Multiple Correct Answers MCQs for Sub-Topics of Topic 16: Statistics & Probability

Question 1. Which of the following are considered characteristics of 'Data' in statistics?

(A) It is a collection of facts and figures.

(B) It is always presented in a table format.

(C) It can be numerical or descriptive.

(D) It is always derived from primary sources only.

Question 2. Raw data refers to data that:

(A) Has been collected but not yet processed or organized.

(B) Is always collected from government sources.

(C) Is also known as ungrouped data.

(D) Is obtained after performing calculations like finding the mean.

Question 3. Which of the following are examples of qualitative variables?

(A) Marital status of individuals.

(B) Number of students in a classroom.

(C) Preferred mode of transport (e.g., bus, train, car).

(D) Weight of newborn babies.

Question 4. Which of the following are examples of discrete variables?

(A) Height of students in a school.

(B) Number of defective bulbs in a box.

(C) Temperature of a room.

(D) Number of cars sold by a dealer in a month.

Question 5. The process of 'Data Handling' involves several stages. Which of the following are key stages in data handling?

(A) Data Collection.

(B) Data Organization.

(C) Data Visualization (Presentation).

(D) Data Guessing.

Question 6. Primary data is collected by the investigator for the first time. Which methods can be used to collect primary data?

(A) Survey through questionnaires.

(B) Data from published census reports.

(C) Direct personal interview.

(D) Data from online statistical databases.

Question 7. Secondary data is data that has been collected and processed by someone other than the current investigator. Which of these are sources of secondary data?

(A) Official publications from the Reserve Bank of India.

(B) Diaries and personal records.

(C) Research papers published in academic journals.

(D) Direct observation of a phenomenon.

Question 8. Data organization helps in:

(A) Simplifying the raw data.

(B) Making the data ready for analysis.

(C) Identifying trends and patterns easily.

(D) Making the data harder to interpret.

Question 9. Data interpretation involves:

(A) Summarizing data using tables and graphs.

(B) Drawing conclusions from the analyzed data.

(C) Making sense of the patterns observed in the data.

(D) Collecting new data points.

Question 10. Which of the following characteristics apply to quantitative variables?

(A) Can be measured numerically.

(B) Can be discrete or continuous.

(C) Represents categories or attributes.

(D) Examples include age, income, number of errors.

Question 11. An 'Observation' in statistics is:

(A) The process of collecting data.

(B) A single recorded value of a variable.

(C) Always a numerical value.

(D) A specific instance of data for a particular unit.

Question 12. Grouping data is beneficial when dealing with:

(A) A very small number of data points.

(B) Data with a large range of values.

(C) Continuous data.

(D) Data with many distinct values.

Question 1. A frequency distribution table shows:

(A) How often each value or range of values appears in a dataset.

(B) The total number of observations.

(C) The correlation between variables.

(D) The cumulative count of observations.

Question 2. In an ungrouped frequency distribution table, the 'Frequency' column represents:

(A) The cumulative count of observations.

(B) The number of times each distinct value appears in the data.

(C) The total number of values in the dataset.

(D) The proportion of each value in the total.

Question 3. For grouped frequency distribution, a 'Class Interval' has:

(A) A lower limit and an upper limit.

(B) A single value (the class mark).

(C) A fixed width (class size).

(D) No boundaries.

Question 4. In exclusive class intervals (e.g., 10-20, 20-30):

(A) The upper limit of a class is included in that class.

(B) The upper limit of a class is the lower limit of the next class.

(C) An observation equal to the upper limit is included in the *next* class.

(D) These are commonly used for continuous data.

Question 5. Inclusive class intervals (e.g., 10-19, 20-29) are characterized by:

(A) Including both the lower and upper limits in the class.

(B) Having a gap between the upper limit of one class and the lower limit of the next.

(C) Being suitable for discrete data where values fall exactly on the limits.

(D) The upper limit of one class being the lower limit of the next class.

Question 6. The 'Class Size' or 'Width' of a class interval can be calculated as:

(A) Upper limit - Lower limit (for exclusive intervals).

(B) Upper boundary - Lower boundary.

(C) Difference between consecutive class marks.

(D) Lower limit + Upper limit.

Question 7. Cumulative frequency for a class interval in a 'less than' cumulative frequency table represents:

(A) The number of observations greater than the lower limit of that class.

(B) The number of observations less than the upper limit (or upper boundary) of that class.

(C) The sum of frequencies of all classes up to and including that class.

(D) The frequency of that specific class interval.

Question 8. Cumulative frequency for a class interval in a 'more than' cumulative frequency table represents:

(A) The total frequency minus the 'less than' cumulative frequency of the preceding class.

(B) The number of observations greater than or equal to the lower limit (or lower boundary) of that class.

(C) The sum of frequencies of all classes from that class onwards.

(D) The frequency of that specific class interval.

Question 9. If a frequency table has class intervals 0-10, 10-20, 20-30, and frequencies 5, 12, 8, the 'less than' cumulative frequencies are:

(A) 5 for 'less than 10'.

(B) 17 for 'less than 20'.

(C) 8 for 'less than 30'.

(D) 25 for 'less than 30'.

Question 10. Tally marks are used in frequency distributions for:

(A) Counting the number of observations in each category or class.

(B) Organizing raw data visually during tabulation.

(C) Representing the cumulative frequency.

(D) Calculating the class mark.

Question 1. Which of the following graphical representations are suitable for displaying categorical or discrete data?

(A) Pictograph.

(B) Bar Graph.

(C) Histogram.

(D) Pie Chart.

Question 2. A Bar Graph uses bars to represent data. Key features of a bar graph include:

(A) Bars are always adjacent to each other.

(B) The height of each bar is proportional to the value it represents.

(C) The width of the bars is uniform.

(D) There are equal spaces between the bars.

Question 3. A Pie Chart is useful for:

(A) Showing the distribution of a single variable.

(B) Comparing parts of a whole.

(C) Displaying trends over time.

(D) Representing grouped continuous data.

Question 4. In a Pie Chart, the angle of each sector is calculated as:

(A) (Value of the part / Total value) $\times$ 100.

(B) (Value of the part / Total value) $\times$ $360^\circ$.

(C) (Frequency of the category / Total frequency) $\times$ $360^\circ$.

(D) (Frequency of the category / Total frequency) $\times$ 100.

Question 5. A Double Bar Graph is used to:

(A) Show two different variables on the same axes.

(B) Compare two sets of related data side-by-side.

(C) Display cumulative frequencies.

(D) Represent data using two types of bars simultaneously.

Question 6. Pictographs:

(A) Use symbols to represent data.

(B) Are suitable for presenting data to a general audience.

(C) Can represent exact values precisely using fractions of symbols.

(D) Are commonly used for grouped continuous data.

Question 7. When constructing a bar graph, the categories or discrete values are typically plotted on the:

(A) Vertical axis.

(B) Horizontal axis.

(C) Diagonal axis.

(D) Z-axis.

Question 8. Graphical representations help in:

(A) Making data analysis more complex.

(B) Providing a quick overview of the data's main features.

(C) Facilitating comparison between different categories or periods.

(D) Replacing the need for numerical summaries.

Question 9. If a pie chart shows the distribution of expenses (in $\textsf{₹}$) in a family budget, the sum of the percentages of all sectors must equal:

(A) 100.

(B) 360.

(C) The total income.

(D) The sum of all angles ($360^\circ$).

Question 10. Which of the following statements are true about bar graphs?

(A) They can be drawn vertically or horizontally.

(B) They are suitable for continuous data.

(C) The height/length of the bar represents the frequency or value.

(D) They are effective for showing trends over a continuous interval.

Question 11. When drawing a pictograph, it is important to:

(A) Choose a symbol that is relevant to the data.

(B) Define the value represented by one symbol.

(C) Ensure all symbols are of different sizes.

(D) Include a key explaining the symbol's value.

Question 12. Double bar graphs are particularly useful for:

(A) Comparing performance across categories for two different periods or groups.

(B) Showing the proportion of items in different categories within a single group.

(C) Displaying the frequency distribution of continuous data.

(D) Tracking changes in a single variable over time.

Question 1. A Histogram is a graphical representation used for:

(A) Ungrouped discrete data.

(B) Grouped continuous data.

(C) Frequency distribution.

(D) Comparing categorical frequencies.

Question 2. In a Histogram with equal class widths:

(A) The bars are separated by gaps.

(B) The bars are adjacent.

(C) The height of each bar is proportional to the frequency of the class.

(D) The width of each bar represents the class interval.

Question 3. A Frequency Polygon is constructed by:

(A) Joining the upper limits of bars in a histogram.

(B) Joining the mid-points (class marks) of the tops of adjacent bars in a histogram.

(C) Plotting frequencies against class marks and joining the points.

(D) Plotting cumulative frequencies against class boundaries.

Question 4. Which of the following are true about Histograms?

(A) They are suitable for discrete variables.

(B) The area of each bar is proportional to the class frequency (for equal width classes).

(C) The x-axis represents the class intervals or boundaries.

(D) They help visualize the shape of the distribution.

Question 5. When drawing a frequency polygon directly from a frequency table (without a histogram), you need:

(A) Class limits.

(B) Class marks.

(C) Frequencies.

(D) Cumulative frequencies.

Question 6. The sum of the areas of the rectangles in a histogram is proportional to the:

(A) Number of classes.

(B) Class width.

(C) Total frequency.

(D) Range of the data.

Question 7. Frequency polygons are often used for:

(A) Comparing the shapes of two or more frequency distributions.

(B) Displaying the cumulative distribution.

(C) Visualizing the concentration of data points.

(D) Representing nominal data.

