 0.7

(B) 0.9

(C) 1.1

(D) 0.2

Question 9. The axiomatic approach provides a rigorous mathematical foundation for probability theory. (True/False)

(A) True

(B) False

(C) Only for classical probability

(D) Only for empirical probability

Question 10. If A and B are mutually exclusive events, then $P(A \cap B)$ is always:

(A) 0

(B) $P(A) + P(B)$

(C) $P(A) \times P(B)$

(D) 1

Question 11. The probability of 'A and B' ($A \cap B$) occurring when A and B are not mutually exclusive is involved in the calculation of $P(A \cup B)$. (True/False)

(A) True

(B) False

(C) Only if A and B are independent

(D) Only if A and B are complements

Question 12. If the probability of rain is 0.3, the probability of no rain is:

(A) 0.3

(B) 0.7

(C) 1

(D) 0

Question 13. If A, B, and C are three mutually exclusive events such that $A \cup B \cup C = \Omega$, then $P(A) + P(B) + P(C) =$:

(A) 0

(B) 0.5

(C) 1

(D) Cannot be determined

Question 1. The probability of event A occurring given that event B has already occurred is called:

(A) Joint probability $P(A \cap B)$

(B) Marginal probability $P(A)$

(C) Conditional probability $P(A|B)$

(D) Union probability $P(A \cup B)$

Question 2. The formula for conditional probability $P(A|B)$ is:

(A) $P(A) \times P(B)$

(B) $P(A) + P(B)$

(C) $P(A \cap B) / P(B)$ (provided $P(B) \neq 0$)

(D) $P(A \cup B) / P(B)$

Question 3. If $P(A \cap B) = 0.3$ and $P(B) = 0.5$, then $P(A|B)$ is:

(A) 0.6

(B) 0.8

(C) 0.15

(D) 0.2

Question 4. If $P(B) = 0.7$ and $P(A|B) = 0.4$, then $P(A \cap B)$ is:

(A) 0.28

(B) 0.3

(C) 0.11

(D) 0.4

Question 5. If A and B are independent events, then $P(A|B)$ is equal to:

(A) $P(A)$

(B) $P(B)$

(C) $P(A \cap B)$

(D) $P(A) \times P(B)$

Question 6. If A and B are mutually exclusive events and $P(B) \neq 0$, then $P(A|B)$ is:

(A) 1

(B) 0

(C) $P(A)$

(D) $P(B)$

Question 7. A property of conditional probability states that for any event A and any conditioning event F with $P(F) \neq 0$, $P(A|F) = P(A \cap F) / P(F)$. (True/False)

(A) True

(B) False

(C) Only if A and F are independent

(D) Only if A and F are mutually exclusive

Question 8. If F is the event that a die roll is an even number {2, 4, 6} and E is the event that the roll is a number greater than 3 {4, 5, 6}, what is $P(E|F)$?

(A) $3/6 = 1/2$

(B) $3/6 = 1/2$

(C) $P(E \cap F)/P(F) = P(\{4,6\})/P(\{2,4,6\}) = (2/6) / (3/6) = 2/3$

(D) $P(F|E) = P(E \cap F)/P(E) = (2/6) / (3/6) = 2/3$

Question 9. In a class of 100 students, 40 study Mathematics (M), 30 study Physics (P), and 20 study both. If a student is chosen randomly and it is known they study Physics, what is the probability they also study Mathematics?

(A) $P(M|P) = P(M \cap P) / P(P) = 0.2 / 0.3 = 2/3$

(B) $P(P|M) = P(M \cap P) / P(M) = 0.2 / 0.4 = 1/2$

(C) $P(M \cup P) = P(M) + P(P) - P(M \cap P) = 0.4 + 0.3 - 0.2 = 0.5$

(D) $P(M \cap P) = 0.2$

Question 10. Conditional probability $P(A|B)$ is defined only when:

(A) Event A has occurred

(B) Event B has occurred

(C) $P(A) \neq 0$

(D) $P(B) \neq 0$

Question 11. If A and B are events such that $P(A) > 0$, then $P(\Omega|A)$ (probability of sample space given A) is:

(A) 0

(B) $P(A)$

(C) 1

(D) $1/P(A)$

Question 12. If A and B are two events, $P(A|B) + P(A'|B)$ is equal to:

(A) 0

(B) 1

(C) $P(A)$

(D) $P(B)$

Question 13. Conditional probability can be thought of as updating the sample space to the conditioning event. (True/False)

(A) True

(B) False

(C) Only for mutually exclusive events

(D) Only for independent events

Question 14. If $P(A) = 0.6$, $P(B) = 0.8$, and $P(A \cap B) = 0.4$, find $P(A|B)$.

(A) $0.4 / 0.6 = 2/3$

(B) $0.4 / 0.8 = 1/2 = 0.5$

(C) $0.6 / 0.8 = 3/4 = 0.75$

(D) $0.6 + 0.8 - 0.4 = 1$

Question 15. If $P(A) = 0.6$, $P(B) = 0.8$, and $P(A \cap B) = 0.4$, find $P(B|A)$.

(A) $0.4 / 0.6 = 2/3 \approx 0.667$

(B) $0.4 / 0.8 = 0.5$

(C) $0.6 / 0.8 = 0.75$

(D) $0.6 + 0.8 - 0.4 = 1$

Question 16. Conditional probability helps in understanding how the occurrence of one event influences the probability of another event. (True/False)

(A) True

(B) False

(C) Only for independent events

(D) Only for mutually exclusive events

Question 17. If $P(A|B) = P(A)$, then events A and B are:

(A) Mutually exclusive

(B) Independent

(C) Dependent

(D) Complements

Question 18. In a box, there are 3 red and 2 blue pens. If two pens are drawn without replacement, what is the probability the second pen is red, given the first pen drawn was red?

(A) $2/5$

(B) $3/5$

(C) $2/4 = 1/2$

(D) $3/4$

Question 19. In a box, there are 3 red and 2 blue pens. If two pens are drawn without replacement, what is the probability the second pen is red, given the first pen drawn was blue?

(A) $2/5$

(B) $3/5$

(C) $2/4 = 1/2$

(D) $3/4$

Question 20. If $P(A|B) = P(A \cap B) / P(B)$, then $P(A \cap B) = P(A|B) \times P(B)$. This formula is known as the:

(A) Addition Law of Probability

(B) Law of Total Probability

(C) Multiplication Law of Probability

(D) Bayes' Theorem

Question 21. If $P(A \cap B) = 0.1$ and $P(A) = 0.4$, then $P(B|A)$ is:

(A) $0.1 / 0.4 = 1/4 = 0.25$

(B) $0.4 / 0.1 = 4$

(C) $0.1 \times 0.4 = 0.04$

(D) $0.4 - 0.1 = 0.3$

Question 22. The concept of conditional probability is fundamental to:

(A) Classical probability

(B) Empirical probability

(C) Subjective probability

(D) All of the above, but particularly important for dependent events

Question 23. If $P(A) = 0.5$, $P(B) = 0.6$, and $P(A \cup B) = 0.8$, then $P(A|B)$ is:

(A) $P(A \cap B) = P(A) + P(B) - P(A \cup B) = 0.5 + 0.6 - 0.8 = 0.3$. Then $P(A|B) = P(A \cap B) / P(B) = 0.3 / 0.6 = 1/2 = 0.5$

