Matching Items MCQs for Sub-Topics of Topic 16: Statistics & Probability

(i) Raw Data

(ii) Observation

(iii) Variable

(iv) Statistics

(a) A characteristic that can vary

(b) Data that is yet to be organized

(c) A single data point/measurement

(d) The science of data

(i) Discrete Data

(ii) Continuous Data

(iii) Qualitative Data

(iv) Primary Data

(a) Measured values (can be fractions/decimals)

(b) Countable, distinct values (often integers)

(c) Data collected directly by the researcher

(d) Categorical data (attributes)

(i) Collection

(ii) Organization

(iii) Presentation

(iv) Interpretation

(a) Drawing conclusions

(b) Gathering raw data

(c) Arranging or grouping data

(d) Using tables or graphs

(i) Data from a conducted survey (unprocessed)

(ii) Data arranged in an array or frequency table

(iii) Data classified into class intervals

(iv) Data about colours or preferences

(a) Raw Data

(b) Organized Data

(c) Grouped Data

(d) Qualitative Data

(i) Population

(ii) Sample

(iii) Parameter

(iv) Statistic

(a) A numerical characteristic of a sample

(b) The entire group under study

(c) A subset of the population

(d) A numerical characteristic of a population

(i) Frequency

(ii) Class Interval

(iii) Class Mark

(iv) Class Size

(a) The number of times an observation/value occurs

(b) The difference between upper and lower boundaries

(c) A range of values used for grouping

(d) The midpoint of a class interval

(i) Ungrouped Frequency Distribution

(ii) Grouped Frequency Distribution

(iii) Less Than Cumulative Frequency

(iv) More Than Cumulative Frequency

(a) Shows total occurrences above a value

(b) Shows frequency of each distinct value

(c) Shows frequency for ranges of values

(d) Shows total occurrences below a value

(i) Inclusive Class (e.g., 10-19)

(ii) Exclusive Class (e.g., 10-20)

(iii) Class Boundary

(iv) Class Mark

(a) Point separating two adjacent classes (real limits)

(b) Upper limit not included

(c) Both limits included

(d) Average of upper and lower limits/boundaries

(i) Cumulative Frequency

(ii) Total Frequency

(iii) Frequency of a class (from 'less than' CF)

(iv) Tally Mark $\bcancel{||||}$

(a) Sum of frequencies up to a class

(b) Represents a count of 5

(c) The sum of all class frequencies

(d) Difference between CF of the class and previous class

(i) Suitable for discrete data with few distinct values

(ii) Suitable for continuous data or discrete data with many values

(iii) Shows the number of items below upper boundaries

(iv) Shows the number of items above lower boundaries

(a) Grouped Frequency Distribution

(b) More Than Cumulative Frequency Table

(c) Ungrouped Frequency Distribution

(d) Less Than Cumulative Frequency Table

(i) Bar Graph

(ii) Pie Chart

(iii) Pictograph

(iv) Double Bar Graph

(a) Uses sectors to show parts of a whole

(b) Compares two related datasets using bars

(c) Uses symbols to represent data quantities

(d) Uses separated bars for discrete/categorical data

(i) Height of a bar (Bar Graph)

(ii) Angle of a sector (Pie Chart)

(iii) Number of symbols (Pictograph)

(iv) Space between bars (Bar Graph)

(a) Indicates distinction between categories

(b) Proportional to the value/frequency

(c) Proportional to the proportion of the total

(d) Represents the data quantity based on the symbol's value

(i) Showing expenditure distribution in a budget

(ii) Comparing sales performance over two years

(iii) Representing number of items produced per day (simple count)

(iv) Visualizing favourite colours of students

(a) Bar Graph

(b) Pie Chart

(c) Double Bar Graph

(d) Pictograph (optional, Bar Graph also suitable)

(i) 25%

(ii) 50%

(iii) 75%

(iv) 100%

(a) $180^\circ$

(b) $360^\circ$

(c) $90^\circ$

(d) $270^\circ$

(i) Proportion of the part

(ii) Total angle of the circle

(iii) Formula for angle

(iv) Sum of all sector angles

(a) $360^\circ$

(b) (Value / Total Value)

(c) (Proportion) $\times$ $360^\circ$

(d) $360^\circ$

(i) Histogram

(ii) Frequency Polygon

(iii) Bar Graph

(iv) Ogive

(a) Used for grouped continuous data frequency

(b) Used for cumulative frequency distribution

(c) Used for ungrouped discrete frequency

(d) Formed by joining midpoints of histogram tops

(i) Adjacent bars

(ii) Height of a bar

(iii) Width of a bar

(iv) Area of a bar

(a) Represents the class interval

(b) Indicates continuity of data (no gaps between classes)

(c) Represents the frequency of the class

(d) Proportional to the frequency of the class

(i) Values on x-axis

(ii) Values on y-axis

(iii) To close the polygon at the start

(iv) To close the polygon at the end

(a) Connect to x-axis at midpoint of next class (freq=0)

(b) Frequencies

(c) Class Marks

(d) Connect to x-axis at midpoint of preceding class (freq=0)

(i) Histogram

(ii) Bar Graph

(iii) Frequency Polygon

(iv) Ogive

(a) Ungrouped discrete data frequencies

(b) Grouped continuous data frequencies

(c) Cumulative frequencies

(d) Shape of frequency distribution (grouped data)

(i) Showing the distribution of student heights

(ii) Comparing the marks distribution in two sections

(iii) Showing the frequency of discrete car colors

(iv) Displaying the number of families by size (e.g., 1-2, 3-4, etc.)

