Assertion-Reason MCQs for Sub-Topics of Topic 16: Statistics & Probability

Question 1. Assertion (A): Raw data is data that has just been collected from a source and has not undergone any processing or organization.

Reason (R): Organizing raw data makes it easier to understand and analyze.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 2. Assertion (A): The number of siblings a person has is an example of a discrete variable.

Reason (R): Discrete variables can take any value within a given range.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 3. Assertion (A): Data collected from a government publication is considered primary data.

Reason (R): Primary data is collected directly by the investigator for their specific purpose.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 4. Assertion (A): Data handling involves stages like collection, organization, presentation, analysis, and interpretation.

Reason (R): These stages help transform raw data into meaningful information for decision-making.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 5. Assertion (A): The favourite colour of students in a class is a quantitative variable.

Reason (R): Qualitative variables represent attributes or categories that are not typically measured numerically.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 1. Assertion (A): In an exclusive class interval like 20-30, the value 30 is included in the class 30-40.

Reason (R): Exclusive class intervals are defined such that the upper limit is not included in the class but serves as the lower limit for the next class.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 2. Assertion (A): A grouped frequency distribution table is preferred over an ungrouped one when the data has a very large number of distinct values.

Reason (R): Grouping data simplifies the presentation and analysis for large datasets with wide variation.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 3. Assertion (A): The cumulative frequency for the last class interval in a 'less than' cumulative frequency distribution is equal to the total frequency.

Reason (R): The last class interval includes all observations in the dataset.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 4. Assertion (A): The class mark of the interval 40-50 is 45.

Reason (R): The class mark is the average of the upper and lower limits of a class interval.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 5. Assertion (A): Tally marks are used in frequency tables to count the occurrences of observations.

Reason (R): Tally marks are grouped in fives ($\bcancel{||||}$) to make counting easier.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 1. Assertion (A): A bar graph is a suitable tool for representing the distribution of heights of students in a class.

Reason (R): Bar graphs are typically used for categorical or discrete data where bars are separated by spaces.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 2. Assertion (A): In a pie chart showing the budget allocation, the angle of the sector for 'Rent' expenditure will be proportional to the percentage of the total budget spent on rent.

Reason (R): The total angle at the center of a pie chart is $360^\circ$, representing the whole dataset.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 3. Assertion (A): A double bar graph is effective for comparing the performance of two different sections in a test across multiple subjects.

Reason (R): Double bar graphs allow for side-by-side comparison of two related sets of data.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 4. Assertion (A): Pictographs are primarily used for presenting complex statistical analysis results.

Reason (R): Pictographs use symbols and are easy for a general audience to understand data visually.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 5. Assertion (A): The bars in a bar graph must always have equal width and equal spacing between them.

Reason (R): Equal width ensures that the height alone determines the quantity represented, and equal spacing maintains visual consistency for discrete categories.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 1. Assertion (A): A histogram is drawn with adjacent bars because it represents continuous data.

Reason (R): The bars in a histogram represent frequency over continuous class intervals, implying no gaps between classes.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 2. Assertion (A): A frequency polygon can be drawn by joining the midpoints of the tops of the bars of a histogram.

Reason (R): The class marks of the intervals are plotted on the x-axis, and the frequencies on the y-axis when drawing a frequency polygon directly.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 3. Assertion (A): If the class intervals in a grouped frequency distribution have unequal widths, the heights of the bars in the histogram must be adjusted.

Reason (R): For unequal class widths, the area of each bar, not just the height, must be proportional to the class frequency. This is achieved by plotting frequency density on the y-axis.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 4. Assertion (A): A frequency polygon helps visualize the shape of the frequency distribution.

Reason (R): By connecting the midpoints of the classes, the frequency polygon creates a smoothed representation of how frequencies are distributed across the values.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 5. Assertion (A): A histogram is suitable for representing the frequency distribution of a discrete variable like the number of cars per household.

Reason (R): Histograms are designed for continuous data or grouped discrete data where class intervals are continuous.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 1. Assertion (A): An Ogive is a graphical representation of a cumulative frequency distribution.

Reason (R): Ogives plot the cumulative frequency against the class boundaries.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 2. Assertion (A): The Median of a distribution can be estimated graphically from a single 'less than' ogive.