(B) 0.3 / 0.5 = 0.6

(C) 0.5 + 0.6 = 1.1

(D) 0.8

Question 24. Conditional probability $P(A|B)$ is related to $P(B|A)$ by:

(A) Bayes' Theorem

(B) Addition Law

(C) Multiplication Law

(D) Law of Total Probability

Question 25. If $P(A|B) = 1$, it means that:

(A) A and B are mutually exclusive

(B) If B occurs, A must occur

(C) A and B are independent

(D) A is a subset of B

Question 26. If $P(A|B) = 0$, it means that:

(A) If B occurs, A cannot occur

(B) A and B are independent

(C) A is an impossible event

(D) B is an impossible event

Question 27. For any event A with $P(A) > 0$, $P(A|A)$ is equal to:

(A) 0

(B) 0.5

(C) 1

(D) $P(A)$

Question 1. The Multiplication Theorem of Probability states that for any two events A and B, $P(A \cap B)$ is equal to:

(A) $P(A) + P(B)$

(B) $P(A) P(B)$

(C) $P(A) P(B|A)$ or $P(B) P(A|B)$

(D) $P(A \cup B) - P(A) - P(B)$

Question 2. If events A and B are independent, then $P(A \cap B)$ is equal to:

(A) $P(A) + P(B)$

(B) $P(A) P(B)$

(C) $P(A|B)$

(D) 0

Question 3. Events A and B are independent if the occurrence of A does not affect the probability of B occurring, and vice versa. This means:

(A) $P(A|B) = P(A)$ and $P(B|A) = P(B)$

(B) $A \cap B = \emptyset$

(C) $P(A \cup B) = P(A) + P(B)$

(D) $P(A \cap B) = 0$

Question 4. If a coin is tossed twice, the outcomes of the two tosses are:

(A) Dependent events

(B) Mutually exclusive events

(C) Independent events

(D) Simple events

Question 5. A bag contains 5 red and 3 blue balls. Two balls are drawn without replacement. The outcomes of the two draws are:

(A) Independent events

(B) Dependent events

(C) Mutually exclusive events

(D) Impossible events

Question 6. If $P(A) = 0.5$ and $P(B) = 0.6$, and A and B are independent, then $P(A \cap B)$ is:

(A) 0.1

(B) 0.3

(C) 1.1

(D) 0

Question 7. If A and B are mutually exclusive events with $P(A) > 0$ and $P(B) > 0$, can they be independent?

(A) Yes, always

(B) No, unless $P(A)=0$ or $P(B)=0$ (or both)

(C) Only if $P(A) + P(B) = 1$

(D) Only if $P(A) = P(B)$

Question 8. A collection of events $E_1, E_2, ..., E_n$ forms a partition of the sample space $\Omega$ if:

(A) They are mutually exclusive and their union is $\Omega$.

(B) They are independent.

(C) $P(E_i) > 0$ for all i.

(D) All of the above.

Question 9. The Law of Total Probability states that if $E_1, E_2, ..., E_n$ is a partition of the sample space, and A is any event, then $P(A)$ is given by:

(A) $\sum_{i=1}^n P(A|E_i)$

(B) $\sum_{i=1}^n P(E_i|A)$

(C) $\sum_{i=1}^n P(A \cap E_i) = \sum_{i=1}^n P(A|E_i)P(E_i)$

(D) $\sum_{i=1}^n P(E_i)$

Question 10. The Law of Total Probability is useful for calculating the probability of an event when it can occur under different, mutually exclusive conditions. (True/False)

(A) True

(B) False

(C) Only if the conditions are independent

(D) Only if the event A is impossible

Question 11. A factory has two machines, A and B, producing 60% and 40% of the total output. Machine A produces 2% defective items and Machine B produces 1% defective items. The probability that a randomly selected item is defective is:

(A) $P(Defective) = P(Defective|A)P(A) + P(Defective|B)P(B) = 0.02 \times 0.60 + 0.01 \times 0.40 = 0.012 + 0.004 = 0.016$

(B) $0.02 + 0.01 = 0.03$

(C) $0.60 \times 0.02 = 0.012$

(D) $0.40 \times 0.01 = 0.004$

Question 12. For events A and B, $P(A \cap B) = P(B \cap A)$. This property is consistent with the Multiplication Law. (True/False)

(A) True

(B) False

(C) Only if A and B are independent

(D) Only if A and B are mutually exclusive

Question 13. If $P(A) = 0.3$, $P(B) = 0.5$, and $P(A \cap B) = 0.15$, are A and B independent?

(A) Yes, because $P(A \cap B) = P(A)P(B) = 0.3 \times 0.5 = 0.15$

(B) No, because $P(A \cap B) \neq 0$

(C) Yes, because they are not mutually exclusive

(D) Cannot be determined from the given information

Question 14. If two events A and B are such that $P(A) \neq 0$ and $P(B) \neq 0$, and $P(A \cap B) = 0$, then A and B are:

(A) Independent

(B) Dependent

(C) Mutually exclusive

(D) Both B and C are correct

Question 15. The Law of Total Probability requires the partitioning events ($E_i$) to be exhaustive, meaning:

(A) They are mutually exclusive

(B) $P(E_i) > 0$ for all i

(C) Their union covers the entire sample space ($\cup E_i = \Omega$)

(D) They are independent

Question 16. If A and B are independent events, then A' and B are also independent. (True/False)

(A) True

(B) False

(C) Only if A and B are mutually exclusive

(D) Only if A and B are complements

Question 17. If $E_1, E_2$ form a partition of $\Omega$, and $P(E_1) = 0.6$, $P(E_2) = 0.4$, $P(A|E_1) = 0.1$, $P(A|E_2) = 0.05$, then $P(A)$ is:

(A) $0.1 + 0.05 = 0.15$

(B) $0.6 \times 0.1 + 0.4 \times 0.05 = 0.06 + 0.002 = 0.062$

(C) $0.6 + 0.4 = 1$

(D) $0.6 \times 0.4 = 0.24$

Question 18. Drawing two cards from a deck with replacement makes the two draws:

(A) Dependent

(B) Independent

(C) Mutually exclusive

(D) Impossible

Question 19. If $P(A) = 0.7$, $P(B) = 0.8$. If A and B are independent, $P(A \cup B)$ is:

(A) $P(A)+P(B) = 0.7+0.8 = 1.5$ (Incorrect for probability)

(B) $P(A)+P(B)-P(A \cap B) = P(A)+P(B)-P(A)P(B) = 0.7+0.8 - (0.7)(0.8) = 1.5 - 0.56 = 0.94$

(C) $P(A)P(B) = 0.56$

(D) $0.7 + 0.8 = 1.5$

Question 20. If $P(A|B) = 0.6$ and $P(B) = 0.5$, then $P(A \cap B)$ is:

(A) $0.6 + 0.5 = 1.1$

(B) $0.6 \times 0.5 = 0.3$

(C) $0.6 / 0.5 = 1.2$

(D) $0.5 / 0.6 \approx 0.833$

Question 21. If A and B are events, $P(A \cup B) = P(A) + P(B)$ is only true if A and B are:

(A) Independent

(B) Dependent

(C) Mutually exclusive

(D) Complements

Question 22. In probability, the term 'and' ($\cap$) corresponds to multiplication for independent events and conditional multiplication for dependent events. (True/False)