Question 8. If the class intervals are not of equal width in a histogram, the height of the bars should be adjusted. This adjustment involves using:

(A) Frequency density (Frequency / Class Width).

(B) Cumulative frequency.

(C) Class marks.

(D) Relative frequency.

Question 9. To draw a frequency polygon from a histogram, you mark the midpoints of the tops of the bars and connect them. What else needs to be done to close the polygon?

(A) Connect the first and last points to the origin (0,0).

(B) Connect the first point to the midpoint of the class *before* the first class (with frequency 0) on the x-axis.

(C) Connect the last point to the midpoint of the class *after* the last class (with frequency 0) on the x-axis.

(D) Join the first point to the last point directly.

Question 10. Which of these graphs represent the frequency distribution of data?

(A) Bar Graph (for discrete data).

(B) Histogram (for continuous data).

(C) Frequency Polygon.

(D) Pie Chart.

Question 1. Cumulative frequency graphs are also known as:

(A) Histograms.

(B) Frequency polygons.

(C) Ogives.

(D) Lorenz curves.

Question 2. To construct a 'less than' ogive, you plot cumulative frequencies against the:

(A) Lower limits of the class intervals.

(B) Upper limits of the class intervals.

(C) Lower boundaries of the class intervals.

(D) Upper boundaries of the class intervals.

Question 3. To construct a 'more than' ogive, you plot cumulative frequencies against the:

(A) Lower limits of the class intervals.

(B) Upper limits of the class intervals.

(C) Lower boundaries of the class intervals.

(D) Upper boundaries of the class intervals.

Question 4. Which measures of central tendency and position can be estimated graphically from ogives?

(A) Mean.

(B) Median.

(C) Quartiles (Q1, Q3).

(D) Mode.

Question 5. The intersection point of the 'less than' ogive and the 'more than' ogive gives the:

(A) Mean on the x-axis.

(B) Median on the x-axis.

(C) Half of the total frequency (N/2) on the y-axis.

(D) Mode on the x-axis.

Question 6. A 'less than' ogive is an increasing curve. A 'more than' ogive is a decreasing curve. (True/False)

(A) True for 'less than' ogive.

(B) False for 'less than' ogive.

(C) True for 'more than' ogive.

(D) False for 'more than' ogive.

Question 7. To estimate the first quartile (Q1) graphically from a 'less than' ogive with total frequency N, you would:

(A) Locate N/2 on the y-axis and find the corresponding value on the x-axis.

(B) Locate N/4 on the y-axis and find the corresponding value on the x-axis.

(C) Locate 3N/4 on the y-axis and find the corresponding value on the x-axis.

(D) Find the intersection of the two ogives.

Question 8. Ogives help in determining:

(A) The number of observations below a certain value.

(B) The number of observations above a certain value.

(C) The frequency of a specific class interval.

(D) The total number of observations.

Question 9. If a 'less than' ogive passes through the point (50, 30), it means:

(A) 30 observations have a value equal to 50.

(B) 30 observations have a value less than 50.

(C) 50 observations have a value less than 30.

(D) The cumulative frequency up to 50 is 30.

Question 10. Which types of ogives are commonly used in statistics?

(A) Frequency Ogive.

(B) Less Than Ogive.

(C) More Than Ogive.

(D) Mean Ogive.

Question 1. Measures of Central Tendency provide:

(A) A single value that represents the typical or central value of a dataset.

(B) Information about the spread of the data.

(C) Different ways to describe the average of the data.

(D) The minimum and maximum values.

Question 2. The Arithmetic Mean is calculated by:

(A) Finding the middle value after ordering the data.

(B) Finding the most frequent value.

(C) Summing all observations and dividing by the number of observations.

(D) $\frac{\sum x_i}{n}$ for ungrouped data.

Question 3. For grouped data, the Mean can be calculated using:

(A) Direct Method.

(B) Assumed Mean Method.

(C) Step-Deviation Method.

(D) Median Method.

Question 4. If a constant value 'c' is added to each observation in a data set:

(A) The Mean increases by 'c'.

(B) The Variance remains unchanged.

(C) The Standard Deviation remains unchanged.

(D) The Median increases by 'c'.

Question 5. If each observation in a data set is multiplied by a constant value 'k':

(A) The Mean is multiplied by 'k'.

(B) The Variance is multiplied by 'k'.

(C) The Variance is multiplied by 'k$^2$'.

(D) The Standard Deviation is multiplied by $|k|$.

Question 6. The Mean is affected by:

(A) Every observation in the dataset.

(B) Extreme values (outliers).

(C) The ordering of the data.

(D) The frequencies of each value.

Question 7. In the context of grouped data, the class mark ($x_i$) is used in the calculation of the Mean. It represents the:

(A) Lower limit of the class.

(B) Upper limit of the class.

(C) Mid-point of the class interval.

(D) Average value for all observations within that class.

Question 8. The sum of the deviations of individual observations from their arithmetic mean ($\sum (x_i - \bar{x})$) is always:

(A) Positive.

(B) Negative.

(C) Zero.

(D) Depends on the data values.

Question 9. If the weighted mean is used, the weights assigned to each observation or group should typically be:

(A) Equal for all observations.

(B) Proportional to their importance or frequency.

(C) Arbitrary values.

(D) Negative values.

Question 10. The Mean is a measure of central tendency that:

(A) Requires numerical data.

(B) Can be used for open-ended distributions.

(C) Is unique for a given data set.

(D) Represents the arithmetic average.

Question 1. The Median is the middle value of a data set when the data is arranged in order. Which of the following are properties of the Median?

(A) It divides the data into two equal halves.

(B) It is affected by extreme values.

(C) It is a positional average.

(D) It can be estimated graphically from ogives.

Question 2. To find the Median of ungrouped data:

(A) Calculate the sum of observations divided by the number of observations.

(B) Arrange the data in ascending or descending order.

(C) If the number of observations (n) is odd, the median is the value at the $(n+1)/2$ position.

(D) If the number of observations (n) is even, the median is the average of the values at the $n/2$ and $(n/2)+1$ positions.

Question 3. The Median is generally preferred over the Mean when:

(A) The data is perfectly symmetric.

(B) The data set contains significant outliers.

(C) The distribution is highly skewed.

(D) The data is nominal.

Question 4. For grouped data, calculating the Median requires:

(A) Finding the modal class.

(B) Calculating cumulative frequencies.

(C) Identifying the median class.

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

Question 5. In the Median formula for grouped data, $M = L + \frac{(N/2 - cf)}{f} \times h$:

(A) L is the lower limit of the median class.

(B) N is the total frequency.

(C) cf is the cumulative frequency of the class *preceding* the median class.

(D) f is the frequency of the median class.

Question 6. The Median can be estimated graphically from:

(A) A single 'less than' ogive.

(B) A single 'more than' ogive.

(C) The intersection point of the 'less than' and 'more than' ogives.

(D) A histogram.

Question 7. If a constant 'c' is added to every observation in a data set, the Median of the new data set will be:

(A) The original Median plus c.

(B) Unchanged.

(C) Multiplied by c.

(D) The middle value of the new ordered data.

Question 8. If every observation in a data set is multiplied by a positive constant 'k', the Median of the new data set will be:

(A) The original Median plus k.

(B) The original Median multiplied by k.

(C) Unchanged.

(D) The middle value of the new ordered data.

Question 9. The Median is a suitable measure of central tendency for:

(A) Nominal data.

(B) Ordinal data.

(C) Skewed numerical data.

(D) Symmetric numerical data.

Question 10. For grouped data with inclusive classes, before finding the median class, you might need to:

(A) Calculate class marks.

(B) Convert classes to exclusive form (find class boundaries).

(C) Calculate cumulative frequencies.

(D) Find the total frequency (N).

Question 1. The Mode is the value that occurs most frequently in a data set. Properties of the Mode include:

(A) It is the most stable measure of central tendency.

(B) A dataset can have more than one mode.

(C) It is suitable for qualitative data.

(D) It is affected by extreme values.

Question 2. For grouped data, the Modal class is identified as the class with the:

(A) Smallest frequency.

(B) Largest frequency.

(C) Largest class mark.

(D) Largest cumulative frequency.

Question 3. The empirical relationship between Mean, Median, and Mode for moderately skewed distributions suggests:

(A) Mean, Median, and Mode are approximately equal.

(B) Mode $\approx$ 3 Median - 2 Mean.

(C) Mean - Mode $\approx$ 3 (Mean - Median).

(D) The relationship is exact for all distributions.

Question 4. For a positively skewed distribution, which of the following relationships between Mean, Median, and Mode are typically observed?

(A) Mean $=$ Median $=$ Mode.

(B) Mean $>$ Median.

(C) Median $>$ Mode.

(D) Mean $>$ Median $>$ Mode.

Question 5. For a negatively skewed distribution, which of the following relationships between Mean, Median, and Mode are typically observed?

(A) Mean $=$ Median $=$ Mode.

(B) Mean $<$ Median.

(C) Median $<$ Mode.

(D) Mean $<$ Median $<$ Mode.

Question 6. The Mode can be estimated graphically from a:

(A) Pie chart.

(B) Histogram.

(C) Frequency polygon.

(D) Ogive.

Question 7. In the Mode formula for grouped data, $Mode = L + \frac{(f_1 - f_0)}{(2f_1 - f_0 - f_2)} \times h$:

(A) L is the lower limit of the modal class.

(B) $f_1$ is the frequency of the modal class.

(C) $f_0$ is the frequency of the class preceding the modal class.

(D) $f_2$ is the frequency of the class succeeding the modal class.

Question 8. Which measure(s) of central tendency is/are suitable for identifying the most popular choice in a survey about favourite ice cream flavours?

(A) Mean.