(a) Bar Graph

(b) Histogram

(c) Overlaid Frequency Polygons

(d) Histogram or Bar Graph (if classes are treated as categories)

(i) Ogive

(ii) Less Than Ogive

(iii) More Than Ogive

(iv) Median Estimation

(a) Graphical method from Ogives

(b) Plots cumulative frequencies below values

(c) Plots cumulative frequencies above values

(d) A cumulative frequency curve

(i) Less Than Ogive

(ii) More Than Ogive

(iii) Y-axis for any Ogive

(iv) Intersection of both Ogives (x-coordinate)

(a) Cumulative Frequency

(b) Upper Boundaries

(c) Lower Boundaries

(d) Median

(i) Less Than Ogive shape

(ii) More Than Ogive shape

(iii) Intersection of both Ogives

(iv) Y-coordinate of intersection point

(a) Decreasing curve

(b) Value equal to $N/2$

(c) Increasing curve

(d) Point showing the Median

(i) Median

(ii) First Quartile (Q1)

(iii) Third Quartile (Q3)

(iv) Number of observations below value 'X' (from 'less than' ogive)

(a) Locate $N/4$ on y-axis, find x-value

(b) Locate $N/2$ on y-axis, find x-value

(c) Locate value 'X' on x-axis, find y-value

(d) Locate $3N/4$ on y-axis, find x-value

(i) Less Than Ogive (start)

(ii) Less Than Ogive (end)

(iii) More Than Ogive (start)

(iv) More Than Ogive (end)

(a) Cumulative Frequency = 0

(b) Cumulative Frequency = Total Frequency ($N$)

(c) Cumulative Frequency = Total Frequency ($N$)

(d) Cumulative Frequency = 0

(i) Measures of Central Tendency

(ii) Arithmetic Mean

(iii) Average (in general sense)

(iv) Representative Value

(a) A single value summarizing data center

(b) A number indicating the center of data

(c) Sum of values divided by count

(d) Statistics describing the center of a dataset

(i) Direct Method

(ii) Assumed Mean Method

(iii) Step-Deviation Method

(iv) Mean for ungrouped data

(a) $\bar{x} = \sum x_i / n$

(b) Simplification using deviations divided by class size

(c) Uses midpoint $\times$ frequency sum

(d) Uses an arbitrary value to simplify calculations

(i) Add a constant 'c' to each observation

(ii) Multiply each observation by 'k'

(iii) Subtract a constant 'c' from each observation

(iv) Divide each observation by 'k'

(a) New mean is original mean $\times$ k

(b) New mean is original mean + c

(c) New mean is original mean - c

(d) New mean is original mean / k

(i) Sum of deviations from Mean

(ii) Effect of outliers on Mean

(iii) Usefulness for skewed data

(iv) Usefulness for symmetric data

(a) Significant

(b) Zero

(c) Preferred measure

(d) Less preferred than Median/Mode

(i) $x_i$

(ii) $f_i$

(iii) $\sum f_i x_i$

(iv) $\sum f_i$

(a) Total frequency ($N$)

(b) Class mark

(c) Frequency of the class

(d) Sum of products of frequency and class mark

(i) Positional Average

(ii) Effect of extreme values

(iii) Divides data into two halves

(iv) Graphical estimation

(a) Possible using Ogives

(b) It is based on position, not values of all items

(c) Less affected

(d) 50% observations are below it, 50% above

(i) Finding the Median

(ii) Data set with odd number of observations ($n$)

(iii) Data set with even number of observations ($n$)

(iv) Requirement before finding Median

(a) Arrange data in order

(b) The middle value(s)

(c) Median is the value at the $(n+1)/2$ position

(d) Median is the average of values at $n/2$ and $(n/2)+1$ positions

(i) L

(ii) $N/2$

(iii) cf

(iv) f

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

(b) Lower boundary of the median class

(c) Position of the median observation

(d) Frequency of the median class

(i) Ungrouped data

(ii) Grouped data formula

(iii) Graphical estimation

(iv) Finding Median class

(a) Using cumulative frequencies ($N/2$ position)

(b) Ordering and finding the middle value

(c) Using the formula $L + \frac{(N/2 - cf)}{f} \times h$

(d) Using the intersection of 'less than' and 'more than' ogives

(i) Median class

(ii) $N/2$ value

(iii) Cumulative frequency used to find median class

(iv) Class boundary for formula 'L'