(A) True

(B) False

(C) Only for addition

(D) Only for complements

Question 1. Bayes' Theorem is used to calculate:

(A) The probability of the union of two events

(B) The marginal probability of an event

(C) The posterior probability of a cause, given an effect

(D) The probability of the intersection of independent events

Question 2. The formula for Bayes' Theorem for two events A and B is:

(A) $P(A|B) = P(B|A) \times P(A)$

(B) $P(A|B) = P(A \cap B) / P(B)$

(C) $P(A|B) = [P(B|A) \times P(A)] / P(B)$

(D) $P(A|B) = P(A) + P(B) - P(A \cap B)$

Question 3. In Bayes' Theorem, $P(A)$ is often called the:

(A) Posterior probability

(B) Prior probability

(C) Likelihood

(D) Marginal probability of the effect

Question 4. In Bayes' Theorem, $P(B|A)$ is often called the:

(A) Posterior probability

(B) Prior probability

(C) Likelihood

(D) Marginal probability of the effect

Question 5. In Bayes' Theorem, $P(A|B)$ is often called the:

(A) Posterior probability

(B) Prior probability

(C) Likelihood

(D) Marginal probability of the effect

Question 6. Bayes' Theorem is an extension of the concept of:

(A) Mutually exclusive events

(B) Independent events

(C) Conditional probability

(D) Union of events

Question 7. Consider the machine problem (Q11). If a randomly selected item is found to be defective, what is the probability it was produced by Machine A?

(A) $P(A|Defective) = [P(Defective|A)P(A)] / P(Defective) = (0.02 \times 0.60) / 0.016 = 0.012 / 0.016 = 12/16 = 3/4 = 0.75$

(B) $P(Defective|A) = 0.02$

(C) $P(A) = 0.60$

(D) $P(Defective|B) = 0.01$

Question 8. Bayes' Theorem is particularly useful in updating probabilities based on new evidence. (True/False)

(A) True

(B) False

(C) Only for prior probabilities

(D) Only for likelihoods

Question 9. If $E_1, E_2, ..., E_n$ is a partition of $\Omega$ and A is an event, the general form of Bayes' Theorem to find $P(E_i|A)$ is:

(A) $P(E_i|A) = P(A|E_i) P(E_i)$

(B) $P(E_i|A) = [P(A|E_i) P(E_i)] / P(A)$

(C) $P(E_i|A) = [P(A|E_i) P(E_i)] / \sum_{j=1}^n P(A|E_j)P(E_j)$

(D) $P(E_i|A) = P(E_i) / P(A)$

Question 10. Bayes' Theorem is used in areas like medical diagnosis, spam filtering, and machine learning. (True/False)

(A) True

(B) False

(C) Only for simple probability calculations

(D) Only for theoretical probability

Question 11. Suppose a rare disease affects 1 in 1000 people ($P(\text{Disease}) = 0.001$). A test for the disease has a 99% true positive rate ($P(\text{Positive}|\text{Disease}) = 0.99$) and a 2% false positive rate ($P(\text{Positive}|\text{No Disease}) = 0.02$). If a person tests positive, what is the probability they actually have the disease? (Use Bayes' Theorem)

(A) $P(\text{Disease}|\text{Positive}) = [P(\text{Positive}|\text{Disease}) P(\text{Disease})] / P(\text{Positive})$

(B) $P(\text{Positive}) = P(\text{Positive}|\text{Disease}) P(\text{Disease}) + P(\text{Positive}|\text{No Disease}) P(\text{No Disease})$

(C) $P(\text{No Disease}) = 1 - P(\text{Disease}) = 1 - 0.001 = 0.999$

(D) $P(\text{Positive}) = (0.99 \times 0.001) + (0.02 \times 0.999) = 0.00099 + 0.01998 = 0.02097$. $P(\text{Disease}|\text{Positive}) = 0.00099 / 0.02097 \approx 0.0472$ (Around 4.7%)

Question 12. In the disease example (Q11), if a person tests negative, what is the probability they actually have the disease? ($P(\text{Negative}|\text{Disease}) = 1 - P(\text{Positive}|\text{Disease}) = 1 - 0.99 = 0.01$, $P(\text{Negative}|\text{No Disease}) = 1 - P(\text{Positive}|\text{No Disease}) = 1 - 0.02 = 0.98$)

(A) $P(\text{Disease}|\text{Negative}) = [P(\text{Negative}|\text{Disease}) P(\text{Disease})] / P(\text{Negative})$

(B) $P(\text{Negative}) = P(\text{Negative}|\text{Disease}) P(\text{Disease}) + P(\text{Negative}|\text{No Disease}) P(\text{No Disease})$

(C) $P(\text{Negative}) = (0.01 \times 0.001) + (0.98 \times 0.999) = 0.00001 + 0.97902 = 0.97903$. $P(\text{Disease}|\text{Negative}) = 0.00001 / 0.97903 \approx 0.0000102$ (Very small probability)

(D) $0.98 \times 0.999 \approx 0.979$

Question 13. Bayes' theorem is a way to calculate the probability of a hypothesis being true given some evidence. (True/False)

(A) True

(B) False

(C) It calculates the probability of the evidence given the hypothesis

(D) It calculates the joint probability

Question 14. If $P(A|B) = P(B|A)$, what must be true about $P(A)$ and $P(B)$?

(A) $P(A) = P(B)$ (Assuming $P(A), P(B) \neq 0$)

(B) $P(A) > P(B)$

(C) $P(A) < P(B)$

(D) They must be independent

Question 15. In the context of Bayes' Theorem, the term $P(B)$ in the denominator acts as a normalizing constant, ensuring the posterior probabilities sum to 1. (True/False)

(A) True

(B) False

(C) It is the prior probability of A

(D) It is the likelihood

Question 16. If $E_1$ and $E_2$ are mutually exclusive and exhaustive events, $P(E_1) = 0.7$, $P(E_2) = 0.3$. If $P(A|E_1) = 0.2$, $P(A|E_2) = 0.4$. Find $P(E_1|A)$.

(A) $[P(A|E_1)P(E_1)] / [P(A|E_1)P(E_1) + P(A|E_2)P(E_2)] = (0.2 \times 0.7) / (0.2 \times 0.7 + 0.4 \times 0.3) = 0.14 / (0.14 + 0.12) = 0.14 / 0.26 = 14/26 = 7/13 \approx 0.538$

(B) $0.14 / 0.12$

(C) $0.2 \times 0.7 = 0.14$

(D) $0.14 / (0.7+0.3)$

Question 17. In the above question (Q16), find $P(E_2|A)$.

(A) $1 - P(E_1|A) = 1 - 7/13 = 6/13 \approx 0.462$

(B) $[P(A|E_2)P(E_2)] / P(A) = (0.4 \times 0.3) / 0.26 = 0.12 / 0.26 = 12/26 = 6/13 \approx 0.462$

(C) $0.12 / 0.14$

(D) Both A and B are correct

Question 18. Bayes' Theorem allows us to revise our initial beliefs (prior probabilities) in light of new data (likelihoods) to get updated beliefs (posterior probabilities). (True/False)

(A) True

(B) False

(C) It only confirms prior beliefs

(D) It only calculates likelihoods

Question 19. Which of the following probabilities is NOT directly used in the numerator of the general Bayes' Theorem formula for $P(E_i|A)$?