Reason (R): The median is the value corresponding to the cumulative frequency of $N/2$ (half of the total frequency).

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 3. Assertion (A): A 'more than' ogive is always an increasing curve.

Reason (R): In a 'more than' cumulative frequency distribution, the cumulative frequency decreases as the value on the x-axis increases.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 4. Assertion (A): The intersection point of the 'less than' and 'more than' ogives gives the Median of the distribution.

Reason (R): At the Median, the number of observations less than or equal to the value is equal to the number of observations greater than or equal to the value (or $N/2$).

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 5. Assertion (A): Ogives are primarily used for estimating measures like Mean and Mode graphically.

Reason (R): Ogives are graphical representations of cumulative frequency distributions and are used to estimate positional values like Median and Quartiles.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 1. Assertion (A): The Arithmetic Mean is the most commonly used measure of central tendency.

Reason (R): The Mean is easily calculated and takes into account every observation in the dataset.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 2. Assertion (A): If a constant 'c' is added to every observation in a dataset, the mean of the new dataset increases by 'c'.

Reason (R): The sum of the deviations of observations from their mean is always zero.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 3. Assertion (A): For grouped data, the Direct Method for calculating the mean uses class marks ($x_i$) and frequencies ($f_i$).

Reason (R): The formula for the Direct Method is $\bar{x} = \frac{\sum f_i x_i}{\sum f_i}$.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 4. Assertion (A): The arithmetic mean is a suitable measure of central tendency for datasets containing extreme outliers.

Reason (R): The mean is sensitive to extreme values, which can significantly pull its value away from the typical center of the majority of the data points.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 5. Assertion (A): The sum of deviations of individual observations from an assumed mean is always zero.

Reason (R): The sum of deviations of individual observations from the actual arithmetic mean is always zero.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 1. Assertion (A): The Median is a positional average.

Reason (R): The value of the Median is determined by its position in the ordered data, not by the exact values of all observations.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 2. Assertion (A): For grouped data, calculating the Median requires finding the cumulative frequencies.

Reason (R): Cumulative frequencies help in identifying the median class, which contains the middle observation.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 3. Assertion (A): The Median is less affected by extreme outliers than the Mean.

Reason (R): The Median only depends on the value(s) at the center of the ordered dataset, not on the magnitude of extreme values.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 4. Assertion (A): For an ungrouped dataset with an even number of observations, the Median is the average of the two middle values after ordering.

Reason (R): There is no single middle value in a dataset with an even number of observations.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 5. Assertion (A): The Median can be estimated graphically from the intersection point of the 'less than' and 'more than' ogives.

Reason (R): This intersection point corresponds to the value on the x-axis where the cumulative frequency from below equals the cumulative frequency from above (or $N/2$).

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 1. Assertion (A): The Mode is the most frequent value in a data set.

Reason (R): A data set can have more than one mode or no mode at all.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 2. Assertion (A): For a positively skewed distribution, the Mean is typically greater than the Median.

Reason (R): In a positively skewed distribution, the long tail is on the right, and the Mean is pulled towards this tail by higher values, while the Median remains relatively central.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 3. Assertion (A): The empirical formula Mean = 3 Median - 2 Mode holds exactly for all types of distributions.

Reason (R): This empirical relationship is an approximation that works well for moderately skewed distributions but not necessarily for highly skewed or multi-modal distributions.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 4. Assertion (A): For grouped data, the modal class is the class with the highest frequency.

Reason (R): The Mode represents the value (or class interval) with the highest concentration of observations, which is indicated by the highest frequency in a frequency distribution.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 5. Assertion (A): The Mode is the only measure of central tendency that can be used for nominal data.

Reason (R): Nominal data consists of categories without a natural order or numerical value, and the Mode identifies the most frequent category.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 1. Assertion (A): Measures of dispersion are used to find the central location of a data set.

Reason (R): Measures of dispersion quantify the spread or variability of the data around a central value.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 2. Assertion (A): The Range is the simplest measure of dispersion to calculate.