(B) Median.

(C) Mode.

(D) All of the above.

Question 9. If a data set has values {1, 1, 2, 3, 4, 4, 5, 5, 5, 6}, the Mode(s) is/are:

(A) 1.

(B) 4.

(C) 5.

(D) There is no mode.

Question 10. For a symmetric distribution:

(A) Mean $=$ Median.

(B) Median $=$ Mode.

(C) The distribution is skewed.

(D) The empirical formula holds exactly.

Question 1. Measures of Dispersion quantify the:

(A) Central location of the data.

(B) Spread or variability of the data.

(C) Closeness of the data points to each other or to an average.

(D) Shape of the distribution.

Question 2. The Range of a data set is:

(A) The difference between the maximum and minimum values.

(B) A simple measure of dispersion.

(C) Highly affected by extreme values.

(D) Always positive.

Question 3. Mean Deviation is calculated as the average of the absolute deviations from a central value. The central value can be the:

(A) Mean.

(B) Median.

(C) Mode.

(D) Range.

Question 4. When calculating Mean Deviation, absolute values are taken because:

(A) It simplifies the arithmetic.

(B) The sum of deviations from the Mean is zero, hiding the dispersion.

(C) It treats deviations above and below the average equally in magnitude.

(D) It always results in a larger value for dispersion.

Question 5. If a constant 'c' is added to every observation in a data set:

(A) The Range remains unchanged.

(B) The Mean Deviation remains unchanged.

(C) The Mean Deviation increases by 'c'.

(D) The Standard Deviation remains unchanged.

Question 6. If every observation in a data set is multiplied by a positive constant 'k':

(A) The Range is multiplied by 'k'.

(B) The Mean Deviation is multiplied by 'k'.

(C) The Variance is multiplied by 'k$^2$'.

(D) The Standard Deviation is multiplied by 'k'.

Question 7. Mean Deviation from the Median is always less than or equal to Mean Deviation from the Mean. (True/False)

(A) True, Mean Deviation is minimized when calculated from the Median.

(B) False, it is minimized from the Mean.

(C) True, it is a property of mean deviation.

(D) False, the relationship depends on the distribution shape.

Question 8. For grouped data, the Mean Deviation is calculated using the class marks ($x_i$) and frequencies ($f_i$). The formula involves:

(A) Absolute deviations $|x_i - A|$, where A is Mean or Median.

(B) Squaring the deviations $(x_i - A)^2$.

(C) Summing the product of frequencies and absolute deviations: $\sum f_i |x_i - A|$.

(D) Dividing the sum by the number of observations: $\sum f_i$.

Question 9. Which of the following are disadvantages of using the Range as a measure of dispersion?

(A) It only considers the two extreme values.

(B) It ignores the distribution of values in between.

(C) It is not affected by extreme values.

(D) It is difficult to calculate.

Question 10. The Mean Deviation is less sensitive to extreme values compared to Standard Deviation. (True/False)

(A) True, because it uses absolute values instead of squares.

(B) False, squares amplify deviations more.

(C) True, absolute values are less impacted by outliers.

(D) False, both are equally sensitive.

Question 1. Variance and Standard Deviation are measures of dispersion that:

(A) Are based on the concept of deviations from the Mean.

(B) Use the square of deviations (for Variance) or its square root (for Standard Deviation).

(C) Are zero when all data points are identical.

(D) Are expressed in the same units as the original data (for Variance).

Question 2. The Variance is calculated as the average of the squared deviations from the Mean. For ungrouped data, the population variance $\sigma^2 = \frac{\sum (x_i - \mu)^2}{N}$. The sample variance $s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1}$. Which denominator is used for an unbiased estimate of population variance ($\sigma^2$)?

(A) N (population size).

(B) n (sample size).

(C) n-1 (degrees of freedom).

(D) $\sum f_i$ (for grouped data).

Question 3. The Standard Deviation ($\sigma$ or s) is preferred over Variance as a measure of dispersion because:

(A) It is easier to calculate.

(B) It is in the same units as the original data.

(C) It is the square root of the variance.

(D) It gives a measure of spread that is directly comparable to the mean.

Question 4. If a constant 'c' is added to every observation in a data set:

(A) The Mean increases by 'c'.

(B) The Variance remains unchanged.

(C) The Standard Deviation remains unchanged.

(D) The sum of squared deviations from the new mean remains unchanged.

Question 5. If every observation in a data set is multiplied by a constant 'k':

(A) The Variance is multiplied by 'k$^2$'.

(B) The Standard Deviation is multiplied by 'k' (if k>0).

(C) The Mean is multiplied by 'k'.

(D) The sum of squared deviations from the mean is multiplied by 'k$^2$'.

Question 6. For grouped data, the formula for variance involves:

(A) Class marks ($x_i$).

(B) Frequencies ($f_i$).

(C) Squared deviations from the mean $(x_i - \bar{x})^2$.

(D) Cumulative frequencies.

Question 7. The Variance and Standard Deviation are always non-negative. (True/False)

(A) True for Variance.

(B) False for Variance.

(C) True for Standard Deviation.

(D) False for Standard Deviation.

Question 8. A higher standard deviation implies:

(A) Data points are closer to the mean.

(B) Data points are more spread out from the mean.

(C) Greater variability in the dataset.

(D) Lower dispersion.

Question 9. Standard Deviation is a suitable measure of dispersion for:

(A) Nominal data.

(B) Ordinal data.

(C) Interval or Ratio scale data.

(D) Distributions with a clearly defined mean.

Question 10. The sum of squared deviations from the Mean is minimum compared to the sum of squared deviations from any other value. (True/False)

(A) True, this is a mathematical property.

(B) False, it is minimum from the median.

(C) True, this is the principle behind least squares.

(D) False, it is minimum from the mode.

Question 1. Measures of relative dispersion are used when:

(A) Comparing the variability of two or more datasets with different units of measurement.

(B) Comparing the variability of two or more datasets with significantly different means.

(C) Describing the spread of a single dataset.

(D) The scale of measurement is important for comparison.

Question 2. The Coefficient of Variation (CV) is a common measure of relative dispersion. Which of the following are true about CV?

(A) It is usually expressed as a percentage.

(B) It has the same units as the original data.

(C) It is calculated as (Standard Deviation / Mean) $\times$ 100.

(D) A higher CV indicates greater relative variability.

Question 3. Comparing the price variability of gold (selling at $\textsf{₹}60,000$ per 10g with SD $\textsf{₹}1000$) and silver (selling at $\textsf{₹}700$ per 10g with SD $\textsf{₹}50$). Which metal has higher relative price variability?

(A) Gold ($CV_{Gold} = (1000/60000) \times 100 \approx 1.67\%$)

(B) Silver ($CV_{Silver} = (50/700) \times 100 \approx 7.14\%$)

(C) Gold and Silver have equal relative variability.

(D) Cannot be determined using CV alone.

Question 4. Moments are mathematical concepts used to describe the characteristics of a distribution. Raw moments are calculated about the origin (0). Central moments are calculated about the:

(A) Median.

(B) Mode.

(C) Mean.

(D) Standard Deviation.

Question 5. The first raw moment about the origin is equal to the:

(A) Mean.

(B) Variance.

(C) Standard Deviation.

(D) Sum of observations.

Question 6. The second central moment (moment about the mean) is equal to the:

(A) Mean.

(B) Variance.

(C) Skewness.

(D) Kurtosis.

Question 7. The Coefficient of Variation is useful when comparing the consistency of performance, e.g., comparing the batting consistency of two cricket players with different average scores. A lower CV indicates greater consistency. (True/False)

(A) True.

(B) False.

(C) A higher CV indicates greater consistency.

(D) CV is not used for consistency.

Question 8. If the Mean of a dataset is zero, the Coefficient of Variation is:

(A) Zero.

(B) 100.

(C) Undefined or Infinite.

(D) Equal to the Standard Deviation.

Question 9. Measures of relative dispersion are dimensionless (they have no units). (True/False)

(A) True.

(B) False.

(C) Only if the mean is 1.

(D) Only if the standard deviation is 1.

Question 10. Moments can be used to calculate measures of central tendency, dispersion, skewness, and kurtosis. (True/False)

(A) True, the first four central moments are used for these.

(B) False, moments only describe dispersion.

(C) True, specific moments relate to these properties.

(D) False, moments are unrelated to these measures.

Question 1. Skewness is a measure of the asymmetry of a probability distribution. Which of the following are true about skewness?

(A) A symmetric distribution has zero skewness.

(B) A distribution with a long tail to the right is positively skewed.

(C) A distribution with a long tail to the left is negatively skewed.

(D) Skewness is always positive.

Question 2. For a positively skewed distribution, the typical relationship between Mean, Median, and Mode is:

(A) Mean $<$ Median $<$ Mode.

(B) Mean $>$ Median $>$ Mode.

(C) Mode is generally the lowest value among the three.

(D) Mean is generally the highest value among the three.

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

(A) Mean $>$ Median $>$ Mode.

(B) Mean $<$ Median $<$ Mode.

(C) Mode is generally the highest value among the three.

(D) Mean is generally the lowest value among the three.

Question 4. Methods of measuring skewness include:

(A) Karl Pearson's coefficient of skewness.

(B) Bowley's coefficient of skewness.

(C) Using the first central moment.

(D) Using the third central moment ($\mu_3$).

Question 5. Kurtosis is a measure of the peakedness and tail heaviness of a distribution relative to the normal distribution. Which of the following are true?

(A) A leptokurtic distribution is more peaked than the normal distribution.

(B) A platykurtic distribution is flatter than the normal distribution.

(C) The normal distribution is mesokurtic.

(D) Kurtosis measures the symmetry of the distribution.