(A) $P(A|E_i)$

(B) $P(E_i)$

(C) $P(A)$

(D) $P(A \cap E_i)$

Question 20. If $P(A|E_1) > P(A|E_2)$, it suggests that event A is more likely to occur when $E_1$ is true than when $E_2$ is true. (True/False)

(A) True

(B) False

(C) It implies $P(E_1) > P(E_2)$

(D) It implies $P(E_1|A) > P(E_2|A)$

Question 1. A function that assigns a real number to each outcome in the sample space of a random experiment is called a:

(A) Probability function

(B) Random variable

(C) Sample point

(D) Probability distribution

Question 2. A random variable that can take only a finite or countably infinite number of distinct values is called a:

(A) Continuous random variable

(B) Discrete random variable

(C) Qualitative variable

(D) Sample space

Question 3. A random variable that can take any value within a given range is called a:

(A) Discrete random variable

(B) Continuous random variable

(C) Categorical variable

(D) Frequency variable

Question 4. The number of heads obtained in tossing a coin 3 times is an example of a:

(A) Continuous random variable

(B) Discrete random variable

(C) Qualitative variable

(D) Deterministic variable

Question 5. The height of students in a class is an example of a:

(A) Discrete random variable

(B) Continuous random variable

(C) Qualitative variable

(D) Countable variable

Question 6. A probability distribution of a discrete random variable X lists all possible values that X can take, along with their corresponding:

(A) Frequencies

(B) Cumulative frequencies

(C) Probabilities

(D) Class marks

Question 7. For a valid probability distribution of a discrete random variable X, the following property must hold:

(A) $\sum P(X=x) = 0$

(B) $\sum P(X=x) = 1$

(C) $P(X=x) < 0$ for some x

(D) $P(X=x) > 1$ for some x

Question 8. For a discrete random variable X, the probability of any specific value $x$ is given by the:

(A) Probability Density Function (PDF)

(B) Cumulative Distribution Function (CDF)

(C) Probability Mass Function (PMF)

(D) Frequency distribution

Question 9. For a continuous random variable, the probability of it taking any *specific* value is:

(A) 0

(B) Between 0 and 1

(C) 1

(D) Undefined

Question 10. For a continuous random variable, probability is measured by the area under the curve of the:

(A) Probability Mass Function (PMF)

(B) Probability Density Function (PDF)

(C) Cumulative Distribution Function (CDF)

(D) Frequency polygon

Question 11. Which of the following cannot be the value of $P(X=x)$ for a discrete random variable X?

(A) 0.2

(B) -0.1

(C) 0

(D) 1

Question 12. If the probability distribution of X is:

(A) 0.2

(B) 0.3

(C) 0.5

(D) 1

Question 13. A random variable is different from a variable used in algebra because its value is determined by the outcome of a random event. (True/False)

(A) True

(B) False

(C) They are the same

(D) Only discrete random variables are different

Question 14. The sum of probabilities for all possible values of a discrete random variable must equal:

(A) The number of values

(B) The maximum value

(C) 1

(D) 0

Question 15. The Cumulative Distribution Function (CDF) F(x) for a discrete random variable X is defined as:

(A) $F(x) = P(X=x)$

(B) $F(x) = P(X \le x)$

(C) $F(x) = P(X \ge x)$

(D) $F(x) = P(X=x) / \sum P(X=x)$

Question 16. For a valid probability distribution, $P(X=x)$ for any value x must be:

(A) Negative

(B) Greater than 1

(C) Between 0 and 1 (inclusive)

(D) An integer

Question 17. Which of the following scenarios would typically involve a continuous random variable?

(A) Number of cars passing a point on a road in an hour

(B) Weight of a newborn baby

(C) Number of defective items in a sample of 100

(D) Result of rolling a die

Question 18. Which of the following scenarios would typically involve a discrete random variable?

(A) Time taken to complete a task

(B) Temperature of a city

(C) Number of customers arriving at a shop per day

(D) Height of a plant

Question 19. For a continuous random variable X, $P(X=c)$ for any constant c is always:

(A) $f(c)$ (where f is PDF)

(B) $\int_{-\infty}^c f(x) dx$

(C) 0

(D) 1

Question 20. The total area under the Probability Density Function (PDF) curve for a continuous random variable is always:

(A) 0

(B) 0.5

(C) 1

(D) Undefined

Question 21. A probability distribution is essentially a model that describes the likelihood of different outcomes of a random variable. (True/False)

(A) True

(B) False

(C) It only describes the mean

(D) It only describes the variance

Question 22. In Applied Mathematics, the study of probability distributions is crucial for statistical modeling and inference. (True/False)

(A) True

(B) False

(C) It is only theoretical

(D) It is only for data visualization

Question 23. The values of a random variable are determined by chance. (True/False)

(A) True

(B) False

(C) They are fixed values

(D) They are chosen arbitrarily

Question 24. The range of a discrete random variable is:

(A) An interval of real numbers

(B) A finite or countably infinite set of specific values

(C) Always positive

(D) Always integers

Question 25. Which of the following is NOT a property of a probability distribution for a discrete random variable?

(A) $0 \le P(X=x) \le 1$ for all possible values of x

(B) $\sum P(X=x) = 1$

(C) The sum of all possible values of x is 1

(D) The list of values x covers all possible outcomes

Question 1. The Mathematical Expectation $E(X)$ of a discrete random variable X is also known as its:

(A) Variance

(B) Standard Deviation

(C) Mean

(D) Median

Question 2. For a discrete random variable X with probability distribution $P(X=x_i)$, the expected value $E(X)$ is calculated as:

(A) $\sum x_i$

(B) $\sum P(X=x_i)$

(C) $\sum x_i P(X=x_i)$

(D) $\sum x_i / \sum P(X=x_i)$

Question 3. The variance of a random variable measures its:

(A) Central tendency

(B) Expected value

(C) Spread or dispersion

(D) Skewness

Question 4. The formula for the Variance of a discrete random variable X is $Var(X) = E[(X - E(X))^2]$. This can also be calculated as:

(A) $E(X^2) - [E(X)]^2$

(B) $E(X^2)$

(C) $E(X) - E(X^2)$

(D) $\sum (x_i - E(X))$

Question 5. If the probability distribution of X is:

(A) $0 \times 0.2 + 1 \times 0.3 + 2 \times 0.5 = 0 + 0.3 + 1.0 = 1.3$

(B) $0 + 1 + 2 = 3$

(C) $0.2 + 0.3 + 0.5 = 1$

(D) $(0+1+2)/3 = 1$

Question 6. For the random variable X with probability distribution in Q444, $E(X^2)$ is:

(A) $(0^2 \times 0.2) + (1^2 \times 0.3) + (2^2 \times 0.5) = (0 \times 0.2) + (1 \times 0.3) + (4 \times 0.5) = 0 + 0.3 + 2.0 = 2.3$

(B) $(0^2 + 1^2 + 2^2) = 5$

(C) $(0.2^2 + 0.3^2 + 0.5^2) = 0.04 + 0.09 + 0.25 = 0.38$

(D) $1.3^2 = 1.69$

Question 7. For the random variable X with probability distribution in Q444, the Variance $Var(X)$ is:

(A) $E(X^2) - [E(X)]^2 = 2.3 - (1.3)^2 = 2.3 - 1.69 = 0.61$

(B) 2.3

(C) 1.3

(D) $2.3 + 1.69 = 3.99$

Question 8. The Standard Deviation of a random variable is the square root of its:

(A) Expectation

(B) Variance

(C) Mean

(D) Median

Question 9. If $E(X) = 10$ and $Var(X) = 4$, the standard deviation of X is:

(A) 2

(B) 4

(C) 10

(D) 16

Question 10. If X is a random variable and 'a' is a constant, then $E(aX)$ is equal to:

(A) $a + E(X)$

(B) $a \times E(X)$

(C) $E(X)$

(D) $a^2 E(X)$

Question 11. If X is a random variable and 'b' is a constant, then $E(X + b)$ is equal to:

(A) $E(X) + b$

(B) $E(X) \times b$

(C) $E(X)$

(D) $E(X) / b$

Question 12. If X is a random variable and 'a' is a constant, then $Var(aX)$ is equal to:

(A) $a + Var(X)$

(B) $a \times Var(X)$

(C) $a^2 \times Var(X)$

(D) $Var(X)$

Question 13. If X is a random variable and 'b' is a constant, then $Var(X + b)$ is equal to:

(A) $Var(X) + b$

(B) $Var(X)$

(C) $Var(X) + b^2$

(D) $Var(X) \times b$

Question 14. The expected value $E(X)$ represents the long-run average value of the random variable if the experiment is repeated many times. (True/False)

(A) True

(B) False

(C) It represents the mode

(D) It represents the median

Question 15. If $E(X) = 5$, then $E(2X + 3)$ is:

(A) $2 \times 5 + 3 = 13$

(B) $2 \times 5 = 10$

(C) $5 + 3 = 8$

(D) $5$

Question 16. If $Var(X) = 4$, then $Var(2X + 3)$ is:

(A) $2 \times 4 + 3 = 11$

(B) $2^2 \times 4 = 4 \times 4 = 16$

(C) $4 + 3 = 7$

(D) $4$

Question 17. For a discrete random variable, the expected value is always one of the possible values that the variable can take. (True/False)

(A) True

(B) False (e.g., Expected number of heads in 2 coin tosses is 1, which is a possible value, but the expected value can be a non-integer even if values are integers)

(C) Only if the distribution is symmetric

(D) Only if the variable is binary

Question 18. The variance is always:

(A) Positive

(B) Negative

(C) Non-negative

(D) An integer

Question 19. If $E(X) = 0$, it means the random variable always takes the value 0. (True/False)

(A) True

(B) False (e.g., $X$ takes values -1 and 1 with probability 0.5 each, $E(X)=0$)

(C) Only for discrete variables

(D) Only for continuous variables

Question 20. If $Var(X) = 0$, it means the random variable is a constant (takes only one value with probability 1). (True/False)

(A) True

(B) False

(C) Only if the mean is also 0

(D) Only for discrete variables

Question 1. A sequence of independent trials, each with only two possible outcomes (Success or Failure), and the probability of success is constant for each trial, is called:

(A) Poisson process

(B) Normal distribution

(C) Bernoulli trials

(D) Continuous distribution

Question 2. The number of successes in a fixed number of independent Bernoulli trials follows a:

(A) Poisson distribution

(B) Normal distribution

(C) Uniform distribution

(D) Binomial distribution

Question 3. The parameters of a Binomial distribution are:

(A) Mean ($\mu$) and Standard Deviation ($\sigma$)

(B) Number of trials (n) and probability of success (p)

(C) Lambda ($\lambda$)

(D) Mean and Variance

Question 4. For a Binomial distribution with n trials and probability of success p, the probability of getting exactly k successes is given by $P(X=k) = \binom{n}{k} p^k q^{n-k}$, where q is:

(A) $p^2$

(B) $1-p$

(C) $n-k$

(D) $k/n$

Question 5. The mean of a Binomial distribution with parameters n and p is:

(A) np

(B) npq

(C) $\sqrt{npq}$

(D) n

Question 6. The variance of a Binomial distribution with parameters n and p is:

(A) np

(B) npq

(C) $\sqrt{npq}$

(D) n

Question 7. In a Binomial distribution, the probability of success (p) must be constant for each trial. (True/False)

(A) True

(B) False

(C) Only for small n

(D) Only for large n

Question 8. The number of trials (n) in a Binomial distribution must be:

(A) Variable

(B) Infinite

(C) Fixed

(D) Always greater than k

Question 9. If a Binomial distribution has parameters n=10 and p=0.4, the mean is:

(A) $10 \times 0.4 = 4$

(B) $10 \times 0.4 \times 0.6 = 2.4$

(C) $\sqrt{2.4} \approx 1.55$

(D) 10

Question 10. If a Binomial distribution has parameters n=10 and p=0.4, the variance is:

(A) 4

(B) $10 \times 0.4 \times (1-0.4) = 10 \times 0.4 \times 0.6 = 2.4$

(C) $\sqrt{2.4} \approx 1.55$

(D) 10

Question 11. In a quality control process, items are checked as defective or non-defective. If the probability of a defective item is 0.05, and 20 items are randomly selected, the number of defective items follows a Binomial distribution with parameters:

(A) n=20, p=0.05

(B) n=0.05, p=20

(C) n=20, p=0.95

(D) n=1, p=0.05

Question 12. For a Binomial distribution, as n becomes large, and p is not too close to 0 or 1, the distribution can be approximated by the:

(A) Poisson distribution

(B) Uniform distribution

(C) Normal distribution

(D) Exponential distribution

Question 13. If $np < 5$ and $n(1-p) < 5$, the approximation of the Binomial distribution by the Normal distribution may not be accurate. In such cases, the __________ distribution might be more appropriate if n is large and p is small.

(A) Uniform

(B) Exponential

(C) Poisson

(D) Student's t

Question 14. The shape of a Binomial distribution is symmetric when:

(A) p < 0.5

(B) p > 0.5

(C) p = 0.5

(D) n is very small

Question 15. As n increases, the Binomial distribution tends to become more:

(A) Skewed

(B) Peaked

(C) Symmetric (if p is not extreme)

(D) Uniform

Question 16. The outcomes of a Binomial experiment are always integers (number of successes). (True/False)

(A) True

(B) False

(C) Only for continuous random variables

(D) Only for mean and variance

Question 17. Which of the following is a key characteristic of Bernoulli trials?

(A) More than two outcomes

(B) Dependence between trials

(C) Constant probability of success

(D) Variable number of trials

Question 18. If X ~ B(n, p), the possible values of X are:

(A) All real numbers

(B) All non-negative real numbers

(C) Integers from 0 to n

(D) Integers from 1 to n

Question 19. For a Binomial distribution with n=10 and p=0.1, is it symmetric, positively skewed, or negatively skewed?

(A) Symmetric

(B) Positively skewed (since p < 0.5)

(C) Negatively skewed

(D) Cannot be determined

Question 20. The maximum value of the probability mass function $P(X=k)$ for a Binomial distribution is at the:

(A) Mean (or close to the mean)

(B) Minimum value (0)

(C) Maximum value (n)

(D) Median

Question 1. The Poisson distribution is a __________ probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event.