(a) Upper boundary of previous class and lower boundary of current class are same

(b) The value identifying the position of the median

(c) The class interval containing the median observation

(d) 'Less than' cumulative frequency

(i) Definition

(ii) Bimodal data

(iii) Suitability for nominal data

(iv) Effect of extreme values

(a) Has two modes

(b) Not affected

(c) Most frequent value

(d) Only measure applicable

(i) L

(ii) $f_1$

(iii) $f_0$

(iv) $f_2$

(a) Frequency of the class preceding the modal class

(b) Frequency of the modal class

(c) Lower boundary of the modal class

(d) Frequency of the class succeeding the modal class

(i) Symmetric distribution

(ii) Positively skewed distribution

(iii) Negatively skewed distribution

(iv) Empirical formula

(a) Mode $\approx$ 3 Median - 2 Mean

(b) Mean = Median = Mode

(c) Mean $<$ Median $<$ Mode

(d) Mean $>$ Median $>$ Mode

(i) Mean

(ii) Median

(iii) Mode

(iv) Most appropriate for wage data with few very high earners

(a) Sensitive to outliers

(b) Positional average

(c) Median

(d) Most frequent value

(i) Mean

(ii) Median

(iii) Mode

(iv) Quartiles

(a) From Ogives (intersection or $N/k$ position)

(b) Difficult or not possible directly from simple graphs

(c) From Histogram (peak estimation)

(d) From Ogives ($N/4$ and $3N/4$ positions)

(i) Range

(ii) Mean Deviation

(iii) Measures of Dispersion

(iv) Variability

(a) The spread or scatter of data

(b) The difference between maximum and minimum values

(c) Statistics quantifying the spread

(d) Average of absolute deviations from a central point

(i) Calculation method

(ii) Affected by extreme values

(iii) Simplicity

(iv) Considers all data points

(a) Yes, heavily

(b) Difference between Max and Min

(c) No, only extremes

(d) Easiest measure to calculate

(i) Use of absolute values

(ii) Central value for calculation

(iii) Minimum value occurs from

(iv) Calculation involves

(a) Mean or Median (usually)

(b) $\sum |x_i - A| / n$ or $\sum f_i |x_i - A| / \sum f_i$

(c) To prevent deviations from summing to zero

(d) The Median

(i) Add a constant to data (Range)

(ii) Add a constant to data (Mean Deviation)

(iii) Multiply data by positive constant 'k' (Range)

(iv) Multiply data by positive constant 'k' (Mean Deviation)

(a) Multiplied by k

(b) Remains unchanged

(c) Multiplied by k

(d) Remains unchanged

(i) Range

(ii) Mean Deviation

(iii) Variance

(iv) Standard Deviation

(a) Square root of average squared deviations from Mean

(b) Only the extreme values

(c) Average of absolute deviations from Mean or Median

(d) Average of squared deviations from Mean

(i) Variance

(ii) Standard Deviation

(iii) Population Variance ($\sigma^2$)

(iv) Sample Variance ($s^2$)

(a) $\frac{\sum (x_i - \bar{x})^2}{n-1}$

(b) Square root of variance

(c) Average of squared deviations from the mean

(d) $\frac{\sum (x_i - \mu)^2}{N}$

(i) Add a constant 'c' (Variance)

(ii) Add a constant 'c' (Standard Deviation)

(iii) Multiply by 'k' (Variance)

(iv) Multiply by 'k' (Standard Deviation)

(a) Multiplied by $k^2$

(b) Remains unchanged

(c) Multiplied by $|k|$

(d) Remains unchanged

(i) Units of Variance

(ii) Units of Standard Deviation

(iii) Condition for Variance = 0

(iv) Condition for Standard Deviation = 0

(a) Same as data units

(b) All observations have the same value

(c) Square of data units

(d) All observations have the same value

(i) Average of squared deviations

(ii) $\sqrt{Var(X)}$

(iii) Considers the distance of values from the mean

(iv) Basis for many statistical tests

(a) Standard Deviation

(b) Variance

(c) Standard Deviation

(d) Both Variance and Standard Deviation

(i) Population Variance ($\sigma^2$)

(ii) Sample Variance ($s^2$)

(iii) Sum of frequencies ($\sum f_i$)

(iv) Degrees of freedom ($n-1$)

(a) For grouped data

(b) For calculating unbiased sample variance

(c) Dividing by total population size

(d) Dividing by $n-1$

(i) Measures of Relative Dispersion

(ii) Coefficient of Variation (CV)

(iii) Measures of Absolute Dispersion

(iv) Moments

(a) Quantify spread in original units

(b) Used to compare variability across datasets with different scales

(c) Average of powers of deviations

(d) (Standard Deviation / Mean) $\times$ 100

(i) Formula

(ii) Interpretation of higher CV

(iii) Interpretation of lower CV

(iv) Units

(a) Dimensionless (usually percentage)