Question 6. For a symmetric distribution:

(A) Skewness is zero.

(B) Mean $=$ Median $=$ Mode.

(C) The distribution shape is necessarily normal.

(D) The tails are equally distributed on both sides of the center.

Question 7. If a distribution is positively skewed, it means:

(A) There are more values concentrated on the left side of the peak.

(B) There are more values concentrated on the right side of the peak.

(C) The Mean is pulled towards the right tail.

(D) The median is less than the mean.

Question 8. Kurtosis specifically describes features in the __________ of the distribution.

(A) Center (peak).

(B) Left tail.

(C) Right tail.

(D) Tails (both left and right).

Question 9. Bowley's coefficient of skewness uses the quartiles. It is defined as $\frac{Q_1 + Q_3 - 2Q_2}{Q_3 - Q_1}$. Which of the following are true?

(A) If $Q_1 + Q_3 = 2Q_2$, skewness is 0 (symmetric).

(B) If $Q_1 + Q_3 > 2Q_2$, skewness is positive (positively skewed).

(C) If $Q_1 + Q_3 < 2Q_2$, skewness is negative (negatively skewed).

(D) It is also known as the interquartile range.

Question 10. Leptokurtic distributions are characterized by:

(A) A higher peak than the normal distribution.

(B) Fatter tails than the normal distribution.

(C) Thinner tails than the normal distribution.

(D) Zero skewness.

Question 11. If a distribution is platykurtic, it means:

(A) Its peak is lower and flatter than the normal distribution.

(B) It has thinner tails compared to the normal distribution.

(C) It has heavier tails compared to the normal distribution.

(D) It is symmetric.

Question 12. Measures of skewness and kurtosis provide information about the __________ of a distribution.

(A) Location.

(B) Shape.

(C) Dispersion.

(D) Asymmetry and tailedness.

Question 1. Quartiles are measures that divide an ordered dataset into four equal parts. Which of the following are quartiles?

(A) Q1 (First Quartile).

(B) Q2 (Second Quartile).

(C) Q3 (Third Quartile).

(D) Q4 (Fourth Quartile).

Question 2. The Second Quartile (Q2) is always equal to the:

(A) Mean.

(B) Median.

(C) 50th Percentile (P50).

(D) Average of Q1 and Q3.

Question 3. Percentiles divide an ordered dataset into 100 equal parts. The $k^{th}$ percentile ($P_k$) is the value below which approximately __________ percent of the observations fall.

(A) k.

(B) 100-k.

(C) k/100.

(D) 100/k.

Question 4. Which of the following relationships between quartiles and percentiles are true?

(A) Q1 = P25.

(B) Q2 = P50.

(C) Q3 = P75.

(D) Q1 = P75.

Question 5. The Interquartile Range (IQR) is a measure of dispersion. Which of the following are true about IQR?

(A) It is calculated as Q3 - Q1.

(B) It measures the spread of the middle 50% of the data.

(C) It is affected by extreme values.

(D) It is used in constructing box plots.

Question 6. The Quartile Deviation (QD), or Semi-Interquartile Range, is calculated as:

(A) Q3 - Q1.

(B) (Q3 + Q1) / 2.

(C) (Q3 - Q1) / 2.

(D) Half of the Interquartile Range.

Question 7. Quartiles and percentiles are useful for:

(A) Identifying specific points in the distribution.

(B) Understanding the spread of the data, especially for skewed distributions.

(C) Calculating the Mean of the distribution.

(D) Comparing positions within a dataset or between different datasets.

Question 8. For grouped data, quartiles and percentiles can be estimated using formulas similar to the Median formula, involving:

(A) Cumulative frequencies.

(B) Class boundaries.

(C) The position k(N)/100 for the $k^{th}$ percentile.

(D) Class marks.

Question 9. Quartile Deviation is a measure of dispersion that is:

(A) Based on the middle 50% of the data.

(B) Less affected by extreme values compared to Range and Standard Deviation.

(C) Suitable for open-ended distributions.

(D) Calculated from all observations.

Question 10. Percentile rank of a value indicates the percentage of observations in the dataset that are less than or equal to that value. (True/False)

(A) True, this is the definition.

(B) False, it indicates the percentage greater than the value.

(C) True, and it is often used for comparing individual performance.

(D) False, it is only used for grouped data.

Question 1. Correlation measures the degree and direction of the linear relationship between two variables. Which types of correlation are commonly identified?

(A) Positive Correlation.

(B) Negative Correlation.

(C) Zero Correlation.

(D) Perfect Correlation.

Question 2. A Scatter Diagram is a graphical tool used to:

(A) Show the distribution of a single variable.

(B) Visualize the relationship between two quantitative variables.

(C) Identify potential linear relationships.

(D) Determine the frequency of observations.

Question 3. In a scatter diagram, if the points are scattered randomly with no discernible pattern, it indicates:

(A) A strong linear relationship.

(B) A perfect positive correlation.

(C) Little to no linear correlation.

(D) Potential non-linear relationship (though zero linear correlation).

Question 4. If two variables are positively correlated, it means:

(A) As one variable increases, the other also tends to increase.

(B) As one variable decreases, the other also tends to decrease.

(C) The slope of the line of best fit in a scatter diagram is positive.

(D) The correlation coefficient is negative.

Question 5. If two variables are negatively correlated, it means:

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

(B) As one variable decreases, the other tends to increase.

(C) The slope of the line of best fit in a scatter diagram is negative.

(D) The correlation coefficient is positive.

Question 6. Karl Pearson's Coefficient of Correlation (r) measures the strength and direction of the linear relationship. Its value ranges from -1 to +1. Which of the following values of 'r' indicate a strong linear relationship?

(A) r = $0.1$.

(B) r = $-0.9$.

(C) r = $0.85$.

(D) r = $-0.2$.

Question 7. Correlation does NOT imply causation. This means:

(A) Just because two variables are correlated, one does not necessarily cause the other.

(B) There might be a third variable influencing both.

(C) Correlation is a necessary condition for causation, but not sufficient.

(D) Causation always implies correlation (unless the relationship is non-linear and correlation is zero).

Question 8. Spearman's Rank Correlation Coefficient is used when:

(A) The data is ordinal (ranks).

(B) The relationship is monotonic (consistently increasing or decreasing), but not necessarily linear.

(C) The data is highly skewed and Pearson's coefficient is less appropriate.

(D) The data is continuous and normally distributed.

Question 9. If the correlation coefficient between hours of study and exam marks is found to be +0.7, it implies:

(A) Studying more tends to be associated with higher exam marks.

(B) There is a moderately strong positive linear relationship.

(C) Studying causes higher marks.

(D) The relationship is perfect.

Question 10. If the correlation coefficient between price and demand for a product is found to be -0.8, it implies:

(A) As price increases, demand tends to decrease.

(B) There is a strong negative linear relationship.

(C) Price causes demand to decrease.

(D) The relationship is perfect.

Question 11. A correlation coefficient of 0 means:

(A) There is no relationship between the two variables.

(B) There is no *linear* relationship between the two variables.

(C) The variables are independent (for normally distributed data).

(D) The variables are mutually exclusive.

Question 1. In the context of probability, a 'Random Experiment' is one where:

(A) The outcomes cannot be predicted with certainty before the experiment.

(B) All possible outcomes are known in advance.

(C) The experiment can be repeated under similar conditions.

(D) The outcome is always $0$ or $1$.

Question 2. The 'Sample Space' ($\Omega$ or S) of a random experiment is:

(A) A single outcome of the experiment.

(B) The set of all possible outcomes of the experiment.

(C) Always a finite set.

(D) The event that is sure to occur.

Question 3. An 'Event' is defined as:

(A) A random experiment.

(B) A subset of the sample space.

(C) One or more outcomes of an experiment.

(D) A number between $0$ and $1$.

Question 4. If two events A and B are 'Mutually Exclusive', it means:

(A) They cannot occur at the same time in a single trial.

(B) Their intersection is an empty set ($A \cap B = \emptyset$).

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

(D) If A occurs, B cannot, and vice versa.

Question 5. According to the Classical (Theoretical) definition of probability, $P(E) = \frac{\text{Number of favourable outcomes}}{\text{Total number of possible outcomes}}$. This definition assumes that:

(A) All outcomes in the sample space are equally likely.

(B) The sample space is finite.

(C) The experiment can be repeated many times.

(D) Probability is based on observed frequencies.

Question 6. The range of probability for any event E, $P(E)$, is:

(A) $P(E) \ge 0$.

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

(C) $0 < P(E) < 1$ for a non-impossible, non-sure event.

(D) $-1 \le P(E) \le 1$.

Question 7. The probability of an 'Impossible Event' is $0$. The probability of a 'Sure Event' is $1$. (True/False)

(A) True for Impossible Event.

(B) False for Impossible Event.

(C) True for Sure Event.

(D) False for Sure Event.

Question 8. Experimental (Empirical) Probability is calculated based on observed data from experiments. It is given by:

(A) (Number of times the event occurred) / (Total number of trials).

(B) The limiting value of relative frequency as the number of trials becomes very large.

(C) Always equal to theoretical probability.

(D) (Number of favourable outcomes) / (Total outcomes).

Question 9. Which of the following values can represent the probability of an event?

(A) $0.25$.

(B) $1/4$.

(C) $-0.5$.

(D) $1.5$.

Question 10. If the sample space for rolling a six-sided die is $\Omega = \{1, 2, 3, 4, 5, 6\}$, and event A is 'getting an even number', and event B is 'getting a number greater than 4', then:

(A) Event A = {2, 4, 6}.

(B) Event B = {5, 6}.

(C) $A \cap B = \{6\}$.