(A) Continuous variable

(B) Number of occurrences

(C) Probability density

(D) Discrete probability distribution

Question 2. The key parameter of the Poisson distribution is:

(A) n and p

(B) $\mu$ and $\sigma$

(C) $\lambda$ (lambda), the average rate of occurrence

(D) Mean and Variance

Question 3. The probability mass function for a Poisson distribution is $P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!}$, where $\lambda$ is the average number of occurrences, and k is the number of occurrences. The base of the natural logarithm, e, is approximately:

(A) 2.718

(B) 3.141

(C) 1.618

(D) 1.414

Question 4. For a Poisson distribution with parameter $\lambda$, the Mean is:

(A) $\lambda$

(B) $\lambda^2$

(C) $\sqrt{\lambda}$

(D) $e^{-\lambda}$

Question 5. For a Poisson distribution with parameter $\lambda$, the Variance is:

(A) $\lambda$

(B) $\lambda^2$

(C) $\sqrt{\lambda}$

(D) $e^{-\lambda}$

Question 6. In a Poisson distribution, the Mean and Variance are always:

(A) Equal

(B) Different

(C) Positively correlated

(D) Negatively correlated

Question 7. The Poisson distribution is used to model events that are considered to be "rare" occurrences in a large number of trials or over a continuous interval. (True/False)

(A) True

(B) False

(C) Only for frequent occurrences

(D) Only for Binomial distribution

Question 8. The number of customers arriving at a service counter in an hour, if the average arrival rate is known, can often be modeled by a:

(A) Binomial distribution

(B) Normal distribution

(C) Poisson distribution

(D) Uniform distribution

Question 9. The number of typing errors on a page of a book can sometimes be modeled by a Poisson distribution. What would $\lambda$ represent in this case?

(A) The number of pages

(B) The probability of an error

(C) The average number of errors per page

(D) The total number of words on the page

Question 10. The Poisson distribution can be used as an approximation to the Binomial distribution when:

(A) n is small and p is large

(B) n is large and p is large

(C) n is large and p is small (and np is moderate)

(D) n is small and p is small

Question 11. If a Binomial distribution has n=1000 and p=0.002, the corresponding Poisson approximation would have $\lambda$ equal to:

(A) $1000 + 0.002 = 1000.002$

(B) $1000 \times 0.002 = 2$

(C) $\sqrt{1000 \times 0.002} = \sqrt{2} \approx 1.414$

(D) $1000 / 0.002 = 500000$

Question 12. For a Poisson distribution with $\lambda=3$, the probability of observing exactly 0 events is:

(A) $e^{-3}$

(B) $3e^{-3}$

(C) $e^{-3}/0! = e^{-3}$

(D) 0

Question 13. The possible values for a Poisson random variable are:

(A) Any real number

(B) Any non-negative real number

(C) Any positive integer

(D) Any non-negative integer (0, 1, 2, ...)

Question 14. If the average number of calls received by an office per minute is 2, assuming a Poisson process, the parameter $\lambda$ for the number of calls in a 5-minute interval would be:

(A) 2

(B) 5

(C) $2 \times 5 = 10$

(D) $2/5 = 0.4$

Question 15. A key assumption of the Poisson distribution is that events occur independently. (True/False)

(A) True

(B) False

(C) Only for mean calculation

(D) Only for variance calculation

Question 16. For small values of $\lambda$, the Poisson distribution is:

(A) Symmetric

(B) Positively skewed

(C) Negatively skewed

(D) Uniform

Question 17. As $\lambda$ increases, the Poisson distribution becomes more:

(A) Positively skewed

(B) Negatively skewed

(C) Symmetric (approaching Normal distribution)

(D) Peaked

Question 18. The standard deviation of a Poisson distribution with mean 9 is:

(A) 3

(B) 9

(C) 81

(D) $\sqrt{9} = 3$

Question 19. If the variance of a Poisson distribution is 4, its mean is:

(A) 2

(B) 4

(C) 8

(D) 16

Question 20. Which property distinguishes the Poisson distribution from the Binomial distribution?

(A) Discrete nature

(B) Fixed number of trials

(C) Infinite range of possible outcomes (non-negative integers)

(D) Probability of success is constant

Question 1. The Normal distribution is a __________ probability distribution.

(A) Discrete

(B) Continuous

(C) Categorical

(D) Ordinal

Question 2. The graph of the Normal distribution is symmetric and bell-shaped. (True/False)

(A) True

(B) False

(C) Only for specific parameters

(D) Only for large sample sizes

Question 3. For a Normal distribution, the Mean, Median, and Mode are all located at the same point, which is the peak of the curve. (True/False)

(A) True

(B) False

(C) Only for the Standard Normal distribution

(D) Only for discrete distributions

Question 4. The parameters of a Normal distribution are:

(A) n and p

(B) $\lambda$

(C) Mean ($\mu$) and Standard Deviation ($\sigma$)

(D) Skewness and Kurtosis

Question 5. The Normal distribution with a mean of 0 and a standard deviation of 1 is called the:

(A) Standard Binomial distribution

(B) Standard Poisson distribution

(C) Standard Uniform distribution

(D) Standard Normal distribution

Question 6. A z-score measures how many __________ a value is away from the mean.

(A) Means

(B) Variances

(C) Standard deviations

(D) Medians

Question 7. The formula for converting a value x from a Normal distribution with mean $\mu$ and standard deviation $\sigma$ to a z-score is:

(A) $z = (x + \mu) / \sigma$

(B) $z = (x - \mu) / \sigma$

(C) $z = \mu + x \sigma$

(D) $z = \sigma / (x - \mu)$

Question 8. The total area under the curve of the Standard Normal distribution is:

(A) 0

(B) 0.5

(C) 1

(D) Depends on the mean and standard deviation

Question 9. In a Normal distribution, approximately 68% of the data falls within __________ standard deviation(s) of the mean.

(A) 1

(B) 2

(C) 3

(D) 0.5

Question 10. In a Normal distribution, approximately 95% of the data falls within __________ standard deviation(s) of the mean.

(A) 1

(B) 2

(C) 3

(D) 4

Question 11. The tails of the Normal distribution curve theoretically extend to:

(A) The mean

(B) 3 standard deviations from the mean

(C) $-\infty$ and $+\infty$

(D) The maximum and minimum values in the dataset

Question 12. The area under the Normal curve between two z-scores represents the __________ of a random variable falling between the corresponding values.

(A) Frequency

(B) Cumulative frequency

(C) Probability

(D) Expected value

Question 13. The Normal distribution is widely used in statistics because of the Central Limit Theorem, which states that the distribution of sample means approaches a Normal distribution as the sample size increases, regardless of the population distribution. (True/False)

(A) True

(B) False

(C) Only if the population is already Normal

(D) Only for discrete variables

Question 14. If the mean of a Normal distribution is 100 and the standard deviation is 10, what is the z-score for a value of 120?

(A) $z = (120 - 100) / 10 = 20 / 10 = 2$

(B) $z = (120 + 100) / 10 = 220 / 10 = 22$

(C) $z = 120 - 100 = 20$

(D) $z = 10 / (120 - 100) = 10 / 20 = 0.5$

Question 15. If a value has a z-score of -1 in a Standard Normal distribution, it means the value is:

(A) 1 unit above the mean

(B) 1 unit below the mean

(C) 1 standard deviation above the mean

(D) 1 standard deviation below the mean

Question 16. The probability of a value being exactly at the mean in a continuous Normal distribution is:

(A) Maximum

(B) Minimum

(C) 0

(D) 0.5

Question 17. Z-tables (Standard Normal tables) provide the area under the curve to the __________ of a given z-score.