(b) (SD / Mean) $\times$ 100

(c) More variability relative to the mean

(d) More consistency relative to the mean

(i) Raw Moments

(ii) Central Moments

(iii) First Raw Moment

(iv) Second Central Moment

(a) Calculated about the Mean

(b) Calculated about the origin (0)

(c) Is equal to the Mean

(d) Is equal to the Variance

(i) Mean

(ii) Variance

(iii) Skewness (often related to)

(iv) Kurtosis (often related to)

(a) Fourth central moment (standardized)

(b) First central moment (=0, Mean from raw moment)

(c) Second central moment

(d) Third central moment (standardized)

(i) Comparing variability of heights (in cm) and weights (in kg)

(ii) Comparing consistency of batsmen with different average scores

(iii) Describing spread of exam marks in a single class

(iv) When Mean is close to zero

(a) CV is appropriate

(b) Standard Deviation is usually sufficient

(c) CV is appropriate

(d) CV may be unstable or not meaningful

(i) Skewness

(ii) Kurtosis

(iii) Central Tendency

(iv) Dispersion

(a) Asymmetry

(b) Peakedness and tail heaviness

(c) Center of the distribution

(d) Spread of the distribution

(i) Positively Skewed

(ii) Negatively Skewed

(iii) Symmetric

(iv) Skewness Coefficient = 0

(a) Mean $<$ Median $<$ Mode

(b) Tail on the right; Mean $>$ Median $>$ Mode

(c) Mean = Median = Mode

(d) Implies symmetry

(i) Leptokurtic

(ii) Platykurtic

(iii) Mesokurtic

(iv) Kurtosis (Excess) = 0

(a) Flatter peak and thinner tails

(b) Normal distribution

(c) Sharper peak and fatter tails

(d) Implies mesokurtic shape

(i) Karl Pearson's Coefficient (Mode)

(ii) Karl Pearson's Coefficient (Median)

(iii) Bowley's Coefficient

(iv) Skewness based on moments

(a) Based on Quartiles and Median

(b) $\mu_3 / \sigma^3$ (Standardized third central moment)

(c) (Mean - Mode) / Standard Deviation

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

(i) Mean = Median = Mode

(ii) Long tail to the left

(iii) Higher probability in the tails than Normal

(iv) Fatter tails and lower peak than Normal

(a) Negatively Skewed

(b) Platykurtic

(c) Symmetric (zero skew)

(d) Leptokurtic (high kurtosis)

(i) Quartiles

(ii) Percentiles

(iii) Interquartile Range (IQR)

(iv) Quartile Deviation (QD)

(a) Divide ordered data into 100 equal parts

(b) ($Q3$ - $Q1$) / 2

(c) Divide ordered data into 4 equal parts

(d) $Q3$ - $Q1$

(i) Q1

(ii) Q2

(iii) Q3

(iv) P50

(a) 25th Percentile

(b) 75th Percentile

(c) 50th Percentile

(d) Equal to $Q2$

(i) Median

(ii) Quartiles

(iii) Percentiles

(iv) Deciles

(a) Into 100 equal parts

(b) Into 4 equal parts

(c) Into 2 equal parts

(d) Into 10 equal parts

(i) Interquartile Range (IQR)

(ii) Quartile Deviation (QD)

(iii) Suitable for skewed data

(iv) Suitable for open-ended distributions

(a) Yes, both IQR and QD are suitable

(b) Measure of spread of the middle 50%

(c) Semi-IQR

(d) Yes, both IQR and QD are suitable

(i) Percentile Rank of a value X

(ii) Graphical estimation of Percentiles/Quartiles

(iii) Formula for $k^{th}$ Percentile for grouped data

(iv) Interpretation of 80th Percentile

(a) $P_k = L + \frac{(kN/100 - cf)}{f} \times h$

(b) Using Ogives

(c) Percentage of values $\le$ X

(d) 80% of observations are below this value

(i) Correlation

(ii) Positive Correlation

(iii) Negative Correlation

(iv) Zero Correlation

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

(b) Measures the linear association between two variables

(c) No discernible linear relationship

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

(i) Scatter Diagram

(ii) Histogram

(iii) Pie Chart

(iv) Line Graph (for time series)

(a) Shows relationship between two variables

(b) Shows frequency distribution of a single variable

(c) Shows proportion of a whole

(d) Shows trend of a variable over time

(i) Points clustered around an upward sloping line

(ii) Points clustered around a downward sloping line

(iii) Points scattered randomly

(iv) Points forming a perfect straight line upwards

(a) Negative Correlation

(b) Perfect Positive Correlation

(c) Positive Correlation

(d) Zero or Weak Correlation

(i) $r$ = +1

(ii) $r$ = -1

(iii) $r$ = 0

(iv) $r$ = +0.8

(a) No linear correlation

(b) Perfect negative linear correlation

(c) Strong positive linear correlation

(d) Perfect positive linear correlation

(i) Karl Pearson's Coefficient

(ii) Spearman's Rank Correlation

(iii) Suitable for linear relationships with numerical data

(iv) Suitable for ordinal data or monotonic relationships

(a) Pearson's Coefficient

(b) Measures linear association

(c) Measures monotonic association

(d) Spearman's Coefficient

(i) Random Experiment

(ii) Sample Space

(iii) Event

(iv) Outcome

(a) A single result of a random experiment

(b) An experiment with unpredictable results but known possible outcomes

(c) A subset of the sample space

(d) The set of all possible outcomes

(i) Simple Event

(ii) Compound Event

(iii) Impossible Event

(iv) Sure Event

(a) An event that is certain to happen (equal to Sample Space)