(D) A and B are mutually exclusive.

Question 11. A 'Simple Event' consists of a single outcome. A 'Compound Event' consists of more than one outcome. (True/False)

(A) True for Simple Event.

(B) False for Simple Event.

(C) True for Compound Event.

(D) False for Compound Event.

Question 1. The axiomatic approach to probability defines probability based on a set of axioms (rules). These axioms include:

(A) The probability of any event A is between 0 and 1, inclusive ($0 \le P(A) \le 1$).

(B) The probability of the sample space $\Omega$ is 1 ($P(\Omega) = 1$).

(C) For any two mutually exclusive events A and B, the probability of their union is the sum of their probabilities ($P(A \cup B) = P(A) + P(B)$).

(D) Probability must be calculated using the classical definition.

Question 2. The Law of Complementary Events states that for any event A, the probability of its complement A' (not A) is $P(A') = 1 - P(A)$. This law is derived from the axioms and implies that:

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

(B) A and A' are mutually exclusive.

(C) The union of A and A' is the sample space ($A \cup A' = \Omega$).

(D) A and A' are independent.

Question 3. The Addition Law of Probability for any two events A and B (whether mutually exclusive or not) is $P(A \cup B) = P(A) + P(B) - P(A \cap B)$. If A and B ARE mutually exclusive, then:

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

(B) The formula simplifies to $P(A \cup B) = P(A) + P(B)$.

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

(D) The events cannot occur together.

Question 4. If $P(A) = 0.5$ and $P(B) = 0.4$, and A and B are mutually exclusive, then:

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

(B) $P(A \cup B) = 0.9$.

(C) $P(A \cup B) = 0.2$.

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

Question 5. If $P(A) = 0.6$, $P(B) = 0.7$, and $P(A \cup B) = 0.9$, then $P(A \cap B)$ is:

(A) $P(A) + P(B) - P(A \cup B) = 0.6 + 0.7 - 0.9 = 1.3 - 0.9 = 0.4$.

(B) $0.6 \times 0.7 = 0.42$.

(C) $0.4$.

(D) Not possible as $P(A \cup B) > P(A)$ and $P(A \cup B) > P(B)$.

Question 6. For any two events A and B, $P(A \cup B) \le P(A) + P(B)$. (True/False)

(A) True, this is a property derived from the Addition Law.

(B) False, it can be greater sometimes.

(C) True, because $P(A \cap B) \ge 0$.

(D) False, it is only true if they are mutually exclusive.

Question 7. If A is an event, its complement A' consists of all outcomes in the sample space that are NOT in A. Properties of complementary events include:

(A) $A \cup A' = \Omega$.

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

(C) $P(A \cap A') = 1$.

(D) $P(A \cup A') = 1$.

Question 8. If A and B are mutually exclusive, then $P(A \text{ and } B)$ is:

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

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

(C) 0.

(D) Represented by $P(A \cap B)$.

Question 9. Which of the following are fundamental axioms in the axiomatic approach to probability?

(A) $P(A) \ge 0$ for any event A.

(B) $P(\Omega) = 1$.

(C) For a sequence of pairwise mutually exclusive events $A_1, A_2, ...$, $P(\cup_{i=1}^\infty A_i) = \sum_{i=1}^\infty P(A_i)$.

(D) Probability is the ratio of favorable outcomes to total outcomes.

Question 10. If $P(A)=0.3$ and $P(B)=0.4$, and A and B are not mutually exclusive, which of the following could be $P(A \cup B)$?

(A) 0.5 (Requires $P(A \cap B) = 0.3+0.4-0.5 = 0.2$, which is possible)

(B) 0.7 (Requires $P(A \cap B) = 0$, which means they are mutually exclusive, but the question says they are not)

(C) 0.9 (Requires $P(A \cap B) = 0.3+0.4-0.9 = -0.2$, impossible)

(D) 0.6 (Requires $P(A \cap B) = 0.3+0.4-0.6 = 0.1$, which is possible)

Question 1. Conditional probability, denoted as $P(A|B)$, is the probability of event A occurring given that event B has already occurred. Its formula is $P(A|B) = \frac{P(A \cap B)}{P(B)}$. This concept is applicable when:

(A) $P(B) \neq 0$.

(B) Events A and B are independent.

(C) Events A and B are dependent.

(D) Events A and B are mutually exclusive.

Question 2. If $P(A \cap B) = 0.2$ and $P(B) = 0.4$, then $P(A|B)$ is:

(A) $0.2 / 0.4 = 0.5$.

(B) $0.4 / 0.2 = 2$.

(C) $0.5$.

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

Question 3. If $P(A|B) = P(A)$, assuming $P(A) > 0$ and $P(B) > 0$, then events A and B are:

(A) Dependent.

(B) Independent.

(C) Mutually exclusive.

(D) The occurrence of B does not affect the probability of A.

Question 4. If A and B are mutually exclusive events and $P(B) \neq 0$, then $P(A|B)$ is:

(A) 0.

(B) 1.

(C) $P(A)$.

(D) The probability of A given B is impossible (since they cannot occur together).

Question 5. Properties of conditional probability include:

(A) For any event A and conditioning event F ($P(F) \neq 0$), $P(\Omega|F) = 1$.

(B) For any event A and conditioning event F ($P(F) \neq 0$), $P(A|F) = P(A)$.

(C) For any two events A, B and conditioning event F ($P(F) \neq 0$), $P(A \cup B|F) = P(A|F) + P(B|F) - P(A \cap B|F)$.

(D) For any event A and conditioning event F ($P(F) \neq 0$), $0 \le P(A|F) \le 1$.

Question 6. Consider a bag with 4 red and 6 blue marbles. Two marbles are drawn without replacement. Let R1 be the event the first is red, and B2 be the event the second is blue. Which of the following are correct probabilities?

(A) $P(R1) = 4/10 = 2/5$.

(B) $P(B2|R1) = 6/9 = 2/3$ (After R1, 3 red, 6 blue remain out of 9).

(C) $P(R1 \cap B2) = P(R1) \times P(B2|R1) = (4/10) \times (6/9) = (2/5) \times (2/3) = 4/15$.

(D) $P(B2) = P(B2|R1)P(R1) + P(B2|B1)P(B1) = (6/9)(4/10) + (5/9)(6/10) = 24/90 + 30/90 = 54/90 = 3/5$.

Question 7. If $P(A|B) = 1$, it indicates that:

(A) A and B are independent.

(B) Event B is impossible.

(C) If B occurs, A is certain to occur.

(D) B is a subset of A ($B \subseteq A$).

Question 8. If $P(A|B) = 0$, it indicates that:

(A) A and B are independent.

(B) Event A is impossible.

(C) If B occurs, A cannot occur.

(D) A and B are mutually exclusive (if $P(B) \neq 0$).

Question 9. Conditional probability $P(A|B)$ can be calculated even if $P(A)$ is unknown, provided $P(A \cap B)$ and $P(B)$ are known. (True/False)

(A) True, by using the definition $P(A|B) = P(A \cap B) / P(B)$.

(B) False, $P(A)$ is always required.

(C) True, the formula directly uses $P(A \cap B)$ and $P(B)$.

(D) False, this only works for independent events.

Question 10. If $P(A) = 0.6$, $P(B) = 0.4$, and $P(A \cup B) = 0.7$, then $P(A|B)$ is:

(A) $P(A \cap B) = P(A) + P(B) - P(A \cup B) = 0.6 + 0.4 - 0.7 = 0.3$.

(B) $P(A|B) = P(A \cap B) / P(B) = 0.3 / 0.4 = 3/4 = 0.75$.

(C) $0.3$.

(D) $0.75$.

Question 1. The Multiplication Law of Probability states that for any two events A and B, $P(A \cap B) = P(A)P(B|A)$ or $P(A \cap B) = P(B)P(A|B)$. This law is used to calculate the probability of:

(A) A or B occurring.

(B) A and B both occurring.

(C) The intersection of two events.

(D) A given that B has occurred.

Question 2. Events A and B are defined as Independent if $P(A \cap B) = P(A)P(B)$. This condition is equivalent to:

(A) $P(A|B) = P(A)$, provided $P(B) \neq 0$.

(B) $P(B|A) = P(B)$, provided $P(A) \neq 0$.

(C) A and B are mutually exclusive.

(D) The occurrence of one event does not affect the probability of the other.

Question 3. If a bag contains 3 red and 2 blue balls. Two balls are drawn *with replacement*. Let A be the event the first ball is red, and B be the event the second ball is blue. Which of the following is true?

(A) $P(A) = 3/5$.

(B) $P(B) = 2/5$.

(C) Events A and B are independent.

(D) $P(A \cap B) = P(A) \times P(B) = (3/5) \times (2/5) = 6/25$.

Question 4. If a bag contains 3 red and 2 blue balls. Two balls are drawn *without replacement*. Let A be the event the first ball is red, and B be the event the second ball is blue. Which of the following is true?

(A) $P(A) = 3/5$.

(B) $P(B|A) = 2/4 = 1/2$.

(C) Events A and B are dependent.

(D) $P(A \cap B) = P(A) \times P(B|A) = (3/5) \times (1/2) = 3/10$.

Question 5. A collection of events $E_1, E_2, ..., E_n$ forms a 'Partition of the Sample Space' if:

(A) They are mutually exclusive ($E_i \cap E_j = \emptyset$ for $i \neq j$).

(B) They are exhaustive ($\cup_{i=1}^n E_i = \Omega$).

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

(D) They are independent.