Reason (R): Calculating the Range only requires identifying the highest and lowest values in the dataset.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 3. Assertion (A): Mean Deviation from the Median is always less than or equal to Mean Deviation from any other value.

Reason (R): The sum of absolute deviations is minimized when taken from the Median.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 4. Assertion (A): If a constant 'c' is added to every observation in a data set, the Range remains unchanged.

Reason (R): Adding a constant shifts all data points equally, so the difference between the highest and lowest values remains the same.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 5. Assertion (A): Mean Deviation uses the absolute values of deviations from an average.

Reason (R): Using absolute values ensures that positive and negative deviations do not cancel each other out when summed, providing a meaningful measure of spread.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 1. Assertion (A): The Variance is the average of the squared deviations from the Mean.

Reason (R): Squaring the deviations ensures that negative and positive deviations contribute positively to the measure of spread.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 2. Assertion (A): The Standard Deviation is expressed in the same units as the original data.

Reason (R): The Standard Deviation is the square root of the Variance, which is in the square of the original units.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 3. Assertion (A): If a constant 'c' is added to every observation in a dataset, the Variance remains unchanged.

Reason (R): Adding a constant shifts the entire distribution, but the deviations from the mean (which also shifts by 'c') remain the same.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 4. Assertion (A): If all observations in a dataset have the same value, the Standard Deviation is zero.

Reason (R): Standard Deviation measures the spread of data from the mean, and if all values are identical, there is no spread.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 5. Assertion (A): For a sample variance, the sum of squared deviations is divided by $n-1$ instead of $n$ to get an unbiased estimate of the population variance.

Reason (R): Using $n-1$ (degrees of freedom) accounts for the fact that the sample mean is used as an estimate of the unknown population mean, which slightly underestimates the variability.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 1. Assertion (A): Coefficient of Variation is a measure of relative dispersion.

Reason (R): It is used to compare the variability of datasets measured in different units or having significantly different means.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 2. Assertion (A): A lower Coefficient of Variation indicates greater consistency or uniformity in the data.

Reason (R): CV measures the standard deviation relative to the mean, so a smaller value means the spread is smaller compared to the average value.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 3. Assertion (A): The Coefficient of Variation is meaningful only when the mean of the distribution is non-zero and ideally positive.

Reason (R): The formula for CV involves dividing the standard deviation by the mean, which is undefined when the mean is zero and interpretation is difficult for negative means.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 4. Assertion (A): The first moment about the origin is equal to the Mean of the distribution.

Reason (R): Moments about the origin are called central moments.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 5. Assertion (A): The second central moment of a distribution is equal to its Variance.

Reason (R): Central moments are moments calculated about the mean, and the second central moment is the average of squared deviations from the mean.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 1. Assertion (A): Skewness measures the peakedness of a distribution.

Reason (R): Skewness measures the asymmetry of a distribution, indicating whether the tail is longer on one side than the other.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 2. Assertion (A): For a perfectly symmetric distribution, the coefficient of skewness is zero.

Reason (R): In a symmetric distribution, the Mean, Median, and Mode are all equal, leading to a skewness value of zero using measures like Karl Pearson's coefficient.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 3. Assertion (A): A leptokurtic distribution has a sharper peak and fatter tails than a normal distribution.

Reason (R): Kurtosis measures the peakedness and tail heaviness relative to the normal distribution, and leptokurtic distributions have high kurtosis.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 4. Assertion (A): For a negatively skewed distribution, the Mean is typically less than the Median.

Reason (R): The long tail of a negatively skewed distribution is on the left, pulling the mean towards lower values relative to the median and mode.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 5. Assertion (A): Bowley's coefficient of skewness is based on quartiles.

Reason (R): Bowley's coefficient uses the relationship between the first quartile, median, and third quartile to measure skewness.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 1. Assertion (A): Quartiles divide an ordered dataset into four equal parts.

Reason (R): The first quartile ($Q1$) is the value below which 25% of the observations fall, and the third quartile ($Q3$) is the value below which 75% of the observations fall.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 2. Assertion (A): The second quartile ($Q2$) is always equal to the Median.

Reason (R): Both $Q2$ and the Median represent the value that divides the ordered dataset into two equal halves (50% below and 50% above).