(b) An event with only one outcome

(c) An event that cannot happen (empty set)

(d) An event with more than one outcome

(i) Classical Probability

(ii) Experimental Probability

(iii) Based on observed frequencies

(iv) Based on equally likely outcomes

(a) (Favorable Outcomes) / (Total Outcomes)

(b) (Number of times Event Occurred) / (Total Trials)

(c) Classical Probability

(d) Experimental Probability

(i) Probability of an Impossible Event

(ii) Probability of a Sure Event

(iii) Range of probability for any event E

(iv) Probability value of 0

(a) 1

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

(c) Indicates an impossible event

(d) 0

(i) A and B are Mutually Exclusive

(ii) A and B are Independent

(iii) Intersection of mutually exclusive events

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

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

(b) Occurrence of one does not affect the other's probability

(c) The empty set $\emptyset$

(d) Condition for independence

(i) First Axiom ($P(A)$)

(ii) Second Axiom ($P(\Omega)$)

(iii) Addition Law for Mutually Exclusive A, B

(iv) Law of Complementary Event

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

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

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

(d) $P(\Omega) = 1$

(i) $A \cup B$ (A or B) for mutually exclusive events

(ii) $A \cap B$ (A and B) for mutually exclusive events

(iii) $A \cup B$ (A or B) for any events

(iv) $A'$ (not A)

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

(b) $1 - P(A)$

(c) $P(A) + P(B) - P(A \cap B)$

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

(i) $P(A)=0.4, P(B)=0.5$, A & B mutually exclusive. Find $P(A \cup B)$.

(ii) $P(A)=0.6$. Find $P(A')$.

(iii) $P(A)=0.7, P(B)=0.8, P(A \cap B)=0.5$. Find $P(A \cup B)$.

(iv) A & B mutually exclusive. Find $P(A \cap B)$.

(a) 0

(b) $0.7+0.8-0.5 = 1.0$

(c) $0.4+0.5 = 0.9$

(d) $1-0.6 = 0.4$

(i) $P(A) \ge 0$

(ii) $P(\Omega) = 1$

(iii) $P(\emptyset) = 0$

(iv) $P(A \cup B) = P(A) + P(B)$ for mutually exclusive A, B

(a) The probability of an impossible event

(b) The probability of the sample space

(c) Probability cannot be negative

(d) Additivity for disjoint events

(i) $A \cup B$

(ii) $A \cap B$

(iii) $A'$

(iv) $\Omega$

(a) Complement of A (Not A)

(b) A or B

(c) A and B

(d) Sample Space

(i) Conditional Probability $P(A|B)$

(ii) Formula for $P(A|B)$

(iii) $P(A \cap B)$ derived from conditional probability

(iv) $P(B|A)$

(a) $\frac{P(A \cap B)}{P(B)}$

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

(c) Probability of A given B has occurred

(d) Probability of B given A has occurred

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

(ii) $P(A|B) = 0$ (assuming $P(B) \neq 0$)

(iii) $P(A|B) = 1$ (assuming $P(B) \neq 0$)

(iv) $P(A|B) + P(A'|B)$

(a) Mutually Exclusive

(b) Independent

(c) A is certain if B occurs

(d) Equals 1 (for valid B)

(i) $P(A \cap B)=0.3, P(B)=0.6$. Find $P(A|B)$.

(ii) $P(B)=0.5, P(A|B)=0.7$. Find $P(A \cap B)$.

(iii) $P(A)=0.4, P(A \cap B)=0.1$. Find $P(B|A)$.

(iv) A, B mutually exclusive, $P(B)=0.8$. Find $P(A|B)$.

(a) $0.3 / 0.6 = 0.5$

(b) $0.5 \times 0.7 = 0.35$

(c) 0

(d) $0.1 / 0.4 = 0.25$

(i) If $P(A|B) > P(A)$

(ii) If $P(A|B) = P(A)$

(iii) If $P(A|B) < P(A)$

(iv) Relationship between $P(A|B)$ and $P(B|A)$

(a) A and B are Independent

(b) B makes A less likely

(c) B makes A more likely

(d) Connected by Bayes' Theorem

(i) Probability of first ball being Red ($P(R1)$)

(ii) Probability of second ball being Blue given first was Red ($P(B2|R1)$)

(iii) Probability of first Red AND second Blue ($P(R1 \cap B2)$)

(iv) Probability of second ball being Red given first was Blue ($P(R2|B1)$)