Question 6. The Law of Total Probability states that if $E_1, E_2, ..., E_n$ is a partition of $\Omega$ such that $P(E_i) > 0$ for all i, and A is any event, then $P(A) = \sum_{i=1}^n P(A \cap E_i) = \sum_{i=1}^n P(A|E_i)P(E_i)$. This law is used to find the marginal probability of A when:

(A) The conditional probabilities of A given the partitioning events are known.

(B) The prior probabilities of the partitioning events are known.

(C) A is independent of all $E_i$.

(D) The partitioning events are mutually exclusive and exhaustive.

Question 7. If A and B are independent events, then:

(A) $A'$ and $B'$ are independent.

(B) A and $B'$ are independent.

(C) $A'$ and B are independent.

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

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

(A) Independent (since $0.4 \times 0.5 = 0.2$).

(B) Dependent.

(C) Not mutually exclusive (since $P(A \cap B) \neq 0$).

(D) Events for which the multiplication law $P(A \cap B) = P(A)P(B)$ applies.

Question 9. If $P(A) = 0.3$, $P(B) = 0.4$, and A and B are independent, then $P(A \cup B)$ is:

(A) $P(A) + P(B) = 0.3 + 0.4 = 0.7$. (Incorrect, as $P(A \cap B)$ is not 0).

(B) $P(A) + P(B) - P(A \cap B) = P(A) + P(B) - P(A)P(B) = 0.3 + 0.4 - (0.3)(0.4) = 0.7 - 0.12 = 0.58$.

(C) $0.58$.

(D) $P(A) \times P(B) = 0.12$.

Question 10. Drawing cards from a deck without replacement results in dependent events because:

(A) The composition of the deck changes after each draw.

(B) The probability of drawing a specific card changes depending on what was drawn before.

(C) The events are mutually exclusive.

(D) The multiplication law cannot be applied.

Question 1. Bayes' Theorem allows us to revise probabilities based on new information or evidence. It is particularly useful for calculating:

(A) Prior probabilities.

(B) Posterior probabilities.

(C) The probability of a cause given an observed effect.

(D) The probability of a hypothesis being true given the evidence.

Question 2. For two events A and B, Bayes' Theorem is given by $P(A|B) = \frac{P(B|A)P(A)}{P(B)}$. In this formula:

(A) $P(A)$ is the prior probability of A.

(B) $P(B|A)$ is the likelihood of observing B given A is true.

(C) $P(A|B)$ is the posterior probability of A given B has occurred.

(D) $P(B)$ is the joint probability of A and B.

Question 3. The denominator in Bayes' Theorem, $P(B)$, can be calculated using the Law of Total Probability if B can occur under different mutually exclusive conditions $A_1, A_2, ..., A_n$. In this case, $P(B) = \sum_{i=1}^n P(B|A_i)P(A_i)$. This makes Bayes' Theorem applicable when:

(A) The prior probabilities of the conditions ($P(A_i)$) are known.

(B) The likelihoods ($P(B|A_i)$) are known.

(C) The conditions ($A_i$) form a partition of the sample space.

(D) Events $A_i$ and B are independent.

Question 4. Bayes' Theorem is widely used in applications involving:

(A) Updating beliefs based on new evidence.

(B) Classification problems (e.g., spam filtering).

(C) Medical diagnosis.

(D) Calculating simple event probabilities.

Question 5. If $P(A) = 0.6$, $P(B) = 0.5$, and $P(B|A) = 0.4$, which of the following can be calculated?

(A) $P(A \cap B) = P(B|A)P(A) = 0.4 \times 0.6 = 0.24$.

(B) $P(A|B) = [P(B|A)P(A)] / P(B) = 0.24 / 0.5 = 0.48$.

(C) $P(A \cup B) = P(A) + P(B) - P(A \cap B) = 0.6 + 0.5 - 0.24 = 0.86$.

(D) Whether A and B are independent (since $P(A \cap B) = 0.24 \neq P(A)P(B) = 0.6 \times 0.5 = 0.3$, they are dependent).

Question 6. The terms 'prior probability' and 'posterior probability' are central to Bayesian inference. The prior probability is your initial belief, while the posterior probability is the updated belief after considering the evidence. (True/False)

(A) True.

(B) False, prior is after evidence.

(C) True, Bayes' Theorem links these.

(D) False, posterior is unrelated to prior.

Question 7. In the general form of Bayes' Theorem for $P(E_i|A) = \frac{P(A|E_i)P(E_i)}{\sum_{j=1}^n P(A|E_j)P(E_j)}$, the term $P(E_i)$ is the prior probability of $E_i$. (True/False)

(A) True.

(B) False.

(C) It represents the likelihood.

(D) It represents the marginal probability of A.

Question 8. If $P(A|B) > P(A)$, it indicates that the occurrence of event B makes event A:

(A) More likely.

(B) Less likely.

(C) Independent of B.

(D) Dependent on B and positively associated.

Question 9. Bayes' Theorem is valid for:

(A) Both discrete and continuous random variables (when extended appropriately).

(B) Only for mutually exclusive events.

(C) Any events A and B, provided $P(B) \neq 0$.

(D) Calculating prior probabilities.

Question 10. If we have two competing hypotheses, $H_1$ and $H_2$, and we observe some evidence E, Bayes' Theorem can be used to compare the support for $H_1$ versus $H_2$ given E by calculating the ratio $P(H_1|E) / P(H_2|E)$. (True/False)

(A) True.

(B) False.

(C) This is not possible with Bayes' Theorem.

(D) This ratio is equal to $[P(E|H_1) P(H_1)] / [P(E|H_2) P(H_2)]$.

Question 1. A Random Variable is a function that maps the outcomes of a random experiment to real numbers. Which of the following are types of random variables?

(A) Discrete Random Variable.

(B) Continuous Random Variable.

(C) Deterministic Variable.

(D) Qualitative Variable.

Question 2. A Discrete Random Variable can take values that are:

(A) Finite in number.

(B) Countably infinite.

(C) Integers (often, but not exclusively).

(D) Any value within an interval.

Question 3. A Continuous Random Variable can take values that are:

(A) Any real number within a given range or interval.

(B) Only integers.

(C) Infinite in number (uncountably infinite).

(D) Specific, distinct values.

Question 4. A Probability Distribution of a random variable describes:

(A) All possible values the random variable can take.

(B) The probability of the random variable taking each of these values (for discrete) or falling within a range (for continuous).

(C) The average value of the random variable.

(D) The spread of the random variable.

Question 5. For a valid Probability Distribution of a discrete random variable X, the following properties must hold for all possible values $x_i$:

(A) $P(X=x_i) \ge 0$.

(B) $P(X=x_i) \le 1$.

(C) $\sum_{\text{all } i} P(X=x_i) = 1$.

(D) The list of $x_i$ covers all possible outcomes.

Question 6. The 'Probability Mass Function' (PMF) is used to describe the probability distribution of a:

(A) Continuous random variable.

(B) Discrete random variable.

(C) Qualitative variable.

(D) Deterministic variable.

Question 7. The 'Probability Density Function' (PDF) is used to describe the probability distribution of a:

(A) Discrete random variable.

(B) Continuous random variable.

(C) For a continuous random variable X, $P(X=c) = 0$ for any specific value c.

(D) The area under the PDF curve between two points gives the probability.

Question 8. Which of the following are examples of scenarios that can be modeled using a discrete random variable?

(A) The number of heads in 5 coin tosses.

(B) The number of defective items in a sample of 10.

(C) The time taken to complete a task.

(D) The number of emails received in an hour.

Question 9. Which of the following are examples of scenarios that can be modeled using a continuous random variable?

(A) The height of a randomly selected person.

(B) The temperature recorded at noon.

(C) The number of accidents on a highway per day.

(D) The amount of rainfall in a city per year.

Question 10. The Cumulative Distribution Function (CDF), F(x), for any random variable (discrete or continuous) represents:

(A) The probability that the random variable is less than or equal to x, i.e., $P(X \le x)$.

(B) The probability that the random variable is exactly equal to x, i.e., $P(X = x)$.

(C) It is a non-decreasing function.

(D) It ranges from 0 to 1.

Question 1. The Expected Value $E(X)$ of a random variable X is:

(A) The theoretical mean of the distribution.

(B) The long-run average value if the experiment is repeated many times.

(C) Always an integer for a discrete random variable.

(D) Calculated as $\sum x_i P(X=x_i)$ for a discrete random variable.

Question 2. The Variance $Var(X)$ of a random variable X measures the dispersion or spread of the distribution. Which of the following are true about Variance?

(A) It is always non-negative.

(B) It is calculated as $E[(X - E(X))^2]$.

(C) It is calculated as $E(X^2) - [E(X)]^2$.

(D) Its units are the same as the units of X.

Question 3. The Standard Deviation of a random variable is:

(A) The square of the variance.

(B) The positive square root of the variance.

(C) Expressed in the same units as the random variable X.

(D) A measure of spread.

Question 4. If X is a random variable and 'a' and 'b' are constants, then which of the following properties of expectation are true?

(A) $E(aX) = a E(X)$.

(B) $E(X + b) = E(X) + b$.

(C) $E(aX + b) = a E(X) + b$.

(D) $E(a) = a$.

Question 5. If X is a random variable and 'a' and 'b' are constants, then which of the following properties of variance are true?

(A) $Var(aX) = a Var(X)$.

(B) $Var(aX) = a^2 Var(X)$.

(C) $Var(X + b) = Var(X)$.

(D) $Var(aX + b) = a^2 Var(X)$.

Question 6. If the probability distribution of X is:

(A) $\sum P(X=x) = 1$.

(B) $E(X) = (-1)(0.4) + (0)(0.2) + (1)(0.4) = -0.4 + 0 + 0.4 = 0$.