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 3. Assertion (A): The Interquartile Range (IQR) is a measure of dispersion that is heavily influenced by outliers.

Reason (R): IQR is calculated as $Q3 - Q1$ and only considers the spread of the middle 50% of the data, ignoring the extreme values in the tails.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 4. Assertion (A): Percentiles divide an ordered dataset into 100 equal parts.

Reason (R): The $k^{th}$ percentile ($P_k$) is the value below which k% of the observations fall.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 5. Assertion (A): Quartiles and percentiles can be estimated graphically from ogives.

Reason (R): Ogives provide cumulative frequency information against values, which is needed to locate the positional values like quartiles and percentiles.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 1. Assertion (A): Correlation between two variables implies that one variable causes the other.

Reason (R): Correlation only measures the degree and direction of linear association between variables; it does not establish cause and effect.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 2. Assertion (A): A scatter diagram is a useful tool for visualizing the relationship between two quantitative variables.

Reason (R): A scatter diagram plots pairs of data points on a graph, allowing us to visually inspect for patterns like linear trends.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 3. Assertion (A): If Karl Pearson's coefficient of correlation ($r$) is close to $+1$, it indicates a strong positive linear relationship.

Reason (R): The value of 'r' ranges from $-1$ to $+1$, where values near $+1$ indicate strong positive association and values near $-1$ indicate strong negative association.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 4. Assertion (A): Spearman's Rank Correlation Coefficient is used when the data is nominal.

Reason (R): Spearman's coefficient is suitable for ordinal data (ranks) or when the relationship is monotonic but not necessarily linear, unlike Pearson's coefficient which requires quantitative data and measures linearity.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 5. Assertion (A): A correlation coefficient of zero means there is no relationship between the two variables.

Reason (R): A correlation coefficient of zero indicates no *linear* relationship between the variables, but there might still be a non-linear association.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 1. Assertion (A): Tossing a fair coin is a random experiment.

Reason (R): The outcome of tossing a fair coin cannot be predicted with certainty before the toss, although the possible outcomes (Head, Tail) are known.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 2. Assertion (A): The sample space for rolling a standard six-sided die is {1, 2, 3, 4, 5, 6}.

Reason (R): The sample space is the set of all possible outcomes of a random experiment.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 3. Assertion (A): If event A is 'getting an even number' and event B is 'getting an odd number' when rolling a die, then A and B are mutually exclusive events.

Reason (R): Mutually exclusive events cannot occur simultaneously in a single trial.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 4. Assertion (A): According to the classical definition, the probability of an event is the ratio of favorable outcomes to the total number of outcomes, assuming all outcomes are equally likely.

Reason (R): The classical definition can be applied even if the outcomes are not equally likely.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 5. Assertion (A): The probability of any event must be between $0$ and $1$, inclusive.

Reason (R): A probability of $0$ indicates an impossible event, and a probability of $1$ indicates a sure event.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 1. Assertion (A): According to the axiomatic approach, the probability of the sample space is $1$.

Reason (R): The sample space represents the set of all possible outcomes, and it is certain that one of these outcomes will occur.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 2. Assertion (A): If A and B are mutually exclusive events, then $P(A \cup B) = P(A) + P(B)$.

Reason (R): Mutually exclusive events cannot occur together, so $P(A \cap B) = 0$, and the general addition law simplifies.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 3. Assertion (A): If $P(A) = 0.7$, then $P(\text{not } A) = 0.3$.

Reason (R): The Law of Complementary Events states that $P(A') = 1 - P(A)$.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 4. Assertion (A): If $P(A)=0.5, P(B)=0.6$, and A and B are independent, then $P(A \cup B) = 1.1$.

Reason (R): The probability of any event cannot be greater than $1$.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 5. Assertion (A): The axiomatic approach provides a universal framework for probability that encompasses classical and empirical definitions.

Reason (R): The axioms are fundamental rules from which other probability laws and properties can be derived, regardless of how probabilities are assigned initially.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 1. Assertion (A): Conditional probability $P(A|B)$ is the probability of event A occurring given that event B has occurred.

Reason (R): The formula for $P(A|B)$ is $\frac{P(A \cap B)}{P(B)}$, provided $P(B) \neq 0$.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 2. Assertion (A): If A and B are independent events, then $P(A|B) = P(A)$.