(a) $3/5 \times 1/2 = 3/10$

(b) $3/5$

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

(d) $3/4$

(i) Multiplication Law (general)

(ii) Multiplication Law (independent events)

(iii) Law of Total Probability (Partition $E_i$)

(iv) Partition of Sample Space conditions

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

(b) $E_i$ are mutually exclusive and exhaustive

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

(d) $P(A) = \sum_i P(A|E_i)P(E_i)$

(i) Independent events

(ii) Mutually exclusive events

(iii) Dependent events

(iv) Exhaustive events

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

(b) $A \cup B \cup ... = \Omega$

(c) The probability of one is affected by the other

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

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

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

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

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

(a) A and B are Mutually Exclusive

(b) A and B are Independent

(c) A and B are Independent

(d) A and B are Independent

(i) Mutually Exclusive

(ii) Exhaustive

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

(iv) $\sum P(E_i)$

(a) Every outcome is in at least one $E_i$

(b) Necessary for Law of Total Probability formula denominator

(c) No two events can occur simultaneously

(d) Equals 1

(i) Tossing a coin twice

(ii) Drawing two cards from a deck *with replacement*

(iii) Drawing two cards from a deck *without replacement*

(iv) Rolling a die and tossing a coin

(a) Dependent events

(b) Independent events

(c) Independent events

(d) Independent events

(i) $P(A)$

(ii) $P(B|A)$

(iii) $P(A|B)$

(iv) $P(B)$

(a) Likelihood

(b) Prior Probability

(c) Posterior Probability

(d) Marginal Probability of Evidence

(i) Calculating the probability of a cause given an effect

(ii) Updating prior beliefs with new evidence

(iii) $P(B)$ in the denominator of Bayes' formula

(iv) Formula for $P(E_i|A)$ using partition $E_1, ..., E_n$

(a) $P(E_i|A) = \frac{P(A|E_i)P(E_i)}{\sum_j P(A|E_j)P(E_j)}$

(b) Bayes' Theorem

(c) Calculated using Law of Total Probability

(d) Bayes' Theorem

(i) Medical Diagnosis

(ii) Spam Filtering

(iii) Machine Learning Classification

(iv) Crime Scene Investigation

(a) Calculating probability of disease given test results

(b) Determining the probability of guilt given evidence

(c) Classifying emails as spam or not based on word probabilities

(d) Used in Bayesian classifiers

(i) If $P(A|B) > P(A)$

(ii) If $P(A|B) = P(A)$

(iii) If $P(A|B) < P(A)$

(iv) Relationship between $P(A|B)$ and $P(B|A)$

(a) A and B are Independent

(b) B makes A less likely

(c) B makes A more likely

(d) Connected by Bayes' Theorem

(i) Numerator of $P(A|B)$

(ii) Denominator of $P(A|B)$

(iii) Updating belief from prior to posterior

(iv) The role of likelihood in Bayes' Theorem

(a) $P(B|A)P(A)$

(b) Modifies the prior probability

(c) $P(B)$

(d) Is the core function of Bayes' Theorem

(i) Random Variable

(ii) Probability Distribution

(iii) Discrete Random Variable

(iv) Continuous Random Variable

(a) Can take any value in an interval

(b) Maps outcomes of random experiment to numbers

(c) Describes probabilities of possible values/ranges

(d) Can take finite or countable values

(i) Discrete Random Variable

(ii) Continuous Random Variable

(iii) Probability of specific value P(X=c) for continuous RV

(iv) Area under the curve for continuous RV

(a) Probability Density Function (PDF)

(b) 0

(c) Represents probability

(d) Probability Mass Function (PMF)

(i) Probability of any specific value $P(X=x)$

(ii) Sum of probabilities for all possible values $\sum P(X=x)$ (Discrete)

(iii) Total area under the PDF curve (Continuous)

(iv) Value range for $P(X=x)$ or PDF $f(x)$

(a) 1

(b) $\ge 0$ (for PDF $f(x)$, probability for value = 0)

(c) Between 0 and 1 (inclusive for PMF, $f(x)$ can be $>$ 1)

(d) 1

(i) Number of goals scored in a football match

(ii) Time taken to finish a race

(iii) Number of defective items in a sample

(iv) Weight of a person

(a) Continuous RV

(b) Discrete RV

(c) Discrete RV

(d) Continuous RV

(i) Cumulative Distribution Function (CDF)

(ii) $P(X \le x)$

(iii) For discrete RV, $P(X=x)$

(iv) For continuous RV, probability for a range $[a,b]$

(a) PMF

(b) Definition of CDF

(c) $\int_a^b f(x) dx$

(d) F(x)

(i) Expected Value $E(X)$

(ii) Variance $Var(X)$

(iii) Standard Deviation $\sigma_X$

(iv) Mean of the distribution

(a) A measure of the spread

(b) The theoretical average

(c) $E(X)$

(d) $\sqrt{Var(X)}$

(i) Discrete Random Variable

(ii) Continuous Random Variable

(iii) Formula for $E(X)$ (Discrete)

