Multiple choice questions in Probability and Statistics.pdf

Edited by: Achilleas Papatsimpas, Mathematician MSc
Multiple choice questions in Probability and Statistics
1. Which of the following is NOT a measure of central tendency?
A) Mean
B) Median
C) Mode
D) Standard Deviation
2. What type of chart is used to visualize categorical data?
A) Histogram
B) Scatter plot
C) Bar chart
D) Stem-and-leaf plot
3. In a right-skewed distribution, which of the following is true about the
relationship between the mean, median, and mode?
A) Mean > Median > Mode
B) Median > Mean > Mode
C) Mode > Mean > Median
D) Mean = Median = Mode
4. The standard deviation is a measure of:
A) Central tendency
B) Spread or variability
C) Correlation
D) Frequency distribution

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5. Which of the following distributions is used to model the number of
occurrences of an event in a fixed interval of time or space?
A) Binomial distribution
B) Poisson distribution
C) Hypergeometric distribution
D) Normal distribution
6. The probability of an event occurring, given that another event has already
occurred, is known as:
A) Joint probability
B) Marginal probability
C) Conditional probability
D) Independent probability
7. The Central Limit Theorem states that as the sample size increases, the
distribution of the sample mean:
A) Becomes more skewed
B) Approaches a uniform distribution
C) Approaches a normal distribution
D) Approaches a binomial distribution
8. Which statistical test is used to compare the means of two independent
populations?
A) Chi-square test
B) Paired t-test
C) Pooled-variance t-test
D) Z-test for proportions

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9. In hypothesis testing, the probability of rejecting a true null hypothesis is called:
A) Type I error
B) Type II error
C) Confidence level
D) Power of the test
10. A one-way ANOVA is used to compare the means of:
A) Two independent populations
B) Two related populations
C) More than two independent groups
D) Paired data sets
11. Which of the following is true for a normal distribution?
A) It is always skewed to the right.
B) The mean, median, and mode are all equal.
C) It has two modes.
D) It is always negatively skewed.
12. Which measure of variation is the difference between the first and third
quartiles?
A) Range
B) Interquartile range
C) Variance
D) Standard deviation

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13. What is the probability that a continuous random variable takes on an exact
value?
A) 1
B) Between 0 and 1
C) 0
D) Depends on the distribution
14. In hypothesis testing, the p-value represents:
A) The probability of the alternative hypothesis being true.
B) The probability of making a Type II error.
C) The probability of observing a test statistic as extreme as, or more extreme
than, the one calculated, assuming the null hypothesis is true.
D) The confidence level of the test.
15. Which of the following tests is used to analyze the relationship between two
categorical variables?
A) t-test
B) Z-test
C) Chi-square test of independence
D) ANOVA
16. The coefficient of determination (R²) in a regression analysis represents:
A) The slope of the regression line.
B) The proportion of the variance in the dependent variable that is predictable
from the independent variable(s).
C) The strength and direction of the linear relationship between two variables.
D) The standard error of the regression.

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17. Which of the following is NOT an assumption of the one-way ANOVA?
A) The populations have equal variances.
B) The populations are normally distributed.
C) The samples are dependent on each other.
D) The samples are randomly selected and independent.
18. In regression analysis, what does multicollinearity refer to?
A) A strong correlation between the dependent variable and the residuals.
B) High correlation between independent variables in a multiple regression model.
C) A model with more than two dependent variables.
D) When the regression line does not fit the data.
19. Which sampling method involves dividing the population into mutually exclusive
groups and randomly selecting individuals from each group?
A) Simple random sampling
B) Cluster sampling
C) Stratified sampling
D) Systematic sampling
20. The purpose of a confidence interval is to:
A) Test a hypothesis about the population parameter.
B) Estimate a population parameter with a specified level of confidence.
C) Determine if a result is statistically significant.
D) Measure the probability of a Type I error.