(A) Left

(B) Right

(C) Both sides

(D) Center

Question 18. The skewness of a Normal distribution is always:

(A) Positive

(B) Negative

(C) Zero

(D) 1

Question 19. The kurtosis for a Standard Normal distribution is often defined as 3. Excess kurtosis (kurtosis - 3) is used to compare peakedness to the Normal. For the Normal distribution, the excess kurtosis is:

(A) 3

(B) 0

(C) Positive

(D) Negative

Question 20. If the heights of adult males in a region are normally distributed with a mean of 170 cm and a standard deviation of 5 cm, a height of 175 cm has a z-score of:

(A) 1

(B) -1

(C) 5

(D) 170

Question 21. In a Normal distribution, the two tails are symmetric and approach the x-axis asymptotically (never actually touch it). (True/False)

(A) True

(B) False

(C) They touch the x-axis at $\pm 3\sigma$

(D) They only extend to $\pm n$

Question 22. If a value is 1.5 standard deviations below the mean in a Normal distribution, its z-score is:

(A) 1.5

(B) -1.5

(C) Dependent on the mean and SD

(D) $1.5 \times \sigma$

Question 23. The shape of the Normal distribution is solely determined by its mean ($\mu$) and standard deviation ($\sigma$). (True/False)

(A) True

(B) False

(C) It also depends on the sample size

(D) It also depends on the skewness and kurtosis

Question 1. The entire group of individuals or objects that you want to study and draw conclusions about is called the:

(A) Sample

(B) Population

(C) Parameter

(D) Statistic

Question 2. A subset of the population that is selected for study is called the:

(A) Population

(B) Parameter

(C) Statistic

(D) Sample

Question 3. A numerical characteristic of the entire population (e.g., population mean $\mu$, population standard deviation $\sigma$) is called a:

(A) Statistic

(B) Parameter

(C) Sample

(D) Data point

Question 4. A numerical characteristic calculated from the sample data (e.g., sample mean $\bar{x}$, sample standard deviation s) is called a:

(A) Parameter

(B) Population

(C) Statistic

(D) Observation

Question 5. Inferential statistics is concerned with:

(A) Describing the main features of a data set

(B) Making predictions and inferences about a population based on a sample

(C) Organizing and presenting data visually

(D) Calculating measures of central tendency and dispersion for a sample

Question 6. Why do we often study a sample instead of the entire population?

(A) Studying the population is always impossible.

(B) Studying the sample is usually more cost-effective, time-efficient, and practical.

(C) Parameters are easier to calculate from a sample.

(D) Samples are more likely to be representative than populations.

Question 7. Which of the following is a parameter?

(A) Sample mean $\bar{x}$

(B) Population standard deviation $\sigma$

(C) Sample proportion $\hat{p}$

(D) Sample size n

Question 8. Which of the following is a statistic?

(A) Population mean $\mu$

(B) Sample standard deviation s

(C) Population proportion p

(D) Population variance $\sigma^2$

Question 9. The process of selecting a sample from a population is called:

(A) Data analysis

(B) Parameter estimation

(C) Sampling

(D) Hypothesis testing

Question 10. If a researcher wants to study the average income of households in Mumbai, the population is:

(A) The researcher's sample

(B) All households in India

(C) All households in Mumbai

(D) The average income itself

Question 11. If a researcher collects data from 100 households in Mumbai to estimate the average income, this group of 100 households is the:

(A) Population

(B) Parameter

(C) Sample

(D) Statistic

Question 12. The calculated average income from the 100 households in the sample is a:

(A) Parameter

(B) Population

(C) Statistic

(D) True average

Question 13. The actual average income of all households in Mumbai (if it could be known) is a:

(A) Statistic

(B) Parameter

(C) Sample

(D) Random variable

Question 14. A well-chosen sample should be __________ of the population.

(A) Identical to

(B) A perfect replica

(C) Representative

(D) Larger than

Question 15. Simple Random Sampling is a technique where every member of the population has an equal chance of being selected for the sample. (True/False)

(A) True

(B) False

(C) Only for small populations

(D) Only for qualitative data

Question 16. The goal of inferential statistics is to use sample statistics to make educated guesses or decisions about population parameters. (True/False)

(A) True

(B) False

(C) Inferential statistics only describes samples

(D) Inferential statistics proves population parameters directly

Question 17. Sampling variability refers to the fact that different samples from the same population will likely have slightly different statistics. (True/False)

(A) True

(B) False

(C) Only if the population is not normal

(D) Only if the sample size is small

Question 18. If you conduct a census, you are collecting data from the entire:

(A) Sample

(B) Population

(C) Statistic

(D) Parameter

Question 19. The symbol $\bar{x}$ is a notation for a:

(A) Population parameter

(B) Sample statistic

(C) Random variable

(D) Hypothesis

Question 20. The symbol $\mu$ is a notation for a:

(A) Population parameter

(B) Sample statistic

(C) Random variable

(D) Hypothesis

Question 21. A sample must be large enough to be representative, but not so large that it becomes impractical to collect and analyze. (True/False)

(A) True

(B) False

(C) Larger is always better

(D) Sample size doesn't affect representativeness

Question 22. Stratified sampling is a technique where the population is divided into subgroups (strata) based on shared characteristics, and then a sample is drawn from each subgroup. This is done to ensure:

(A) Convenience

(B) That each subgroup is represented proportionally in the sample

(C) Randomness is not necessary

(D) The sample is larger than the population

Question 1. Statistical inference involves using sample data to make conclusions about population parameters, while accounting for the uncertainty inherent in sampling. (True/False)

(A) True

(B) False

(C) It only describes the sample

(D) It eliminates uncertainty

Question 2. Hypothesis testing is a procedure used to:

(A) Estimate population parameters

(B) Test a claim or hypothesis about a population parameter

(C) Summarize data graphically

(D) Calculate sample statistics

Question 3. The hypothesis that there is no significant difference or relationship (often the status quo) is called the:

(A) Alternative hypothesis ($H_1$ or $H_a$)

(B) Null hypothesis ($H_0$)

(C) Research hypothesis

(D) Statistical hypothesis

Question 4. The hypothesis that challenges the null hypothesis, suggesting there is a significant difference or relationship, is called the:

(A) Null hypothesis ($H_0$)

(B) Alternative hypothesis ($H_1$ or $H_a$)

(C) Test statistic

(D) P-value

Question 5. In a hypothesis test, the decision to reject or fail to reject the null hypothesis is based on:

(A) Subjective judgment

(B) The sample data

(C) The population parameters (which are unknown)

(D) The type of test statistic used

Question 6. The probability of rejecting the null hypothesis when it is actually true is called a:

(A) Type II Error ($\beta$)

(B) Type I Error ($\alpha$ - Level of Significance)

(C) Power of the test

(D) P-value

Question 7. The probability of failing to reject the null hypothesis when it is actually false is called a:

(A) Type I Error ($\alpha$)

(B) Type II Error ($\beta$)

(C) Level of significance

(D) Power of the test

Question 8. The level of significance ($\alpha$) is the maximum acceptable probability of making a Type I Error. Common values are 0.05 or 0.01. (True/False)

(A) True

(B) False

(C) It is the probability of Type II error

(D) It is always 0.05

Question 9. A test statistic is a value calculated from the sample data during a hypothesis test. (True/False)

(A) True

(B) False

(C) It is a population parameter

(D) It is the p-value

Question 10. The p-value is the probability of obtaining a test statistic as extreme as, or more extreme than, the one calculated from the sample data, assuming that the __________ is true.