(iv) Formula for $E(X)$ (Continuous)

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

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

(c) Requires integration

(d) Requires summation

(i) Definition of Variance

(ii) Alternative formula for $Var(X)$

(iii) Sum of squared deviations from the mean

(iv) Mean squared deviation from the mean

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

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

(c) $\sum (x_i - \mu)^2$ or $\sum (x_i - \bar{x})^2$ (ungrouped data part)

(d) Variance

(i) $E(aX)$

(ii) $E(X + b)$

(iii) $E(aX + b)$

(iv) $E(c)$ where c is a constant

(a) $a E(X)$

(b) c

(c) $E(X) + b$

(d) $a E(X) + b$

(i) $Var(aX)$

(ii) $Var(X + b)$

(iii) $Var(aX + b)$

(iv) $Var(c)$ where c is a constant

(a) 0

(b) $a^2 Var(X)$

(c) $Var(X)$

(d) $a^2 Var(X)$

(i) Bernoulli Trial

(ii) Binomial Experiment

(iii) Binomial Distribution

(iv) Parameters of Binomial Distribution

(a) $n$ and $p$

(b) A single trial with two outcomes

(c) A fixed number of independent Bernoulli trials

(d) Probability distribution of the number of successes in a Binomial Experiment

(i) $\binom{n}{k}$

(ii) $p$

(iii) $k$

(iv) $1-p$

(a) Probability of failure

(b) Number of successes

(c) Probability of success

(d) Number of combinations (ways to get k successes)

(i) Mean

(ii) Variance

(iii) Standard Deviation

(iv) Possible values of X

(a) $\sqrt{np(1-p)}$

(b) $np$

(c) $np(1-p)$

(d) $0, 1, 2, ..., n$

(i) Tossing a fair coin 10 times, counting heads

(ii) Checking 50 items for defects, probability of defect is 0.1

(iii) Rolling a die 6 times, counting times you get a 6

(iv) A test with 20 True/False questions, probability of guessing correctly is 0.5

(a) $n=6, p=1/6$

(b) $n=10, p=0.5$

(c) $n=20, p=0.5$

(d) $n=50, p=0.1$

(i) Symmetric shape

(ii) Positively skewed shape

(iii) Negatively skewed shape

(iv) Becomes more symmetric as n increases (for fixed p not near 0 or 1)

(a) $p > 0.5$

(b) Property related to Normal approximation

(c) $p = 0.5$

(d) $p < 0.5$

(i) Poisson Distribution

(ii) Parameter $\lambda$

(iii) Events occur independently

(iv) Discrete random variable

(a) A characteristic of a Poisson process

(b) Describes the number of occurrences in an interval

(c) Can take values $0, 1, 2, ...$

(d) Represents the average rate of occurrence

(i) Mean

(ii) Variance

(iii) Standard Deviation

(iv) Relationship between Mean and Variance

(a) $\sqrt{\lambda}$

(b) Mean = Variance

(c) $\lambda$

(d) $\lambda$

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

(ii) $\lambda^k$

(iii) $k!$

(iv) $e$

(a) Factorial of k

(b) The base of the natural logarithm ($\approx 2.718$)

(c) Related to the probability of zero occurrences

(d) Average rate raised to the power of k

(i) Nature of the random variable

(ii) Range of possible values

(iii) When it approximates Binomial B(n,p)

(iv) Relationship between Mean and Variance

(a) Non-negative integers (0, 1, 2, ...)

(b) They are equal

(c) Discrete

(d) n is large and p is small

(i) Number of calls per minute to a helpline

(ii) Number of customers arriving in a time period

(iii) Number of defects per unit area of material

(iv) Probability of a very rare event occurring k times in many trials

(a) Poisson distribution is suitable

(b) Poisson distribution is suitable

(c) Poisson distribution is suitable

(d) Poisson approximation to Binomial is suitable

(i) Normal Distribution

(ii) Standard Normal Distribution

(iii) Z-score

(iv) Bell Curve

(a) A specific Normal distribution with $\mu=0, \sigma=1$

(b) The shape of the Normal distribution graph

(c) A continuous probability distribution

(d) Measures distance from the mean in standard deviations

(i) Symmetry

(ii) Mean = Median = Mode

(iii) Tails

(iv) Parameters

(a) Extend to $\pm \infty$ asymptotically

(b) Location and Spread ($\mu, \sigma$)