(C) $E(X^2) = (-1)^2(0.4) + (0)^2(0.2) + (1)^2(0.4) = 1(0.4) + 0(0.2) + 1(0.4) = 0.4 + 0 + 0.4 = 0.8$.

(D) $Var(X) = E(X^2) - [E(X)]^2 = 0.8 - 0^2 = 0.8$.

Question 7. If $E(X) = 5$ and $Var(X) = 2$, then:

(A) The standard deviation is $\sqrt{2}$.

(B) $E(2X) = 10$.

(C) $Var(2X) = 4$.

(D) $Var(X+1) = 3$.

Question 8. The expected value represents the center of gravity of the probability distribution. The variance represents the spread around this center. (True/False)

(A) True, for expected value.

(B) False, for expected value.

(C) True, for variance.

(D) False, for variance.

Question 9. If $Var(X) = 0$, it implies that:

(A) $E(X) = 0$.

(B) The random variable X is a constant value (i.e., $P(X=c) = 1$ for some constant c).

(C) There is no variability in the outcomes of the random variable.

(D) The standard deviation is also 0.

Question 10. For any two independent random variables X and Y, which properties are true?

(A) $E(X+Y) = E(X) + E(Y)$.

(B) $Var(X+Y) = Var(X) + Var(Y)$.

(C) $E(XY) = E(X)E(Y)$.

(D) $Var(X-Y) = Var(X) - Var(Y)$.

Question 1. A Binomial distribution arises from a sequence of Bernoulli trials. Which of the following are characteristics of Bernoulli trials and the resulting Binomial distribution?

(A) Each trial has only two possible outcomes (Success/Failure).

(B) The probability of success (p) is constant for each trial.

(C) The trials are independent.

(D) The number of trials (n) is fixed.

Question 2. The parameters of a Binomial distribution B($n$, $p$) are:

(A) $n$, the number of trials.

(B) $p$, the probability of success in a single trial.

(C) $\lambda$, the average rate.

(D) $k$, the number of successes.

Question 3. For a Binomial random variable X ~ B($n$, $p$), the probability of getting exactly k successes is given by the formula $P(X=k) = \binom{n}{k} p^k (1-p)^{n-k}$. In this formula:

(A) $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ is the number of ways to choose k successes from n trials.

(B) $p^k$ is the probability of getting k successes.

(C) $(1-p)^{n-k}$ is the probability of getting n-k failures.

(D) k can take any real value between 0 and n.

Question 4. The Mean and Variance of a Binomial distribution B($n$, $p$) are:

(A) Mean $= np$.

(B) Variance $= np$.

(C) Mean $= npq$, where $q = 1-p$.

(D) Variance $= npq$, where $q = 1-p$.

Question 5. The shape of a Binomial distribution is:

(A) Symmetric when $p = 0.5$.

(B) Positively skewed when $p < 0.5$.

(C) Negatively skewed when $p > 0.5$.

(D) Always symmetric.

Question 6. If X ~ B($n$, $p$), the possible values of X are:

(A) All integers from 0 to n.

(B) {0, 1, 2, ..., n}.

(C) Any real number.

(D) Only positive integers up to n.

Question 7. The Binomial distribution can be applied to scenarios such as:

(A) The number of defective items in a batch produced by a machine (assuming constant probability of defect).

(B) The number of students who pass an exam in a class, given the probability of passing for each student is independent and constant.

(C) The height of students in a class.

(D) The number of calls received by a call center in an hour.

Question 8. As the number of trials ($n$) in a Binomial distribution increases, the distribution tends to approach the shape of a:

(A) Poisson distribution (if $p$ is small).

(B) Normal distribution (if $n$ is large and $p$ is not extreme).

(C) Uniform distribution.

(D) Exponential distribution.

Question 9. If a Binomial distribution has Mean $= 5$ and Variance $= 3$, then $np = 5$ and $npq = 3$. This implies:

(A) $q = \text{Variance} / \text{Mean} = 3/5 = 0.6$.

(B) $p = 1 - q = 1 - 0.6 = 0.4$.

(C) $n = \text{Mean} / p = 5 / 0.4 = 12.5$.

(D) $n$ must be an integer, so these parameters might not correspond to a valid Binomial distribution.

Question 10. The Standard Deviation of a Binomial distribution is:

(A) $\sqrt{np}$.

(B) $\sqrt{npq}$.

(C) Always less than the variance.

(D) Always positive (unless $npq=0$).

Question 1. The Poisson distribution is a discrete probability distribution used to model the number of occurrences of events in a fixed interval of time or space. Key characteristics include:

(A) The events occur with a known constant mean rate.

(B) The occurrences are independent.

(C) The possible number of occurrences is infinite (non-negative integers).

(D) It has two parameters, $n$ and $p$.

Question 2. The parameter of the Poisson distribution is $\lambda$ (lambda), which represents:

(A) The probability of success.

(B) The number of trials.

(C) The average rate of occurrence of the event in the given interval.

(D) Both the Mean and the Variance of the distribution.

Question 3. For a Poisson distribution with parameter $\lambda$, which of the following are true?

(A) Mean $= \lambda$.

(B) Variance $= \lambda$.

(C) Standard Deviation $= \sqrt{\lambda}$.

(D) Mean $=$ Variance $= \lambda$.

Question 4. The probability mass function for a Poisson distribution is $P(X=k) = \frac{e^{-\lambda} \lambda^k}{k!}$. This formula gives the probability of observing exactly k events when the average rate is $\lambda$. The possible values for k are:

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

(B) Any real number.

(C) Integers from 0 to n.

(D) Only positive integers.

Question 5. The Poisson distribution can be used as an approximation for the Binomial distribution B($n$, $p$) when:

(A) $n$ is large.

(B) $p$ is small.

(C) $np$ is a moderate value (e.g., $np < 5$ or 10 is a common guideline).

(D) $np$ is very large.

Question 6. Examples of phenomena that can often be modeled using a Poisson distribution include:

(A) The number of accidents at a specific intersection in a month.

(B) The number of flaws in a length of fabric.

(C) The number of calls received by a helpline during a specific time interval.

(D) The height of individuals in a population.

Question 7. If the average number of emails received per day is 5, assuming a Poisson distribution, the parameter $\lambda$ for the number of emails received in two days would be:

(A) $5$.

(B) $10$ ($5 \times 2$).

(C) $\sqrt{5}$.

(D) $\lambda$ is proportional to the length of the interval.

Question 8. For small values of $\lambda$, the Poisson distribution is positively skewed. As $\lambda$ increases, the distribution becomes more symmetric and approaches the shape of a Normal distribution. (True/False)

(A) True for small $\lambda$ skewness.

(B) False for small $\lambda$ skewness.

(C) True for approaching Normal with large $\lambda$.

(D) False for approaching Normal with large $\lambda$.

Question 9. If the mean of a Poisson distribution is 4, then its variance is:

(A) 2.

(B) 4.

(C) 16.

(D) Equal to its mean.

Question 10. If the standard deviation of a Poisson distribution is 3, then its mean is:

(A) 3.

(B) $\sqrt{3}$.

(C) 9.

(D) The square of the standard deviation.

Question 1. The Normal distribution is a continuous probability distribution with a characteristic bell shape. Key properties include:

(A) It is symmetric about its mean.

(B) The Mean, Median, and Mode are all equal.

(C) The tails extend infinitely in both directions, approaching the x-axis but never touching it.

(D) It has a fixed range from -3 to +3.

Question 2. The Normal distribution is determined by two parameters: the Mean ($\mu$) and the Standard Deviation ($\sigma$). These parameters control:

(A) The location of the center of the distribution ($\mu$).

(B) The spread or variability of the distribution ($\sigma$).

(C) The shape of the distribution (skewness and kurtosis).

(D) The total area under the curve.

Question 3. The Standard Normal distribution is a special case of the Normal distribution with specific parameters. Which are they?

(A) Mean ($\mu$) $= 0$.

(B) Standard Deviation ($\sigma$) $= 1$.

(C) Variance ($\sigma^2$) $= 1$.

(D) Range = 1.

Question 4. A z-score (or standard score) standardizes a value from a Normal distribution by indicating how many standard deviations it is away from the mean. The formula for a z-score is $z = \frac{x - \mu}{\sigma}$. Using z-scores allows us to:

(A) Compare values from different Normal distributions.

(B) Use the Standard Normal distribution table (Z-table) to find probabilities.

(C) Change the shape of the distribution.

(D) Directly calculate the mean and standard deviation.

Question 5. The area under the Normal curve represents probability. Which of the following statements about areas under the Normal curve are true?

(A) The total area under the entire curve is 1.

(B) The area under the curve to the left of the mean is 0.5.

(C) The area under the curve to the right of the mean is 0.5.

(D) The area under the curve between two values represents the probability of the random variable falling between those values.

Question 6. The Empirical Rule (or 68-95-99.7 rule) for Normal distributions states that approximately:

(A) 68% of the data falls within 1 standard deviation of the mean.

(B) 95% of the data falls within 2 standard deviations of the mean.

(C) 99.7% of the data falls within 3 standard deviations of the mean.

(D) All data falls within 3 standard deviations of the mean.

Question 7. If a random variable X is normally distributed with mean $\mu$ and standard deviation $\sigma$, then the variable $Z = \frac{X - \mu}{\sigma}$ is normally distributed with:

(A) Mean 0.

(B) Standard deviation 0.

(C) Mean 1.

(D) Standard deviation 1.

Question 8. The Normal distribution is important in statistics because:

(A) Many natural phenomena follow approximately a normal distribution.

(B) The Central Limit Theorem implies that sample means are approximately normally distributed for large sample sizes.

(C) It simplifies many statistical calculations.

(D) It is the only continuous probability distribution.