Reason (R): Independent events are such that the occurrence of one event does not affect the probability of the other event occurring.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 3. Assertion (A): If A and B are mutually exclusive events with $P(B) \neq 0$, then $P(A|B) = 0$.

Reason (R): Mutually exclusive events cannot occur simultaneously, so their intersection $A \cap B$ is impossible, making $P(A \cap B) = 0$.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 4. Assertion (A): Conditional probability $P(A|B)$ is always less than or equal to $P(A)$.

Reason (R): Conditioning on an event B can either increase, decrease, or keep the probability of A the same, depending on the relationship between A and B.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 5. Assertion (A): $P(A|B) + P(A'|B) = 1$ for any events A and B, provided $P(B) \neq 0$.

Reason (R): Given event B has occurred, events A and A' (not A) form a partition of the sample space conditioned on B.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 1. Assertion (A): The Multiplication Law of Probability for any two events A and B is $P(A \cap B) = P(A)P(B|A)$.

Reason (R): This formula is derived directly from the definition of conditional probability.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 2. Assertion (A): If A and B are independent events, then $P(A \cap B) = P(A) + P(B)$.

Reason (R): For independent events, the probability of their joint occurrence is the product of their individual probabilities, i.e., $P(A \cap B) = P(A)P(B)$.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 3. Assertion (A): Drawing two cards from a deck *without replacement* are dependent events.

Reason (R): The outcome of the first draw affects the composition of the deck and thus the probabilities for the second draw.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 4. Assertion (A): If events $E_1, E_2, ..., E_n$ form a partition of the sample space, their union is the entire sample space $\Omega$.

Reason (R): A partition requires the events to be mutually exclusive and exhaustive.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 5. Assertion (A): The Law of Total Probability is used to find the probability of an event A when it can occur under different, mutually exclusive conditions ($E_i$).

Reason (R): The formula sums the probabilities of A occurring with each of the partitioning events $P(A) = \sum P(A \cap E_i)$, which is $\sum P(A|E_i)P(E_i)$.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 1. Assertion (A): Bayes' Theorem is used to update the probability of a hypothesis based on new evidence.

Reason (R): Bayes' Theorem provides a formula to calculate the posterior probability of a hypothesis given the evidence, using the prior probability and the likelihood of the evidence.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 2. Assertion (A): In Bayes' Theorem, $P(A)$ is called the likelihood.

Reason (R): $P(A)$ in the formula $P(A|B) = \frac{P(B|A)P(A)}{P(B)}$ represents the prior probability of event A.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 3. Assertion (A): Bayes' Theorem can be used in medical diagnosis to calculate the probability of having a disease given a positive test result.

Reason (R): This calculation involves the prior probability of the disease and the test's accuracy (true positive and false positive rates).

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 4. Assertion (A): The denominator $P(B)$ in Bayes' Theorem ($P(A|B) = \frac{P(B|A)P(A)}{P(B)}$) is necessary to normalize the result so that $P(A|B)$ is a valid probability between $0$ and $1$.

Reason (R): The denominator $P(B)$ represents the marginal probability of the evidence B occurring, calculated using the Law of Total Probability.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 5. Assertion (A): If $P(A|B) > P(A)$, it means the occurrence of B increases the probability of A.

Reason (R): This inequality indicates a positive association between events A and B.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 1. Assertion (A): The number of heads obtained when tossing a coin 5 times is a continuous random variable.

Reason (R): A continuous random variable can take any value within a given range.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 2. Assertion (A): For a valid probability distribution of a discrete random variable, the sum of all probabilities must equal 1.

Reason (R): This represents the certainty that the random variable will take one of its possible values.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 3. Assertion (A): The Probability Mass Function (PMF) is used to describe the probability distribution of a continuous random variable.

Reason (R): For a continuous random variable, the probability of taking any single specific value is $0$, and probability is described by the Probability Density Function (PDF).

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 4. Assertion (A): The Cumulative Distribution Function (CDF) F(x) for any random variable gives the probability $P(X \le x)$.