(c) The curve is identical on both sides of the center

(d) All central tendencies coincide at the peak

(i) Mean

(ii) Standard Deviation

(iii) Total Area under the curve

(iv) Z-table

(a) 1

(b) Gives areas (probabilities) under the curve

(c) 0

(d) 1

(i) Formula for Z-score

(ii) Probability $P(X \le x)$

(iii) Probability $P(X \ge x)$

(iv) Probability $P(a \le X \le b)$

(a) Area to the right of z-score $(1 - P(Z \le z))$

(b) Area to the left of z-score $P(Z \le z)$

(c) $P(Z \le z_b) - P(Z \le z_a)$

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

(i) Approx 68%

(ii) Approx 95%

(iii) Approx 99.7%

(iv) Exact 50%

(a) Within $\mu \pm 2\sigma$

(b) Within $\mu \pm 1\sigma$

(c) Less than the Mean $\mu$

(d) Within $\mu \pm 3\sigma$

(i) Population

(ii) Sample

(iii) Parameter

(iv) Statistic

(a) A characteristic of the population

(b) The entire group of interest

(c) A characteristic calculated from a sample

(d) A subset of the population

(i) $\mu$

(ii) $\sigma$

(iii) $\bar{x}$

(iv) s

(a) Sample mean

(b) Population mean

(c) Sample standard deviation

(d) Population standard deviation

(i) Inferential Statistics

(ii) Sampling

(iii) Representative Sample

(iv) Sampling Variability

(a) Natural variation in statistics across different samples

(b) Making inferences about population from sample

(c) Accurately reflects population characteristics

(d) Process of selecting a sample

(i) Simple Random Sampling

(ii) Stratified Sampling

(iii) Convenience Sampling

(iv) Cluster Sampling

(a) Dividing population into subgroups based on shared characteristics and sampling from each

(b) Selecting participants based on ease of access (non-random)

(c) Every member has an equal chance of selection

(d) Dividing population into groups and randomly selecting entire groups

(i) Estimating a population parameter

(ii) Testing a hypothesis about a parameter

(iii) Using sample data

(iv) Making conclusions about the population

(a) Basis for inference

(b) Hypothesis testing

(c) Point or interval estimation

(d) Primary purpose of inferential statistics

(i) Hypothesis Testing

(ii) Null Hypothesis ($H_0$)

(iii) Alternative Hypothesis ($H_1$)

(iv) Statistical Inference

(a) A claim that there is a significant difference or effect

(b) Using sample data to make conclusions about a population

(c) A claim of no significant difference or effect

(d) Formal procedure to test a claim about a parameter

(i) Type I Error

(ii) Type II Error

(iii) Probability of Type I Error

(iv) Probability of Type II Error

(a) $\beta$

(b) Failing to reject $H_0$ when $H_0$ is false

(c) $\alpha$ (Level of Significance)

(d) Rejecting $H_0$ when $H_0$ is true

(i) Level of Significance ($\alpha$)

(ii) Test Statistic

(iii) P-value

(iv) Critical Region

(a) Value calculated from sample data

(b) Probability of observing data as extreme as sample data, assuming $H_0$ is true

(c) Boundary for rejecting $H_0$

(d) Threshold for deciding significance

(i) p-value $\le \alpha$

(ii) p-value $> \alpha$

(iii) Small p-value

(iv) Large p-value

(a) Fail to reject $H_0$

(b) Strong evidence against $H_0$

(c) Weak evidence against $H_0$

(d) Reject $H_0$

(i) One-tailed test (e.g., right-tailed)

(ii) One-tailed test (e.g., left-tailed)

(iii) Two-tailed test

(iv) Hypothesis is about a parameter

(a) $H_1: \theta < \theta_0$

(b) $H_1: \theta \neq \theta_0$

(c) $H_1: \theta > \theta_0$

(d) True for all standard hypothesis tests

(i) t-Distribution

(ii) Degrees of Freedom (df)

(iii) Standard Error of the Mean (SEM) for t-test

(iv) T-statistic

(a) A value calculated from the sample data

(b) Used when population standard deviation is unknown

(c) $s / \sqrt{n}$

(d) Determines the specific shape of the t-distribution

(i) One-sample t-test

(ii) Two independent samples t-test

(iii) Paired samples t-test

(iv) Comparing means of more than two groups

(a) Compares mean of one sample to a constant

(b) Compares means of two unrelated samples

(c) Compares means of dependent samples (e.g., before/after)

(d) Typically uses ANOVA, not a t-test

(i) Shape compared to Z-distribution

(ii) Effect of increasing degrees of freedom

(iii) T-distribution for df $\to \infty$

(iv) Symmetry of the t-distribution

(a) Becomes more like Z-distribution

(b) It is symmetric around 0

(c) Fatter tails

(d) Standard Normal distribution

(i) Population standard deviation

(ii) Normality assumption

(iii) Independence assumption (Two-sample test)

(iv) Homogeneity of variances assumption (Two-sample test)

(a) Data from each group is normally distributed

(b) Unknown (typically when using t-test)

(c) Population variances are equal

(d) Samples are not related

(i) Is the average height of students in Class 12 different from 165 cm?

(ii) Do students perform better on a test after a coaching class compared to before?

(iii) Is there a difference in average salary between male and female employees?

(iv) Is the average score in a sample significantly greater than 70?

(a) Two Independent Samples t-test

(b) One-sample t-test

(c) Paired Samples t-test

(d) One-sample t-test (with one-tailed alternative)