Question 9. If a value 'x' has a positive z-score, it means 'x' is:

(A) Less than the mean.

(B) Greater than the mean.

(C) Exactly at the mean.

(D) Above the average value.

Question 10. The area under the Normal curve is interpreted as probability. For example, $P(a \le X \le b)$ for a normal variable X with mean $\mu$ and standard deviation $\sigma$ is found by:

(A) Calculating the z-scores for a and b: $z_a = (a - \mu)/\sigma$, $z_b = (b - \mu)/\sigma$.

(B) Finding the area between $z_a$ and $z_b$ under the Standard Normal curve using Z-tables.

(C) Finding the area to the left of $z_b$ and subtracting the area to the left of $z_a$.

(D) Summing the frequencies between a and b.

Question 1. In inferential statistics, we distinguish between a Population and a Sample. Which of the following statements are true?

(A) A population is the entire group of interest.

(B) A sample is a subset drawn from the population.

(C) We typically study the entire population due to practical limitations.

(D) The sample is used to make inferences about the population.

Question 2. A Parameter is a numerical characteristic of the population, while a Statistic is a numerical characteristic calculated from the sample. Which of the following represent population parameters?

(A) Population Mean ($\mu$).

(B) Population Variance ($\sigma^2$).

(C) Sample Mean ($\bar{x}$).

(D) Population Proportion (p).

Question 3. Which of the following represent sample statistics?

(A) Sample Mean ($\bar{x}$).

(B) Sample Standard Deviation (s).

(C) Population Standard Deviation ($\sigma$).

(D) Sample Proportion ($\hat{p}$).

Question 4. The primary goal of inferential statistics is to:

(A) Summarize and describe sample data.

(B) Estimate population parameters based on sample statistics.

(C) Test hypotheses about population parameters using sample data.

(D) Collect data from the entire population.

Question 5. Sampling is the process of selecting a sample from a population. Effective sampling techniques aim to ensure the sample is:

(A) Biased.

(B) Representative of the population.

(C) As large as the population.

(D) Randomly selected (in some methods like Simple Random Sampling).

Question 6. Sampling variability refers to the natural variation that occurs when different samples are drawn from the same population. This means:

(A) Different samples will likely have different values for a given statistic (e.g., sample mean).

(B) Sample statistics are perfect estimates of population parameters.

(C) This variability needs to be accounted for when making inferences.

(D) A larger sample size generally reduces sampling variability.

Question 7. Which of the following are valid reasons for studying a sample instead of the entire population?

(A) It is often too costly or time-consuming to study the entire population.

(B) The population might be infinite or inaccessible.

(C) Sampling can be destructive (e.g., quality testing).

(D) Sample statistics are always more accurate than population parameters.

Question 8. Examples of different sampling techniques include:

(A) Simple Random Sampling.

(B) Stratified Sampling.

(C) Convenience Sampling (often non-random).

(D) Cluster Sampling.

Question 9. If the population mean height of students in a university is $\mu$, and a researcher calculates the mean height $\bar{x}$ from a sample of students, then:

(A) $\mu$ is a parameter.

(B) $\bar{x}$ is a statistic.

(C) $\mu = \bar{x}$.

(D) $\bar{x}$ is an estimate of $\mu$.

Question 10. A well-defined population is crucial for accurate statistical inference. A population should be defined by:

(A) Its elements (who or what is being studied).

(B) Its geographical boundaries (where).

(C) A time period (when).

(D) The sample size.

Question 1. Statistical inference involves using sample data to draw conclusions about a population. It encompasses:

(A) Estimation of population parameters (e.g., confidence intervals).

(B) Hypothesis testing (testing claims about parameters).

(C) Descriptive statistics (summarizing sample data).

(D) Predicting future data points with certainty.

Question 2. Hypothesis testing is a formal procedure to evaluate a claim about a population parameter. The process typically involves:

(A) Stating the null and alternative hypotheses.

(B) Calculating sample statistics.

(C) Determining the level of significance ($\alpha$).

(D) Making a decision based on the p-value or critical value.

Question 3. The Null Hypothesis ($H_0$) and Alternative Hypothesis ($H_1$ or $H_a$) in hypothesis testing are:

(A) Statements about sample statistics.

(B) Statements about population parameters.

(C) Mutually exclusive.

(D) Collectively exhaustive (cover all possibilities).

Question 4. A Type I Error occurs when you __________ the null hypothesis when it is actually _________.

(A) Reject, true.

(B) Fail to reject, false.

(C) Reject, false.

(D) Fail to reject, true.

Question 5. A Type II Error occurs when you __________ the null hypothesis when it is actually _________.

(A) Reject, true.

(B) Fail to reject, false.

(C) Reject, false.

(D) Fail to reject, true.

Question 6. The Level of Significance ($\alpha$) is:

(A) The maximum acceptable probability of making a Type I Error.

(B) The probability of rejecting $H_0$ when $H_0$ is true.

(C) Usually set at 0.05 or 0.01.

(D) The probability of making a Type II Error.

Question 7. The P-value is a key component of hypothesis testing. It represents:

(A) The probability of observing the sample data (or more extreme data) if the null hypothesis ($H_0$) were true.

(B) The probability that the null hypothesis is true.

(C) The smallest level of significance at which the null hypothesis can be rejected.

(D) The probability of making a Type I Error.

Question 8. When comparing the p-value to the level of significance ($\alpha$) for decision making:

(A) If p-value $\le \alpha$, reject $H_0$.

(B) If p-value $> \alpha$, fail to reject $H_0$.

(C) If p-value is small, there is strong evidence against $H_0$.

(D) A large p-value indicates strong evidence in favor of $H_0$.

Question 9. Which factors influence the decision to reject or fail to reject the null hypothesis ($H_0$)?

(A) The test statistic calculated from the sample.

(B) The p-value of the test.

(C) The chosen level of significance ($\alpha$).

(D) The population parameter value (which is unknown).

Question 10. Rejecting the null hypothesis suggests that:

(A) The null hypothesis is definitely false.

(B) There is statistically significant evidence against the null hypothesis at the chosen alpha level.

(C) The observed sample data is unlikely to occur if the null hypothesis were true.

(D) The alternative hypothesis is true.

Question 1. The t-distribution is a probability distribution that is used in t-tests. Properties of the t-distribution include:

(A) It is symmetric and bell-shaped, similar to the Normal distribution.

(B) It has fatter tails than the Standard Normal distribution, especially for small degrees of freedom.

(C) Its shape depends on the degrees of freedom.

(D) As the degrees of freedom increase, it approaches the Standard Normal distribution.

Question 2. A one-sample t-test is appropriate when:

(A) You want to compare the mean of a single sample to a known or hypothesized population mean.

(B) The population standard deviation ($\sigma$) is unknown.

(C) The sample size is small (though it can be used for large samples too).

(D) The data is sampled from a population assumed to be approximately normally distributed.

Question 3. A two independent samples t-test is used to compare the means of two groups. Which of the following are required or assumed for this test?

(A) The two samples are independent of each other.

(B) The data in each group is sampled from a population assumed to be approximately normally distributed.

(C) The population variances of the two groups are known.

(D) The population variances ($\sigma^2$) of the two groups are assumed to be equal (in the pooled variance version).

Question 4. The degrees of freedom for a t-test are related to the sample size(s). Which of the following are correct?

(A) For a one-sample t-test with sample size $n$, $df = n-1$.

(B) For a two independent samples t-test with sizes $n_1, n_2$ (equal variances assumed), $df = n_1 + n_2 - 2$.

(C) Degrees of freedom determine the specific shape of the t-distribution used.

(D) Degrees of freedom are always equal to the sample size.

Question 5. A paired sample t-test is used when comparing means from dependent samples. This occurs when:

(A) The same subjects are measured under two different conditions (e.g., before and after a treatment).

(B) Pairs of subjects are matched based on similar characteristics and one subject from each pair is assigned to a different group.

(C) The samples are independent.

(D) We are comparing the means of three or more groups.

Question 6. When using a t-test, the p-value is calculated based on the calculated t-statistic and the appropriate t-distribution (determined by the degrees of freedom). If the p-value is less than the chosen level of significance ($\alpha$), we:

(A) Have statistically significant evidence against the null hypothesis.

(B) Reject the null hypothesis ($H_0$).

(C) Fail to reject the null hypothesis ($H_0$).

(D) Accept the alternative hypothesis (in practice, "reject the null" is preferred wording).

Question 7. If a t-test is conducted with $\alpha = 0.05$ and the calculated p-value is 0.028, which of the following conclusions are correct?

(A) Fail to reject $H_0$ because $0.028 < 0.05$.

(B) Reject $H_0$ because $0.028 < 0.05$.

(C) The result is statistically significant at the 5% level.

(D) The result is not statistically significant at the 1% level (since $0.028 > 0.01$).

Question 8. The t-test is considered robust to violations of the normality assumption, especially when:

(A) The sample size is large.

(B) The violations of normality are not severe (e.g., moderate skewness).

(C) The population standard deviation is known.

(D) Comparing two independent groups with equal variances.

Question 9. If the null hypothesis for a one-sample t-test is $H_0: \mu = 100$, which of the following are possible alternative hypotheses?

(A) $H_1: \mu \neq 100$ (Two-tailed test).

(B) $H_1: \mu > 100$ (One-tailed test).

(C) $H_1: \mu < 100$ (One-tailed test).

(D) $H_1: \bar{x} \neq 100$ (Incorrect, hypotheses are about population parameters).

Question 10. Compared to the Standard Normal distribution (Z), the t-distribution for small degrees of freedom:

(A) Has a higher peak.

(B) Has a lower peak.

(C) Has more area in the tails.

(D) Is less spread out.