(A) Alternative hypothesis

(B) Null hypothesis

(C) Sample statistic

(D) Population parameter

Question 11. If the p-value is less than or equal to the level of significance ($\alpha$), we:

(A) Fail to reject the null hypothesis

(B) Reject the null hypothesis

(C) Accept the null hypothesis

(D) Conclude the alternative hypothesis is false

Question 12. If the p-value is greater than the level of significance ($\alpha$), we:

(A) Reject the null hypothesis

(B) Fail to reject the null hypothesis

(C) Conclude the alternative hypothesis is true

(D) Make a Type I Error

Question 13. A larger p-value provides stronger evidence __________ the null hypothesis.

(A) Against

(B) In favour of

(C) Irrelevant to

(D) Directly proportional to

Question 14. The first step in hypothesis testing is usually to:

(A) Calculate the test statistic

(B) Determine the p-value

(C) State the null and alternative hypotheses

(D) Collect sample data

Question 15. A test that examines whether a population parameter is different from a specific value (e.g., $\mu \neq 50$) is called a:

(A) One-tailed test

(B) Two-tailed test

(C) Z-test

(D) T-test

Question 16. A test that examines whether a population parameter is greater than or less than a specific value (e.g., $\mu > 50$ or $\mu < 50$) is called a:

(A) Two-tailed test

(B) One-tailed test

(C) Chi-square test

(D) F-test

Question 17. The region of the sampling distribution of the test statistic that leads to the rejection of the null hypothesis is called the:

(A) Acceptance region

(B) Non-rejection region

(C) Critical region or Rejection region

(D) P-value region

Question 18. When conducting a hypothesis test, a smaller p-value means the observed data is: (assuming the null hypothesis is true)

(A) More likely to occur

(B) Less likely to occur

(C) Equally likely to occur

(D) Cannot be determined

Question 19. Statistical significance at the 5% level ($\alpha = 0.05$) means there is a 5% chance of observing the result if the null hypothesis were true. (True/False)

(A) True

(B) False

(C) It means there is a 95% chance the alternative hypothesis is true

(D) It means there is a 5% chance the alternative hypothesis is true

Question 20. Failing to reject the null hypothesis means that the sample data provides strong evidence that the null hypothesis is true. (True/False)

(A) True

(B) False (It means there is not enough evidence to reject it)

(C) Only if the sample size is large

(D) Only if the p-value is exactly 1

Question 21. The power of a statistical test is the probability of correctly rejecting the null hypothesis when it is false (i.e., $1 - \beta$). (True/False)

(A) True

(B) False

(C) It is the probability of Type I error

(D) It is the probability of Type II error

Question 1. The t-distribution is similar in shape to the Normal distribution but has __________ tails.

(A) Thinner

(B) Fatter

(C) Square

(D) Asymmetric

Question 2. The shape of the t-distribution depends on its:

(A) Mean

(B) Standard Deviation

(C) Degrees of freedom

(D) Sample size (indirectly, through degrees of freedom)

Question 3. As the degrees of freedom increase, the t-distribution becomes more like the:

(A) Uniform distribution

(B) Poisson distribution

(C) Chi-square distribution

(D) Standard Normal distribution (Z-distribution)

Question 4. A one-sample t-test is used to compare the mean of a single sample to a known or hypothesized:

(A) Sample mean

(B) Sample proportion

(C) Population mean

(D) Population standard deviation

Question 5. The one-sample t-test is typically used when the population standard deviation ($\sigma$) is:

(A) Known

(B) Unknown

(C) Equal to the sample standard deviation

(D) Zero

Question 6. The degrees of freedom for a one-sample t-test with sample size n is:

(A) n

(B) n-1

(C) n-2

(D) $n/2$

Question 7. A two independent groups t-test is used to compare the means of:

(A) A single sample before and after an intervention

(B) Two samples that are related or paired

(C) Two samples that are independent of each other

(D) Three or more independent samples

Question 8. The degrees of freedom for a two independent samples t-test (assuming equal variances) with sample sizes $n_1$ and $n_2$ is:

(A) $n_1 + n_2$

(B) $n_1 + n_2 - 1$

(C) $n_1 + n_2 - 2$

(D) $min(n_1, n_2) - 1$

Question 9. When using a t-test, a key assumption is that the data is sampled from a population that is approximately:

(A) Uniformly distributed

(B) Poisson distributed

(C) Normally distributed

(D) Any distribution is fine

Question 10. For large sample sizes, the results of a t-test are very similar to those of a z-test. (True/False)

(A) True

(B) False

(C) Only if the population standard deviation is known

(D) Only for one-tailed tests

Question 11. A t-test is appropriate for testing hypotheses about:

(A) Population proportions

(B) Population means

(C) Population variances

(D) Relationships between categorical variables

Question 12. If you want to test if the average weight of apples from a new farm is significantly different from the standard average weight of 150 grams, and you have a sample of 30 apples, which test is appropriate?

(A) Two independent sample t-test

(B) Paired sample t-test

(C) One-sample t-test

(D) Chi-square test

Question 13. If you want to compare the average test scores of students taught by Method A versus Method B (with two separate groups of students), which test is appropriate?

(A) One-sample t-test

(B) Paired sample t-test

(C) Two independent samples t-test

(D) ANOVA

Question 14. The t-test statistic is calculated as: (Sample Mean - Hypothesized Population Mean) / (Standard Error of the Mean). (True/False)

(A) True

(B) False

(C) Only for two-sample t-tests

(D) Only for z-tests

Question 15. The standard error of the mean for a one-sample t-test is calculated as:

(A) $\sigma / \sqrt{n}$

(B) $s / \sqrt{n}$

(C) $s^2 / n$

(D) $\sigma / n$

Question 16. A paired sample t-test is used when the two samples are dependent, such as measuring the same subjects before and after a treatment. (True/False)

(A) True

(B) False

(C) This is tested with a Z-test

(D) This is tested with ANOVA

Question 17. When performing a two independent sample t-test, an assumption of homogeneity of variances means that the population variances of the two groups are assumed to be equal. (True/False)

(A) True

(B) False

(C) This assumption is never made

(D) This assumption is only for one-sample t-tests

Question 18. If the calculated t-statistic falls into the critical region (determined by $\alpha$ and degrees of freedom), we:

(A) Fail to reject the null hypothesis

(B) Reject the null hypothesis

(C) Accept the null hypothesis

(D) Conclude there is no difference

Question 19. The t-test is robust to violations of the normality assumption, especially with larger sample sizes. (True/False)

(A) True

(B) False

(C) It is very sensitive to normality violations

(D) It is only robust for small sample sizes

Question 20. If a t-test results in a p-value of 0.03 and $\alpha = 0.05$, you would:

(A) Fail to reject $H_0$

(B) Reject $H_0$

(C) Accept $H_0$

(D) Need more information