Reason (R): The CDF is a non-decreasing function ranging from $0$ to $1$.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 5. Assertion (A): A random variable's value is determined by chance.

Reason (R): A random variable is a numerical outcome of a random experiment, hence its value varies based on the experiment's result.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 1. Assertion (A): The Expected Value of a random variable is its theoretical mean.

Reason (R): The expected value represents the long-run average outcome of the random variable over many repetitions of the experiment.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 2. Assertion (A): For a discrete random variable X, $E(X) = \sum x_i P(X=x_i)$.

Reason (R): The expected value is calculated by summing the products of each possible value and its corresponding probability.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 3. Assertion (A): The Variance of a random variable is always a non-negative value.

Reason (R): Variance is the average of squared deviations from the mean, and squared numbers are always non-negative.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 4. Assertion (A): If $Var(X) = 0$, then the random variable X must be a constant.

Reason (R): Zero variance indicates no dispersion, meaning all possible outcomes are the same value.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 5. Assertion (A): For a random variable X and constant 'a', $Var(aX) = a Var(X)$.

Reason (R): When a random variable is multiplied by a constant, the variance is multiplied by the square of the constant, as $Var(aX) = a^2 Var(X)$.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 1. Assertion (A): A Binomial distribution models the number of successes in a fixed number of independent trials, each with two outcomes.

Reason (R): These independent trials with two outcomes are called Bernoulli trials, and the Binomial distribution is based on the number of successes in such trials.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 2. Assertion (A): The parameters of a Binomial distribution are the mean ($\mu$) and variance ($\sigma^2$).

Reason (R): The parameters that define a specific Binomial distribution are the number of trials (n) and the probability of success (p).

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 3. Assertion (A): For a Binomial distribution with n trials and success probability p, the mean is $np$ and the variance is $np(1-p)$.

Reason (R): These formulas for the mean and variance are derived from the properties of expected value and variance of Bernoulli trials.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 4. Assertion (A): The shape of a Binomial distribution is always symmetric.

Reason (R): The Binomial distribution is symmetric only when the probability of success p is equal to $0.5$; otherwise, it is skewed.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 5. Assertion (A): The Binomial distribution can be approximated by the Normal distribution for large n and when p is not too close to $0$ or $1$.

Reason (R): The Central Limit Theorem suggests that the sum of a large number of independent random variables tends towards a Normal distribution.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 1. Assertion (A): The Poisson distribution is a continuous probability distribution.

Reason (R): The Poisson distribution models the number of occurrences of an event in a fixed interval, and the number of occurrences is a discrete variable ($0, 1, 2, ...$).

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 2. Assertion (A): For a Poisson distribution, the Mean and Variance are equal.

Reason (R): The parameter $\lambda$ of the Poisson distribution represents both the average rate of occurrence (Mean) and the variability (Variance) in the number of occurrences.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 3. Assertion (A): The Poisson distribution is suitable for modeling rare events occurring in a large number of trials or over a continuous interval.

Reason (R): The Poisson distribution is often used as an approximation to the Binomial distribution when the number of trials is large and the probability of success is small.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 4. Assertion (A): The number of customers arriving at a shop in an hour can be modeled by a Poisson distribution if the average arrival rate is constant and arrivals are independent.

Reason (R): This scenario fits the conditions of a Poisson process where events occur randomly and independently over time at a constant rate.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 5. Assertion (A): For small values of $\lambda$, the Poisson distribution is symmetric.

Reason (R): As $\lambda$ increases, the Poisson distribution becomes more symmetric and approaches the Normal distribution, but for small $\lambda$, it is positively skewed.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 1. Assertion (A): The Normal distribution is a symmetric, bell-shaped continuous probability distribution.

Reason (R): In a Normal distribution, the Mean, Median, and Mode are all located at the center of the distribution and are equal in value.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 2. Assertion (A): The area under the entire curve of a Normal distribution is equal to $1$.

Reason (R): The total area under the probability density function of any continuous probability distribution represents the total probability, which must be $1$.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 3. Assertion (A): A z-score standardizes a value by indicating how many standard deviations it is away from the mean in a Normal distribution.

Reason (R): The formula for a z-score is $z = \frac{x - \mu}{\sigma}$, where x is the value, $\mu$ is the mean, and $\sigma$ is the standard deviation.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 4. Assertion (A): The Normal distribution is a discrete probability distribution.

Reason (R): The Normal distribution is used to model continuous random variables like height, weight, and temperature.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 5. Assertion (A): According to the Empirical Rule, approximately 95% of the data in a Normal distribution falls within 1 standard deviation of the mean.

Reason (R): Approximately 95% of the data in a Normal distribution falls within 2 standard deviations of the mean.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 1. Assertion (A): A parameter is a numerical characteristic of a population, while a statistic is a numerical characteristic of a sample.

Reason (R): We often use sample statistics to make inferences about unknown population parameters.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 2. Assertion (A): Studying a sample instead of the entire population is often more practical and cost-effective.

Reason (R): A well-chosen sample can be representative of the population, allowing us to draw conclusions about the whole group.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 3. Assertion (A): The symbol $\mu$ represents a sample statistic.

Reason (R): $\mu$ is the standard notation for the population mean, which is a population parameter.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 4. Assertion (A): Sampling variability means that every sample drawn from the same population will have identical statistics.

Reason (R): Sample statistics vary from sample to sample due to the random nature of the sampling process.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 5. Assertion (A): In Stratified Sampling, the population is divided into subgroups, and then a sample is randomly selected from each subgroup.

Reason (R): Stratified sampling is used to ensure that important subgroups within the population are represented in the sample, often proportionally to their size in the population.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 1. Assertion (A): Hypothesis testing is used to estimate population parameters based on sample data.

Reason (R): Hypothesis testing is a procedure used to test a claim or hypothesis about a population parameter based on evidence from a sample.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 2. Assertion (A): The null hypothesis ($H_0$) typically states that there is no significant difference or relationship.

Reason (R): The alternative hypothesis ($H_1$) is a claim that contradicts the null hypothesis, suggesting a significant difference or relationship exists.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 3. Assertion (A): A Type I Error occurs when you fail to reject the null hypothesis when it is actually false.

Reason (R): A Type I Error is the probability of rejecting the null hypothesis when it is true, and its maximum acceptable probability is the level of significance ($\alpha$).

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 4. Assertion (A): If the p-value of a test is less than the level of significance ($\alpha$), you should reject the null hypothesis.

Reason (R): A small p-value indicates that the observed sample data is unlikely to occur if the null hypothesis were true, providing strong evidence against $H_0$.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 5. Assertion (A): A two-tailed test is used when the alternative hypothesis states that the population parameter is greater than a specific value.

Reason (R): A two-tailed test is used when the alternative hypothesis states that the population parameter is *different from* a specific value ($\neq$), while 'greater than' or 'less than' alternatives require a one-tailed test.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 1. Assertion (A): The t-distribution has fatter tails compared to the Standard Normal distribution, especially for small sample sizes.

Reason (R): This accounts for the increased uncertainty when the population standard deviation is unknown and estimated from the sample.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 2. Assertion (A): A one-sample t-test is used when comparing the mean of a single sample to a known population mean and the population standard deviation is unknown.

Reason (R): The t-test is appropriate when the population standard deviation is unknown and the sample standard deviation is used as an estimate, and the data is from an approximately normal population.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 3. Assertion (A): A two independent samples t-test is used to compare the means of two dependent groups, such as measurements taken before and after an intervention on the same subjects.

Reason (R): For comparing dependent groups, a paired samples t-test is used, as the observations within pairs are related.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 4. Assertion (A): As the sample size increases, the t-distribution approaches the Standard Normal distribution.

Reason (R): With larger sample sizes, the sample standard deviation becomes a more reliable estimate of the population standard deviation, reducing the extra variability captured by the t-distribution's fatter tails.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.

Question 5. Assertion (A): A key assumption for conducting a t-test is that the data is sampled from a population that is exactly normally distributed.

Reason (R): The t-test is relatively robust to moderate violations of the normality assumption, especially with larger sample sizes, although strong skewness or outliers can still affect the results.

(A) Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

(B) Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).

(C) Assertion (A) is true but Reason (R) is false.

(D) Assertion (A) is false but Reason (R) is true.