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21. If two variables have a high correlation coefficient (r = 0.9), this means:
A) One variable causes the other.
B) There is a strong linear relationship, but not necessarily causality.
C) The variables are independent of each other.
D) The variance of the two variables is equal.
22. In hypothesis testing, if the p-value is smaller than the significance level (α),
what decision should be made?
A) Fail to reject the null hypothesis.
B) Reject the null hypothesis.
C) Increase the sample size.
D) Decrease the significance level.
23. Which of the following is true regarding the Central Limit Theorem?
A) It applies only if the population distribution is normal.
B) It states that the sample mean will be normally distributed for any sample size.
C) It applies to both the sample mean and sample proportion.
D) It only applies to samples of size greater than 100.
24. Which of the following can lead to an inflated Type I error rate in multiple
hypothesis testing?
A) Using a smaller significance level.
B) Conducting multiple tests without adjustments.
C) Performing a two-tailed test instead of a one-tailed test.
D) Using a paired t-test instead of an independent t-test.

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25. Which of the following tests can be used to assess whether the variances of two
independent populations are equal?
A) Paired t-test
B) F-test
C) Chi-square test
D) Wilcoxon signed-rank test
26. A 95% confidence interval for the population mean is constructed as (14, 20).
Which of the following is correct?
A) There is a 95% probability that the population mean lies between 14 and 20.
B) 95% of the sample data lie between 14 and 20.
C) We are 95% confident that the population mean lies between 14 and 20.
D) 95% of the population data lie between 14 and 20.
27. What is the effect of increasing the sample size on the margin of error in a
confidence interval, assuming all other factors remain constant?
A) The margin of error increases.
B) The margin of error decreases.
C) The margin of error remains the same.
D) The sample mean shifts towards the population mean.
28. When conducting a hypothesis test for the slope in simple linear regression,
which of the following is the null hypothesis (H₀)?
A) The intercept is equal to zero.
B) The slope is equal to zero.
C) The correlation coefficient is zero.
D) The slope is not equal to zero.

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29. Which of the following is true about the adjusted R2 compared to R2 in multiple
regression?
A) Adjusted R2 increases when additional predictors are added, regardless of their
significance.
B) Adjusted R2 accounts for the number of predictors in the model and can
decrease when irrelevant predictors are added.
C) Adjusted R2 is always larger than R2.
D) Adjusted R2 is used only for simple linear regression.
30. Which of the following tests is used to determine whether residuals in a time
series model exhibit autocorrelation?
A) Chi-square test
B) Durbin-Watson test
C) F-test
D) T-test
31. A sample of 50 data points has a mean of 45 and a standard deviation of 8.
What is the 95% confidence interval for the population mean? (Assume the
population is normally distributed, and the sample standard deviation can be
used as an estimate for the population standard deviation.)
A) (42.77, 47.23)
B) (43.59, 46.41)
C) (41.76, 48.24)
D) (43.77, 46.23)

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32. A company claims that 60% of its customers are satisfied with their product. A
random sample of 200 customers reveals that 110 are satisfied. What is the z-
test statistic for testing the company’s claim?
A) -2.58
B) -1.94
C) -1.83
D) -2.24
33. A t-test is conducted on a sample of 25 students to test if the average test score
is different from 75. The sample mean is 72 with a standard deviation of 10. What
is the t-statistic?
A) -1.5
B) -2.5
C) -3.0
D) -4.5
34. A dataset follows a normal distribution with a mean of 100 and a standard
deviation of 15. What percentage of the data falls between 85 and 115?
A) 50%
B) 68%
C) 75%
D) 95%

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35. You perform a two-sample t-test comparing two independent samples. Group 1
has a mean of 78, a standard deviation of 10, and 30 samples, while Group 2 has
a meanof 84, a standard deviationof 12, and 35 samples. What is the test statistic
for the t-test assuming equal variances?
A) -2.10
B) -1.95
C) -2.47
D) -3.12
36. A linear regression equation is given by 𝒚
̂ = 𝟓 + 𝟐𝒙. If 𝒙 = 𝟏𝟎, what is the
predicted value of 𝒚
̂?
A) 15
B) 20
C) 25
D) 30
37. In a random sample of 200 individuals, 140 prefer product A. What is the 95%
confidence interval for the population proportion of individuals who prefer
product A?
A) (0.64, 0.76)
B) (0.65, 0.80)
C) (0.68, 0.82)
D) (0.70, 0.78)

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38. A dataset has a correlation coefficient of r=0.85 between two variables. What is
the coefficient of determination, and what does it represent?
A) 0.7225; 72.25% of the variation in the dependent variable is explained by the
independent variable.
B) 0.85; 85% of the variation in the dependent variable is explained by the
C) 0.15; 15% of the variation in the dependent variable is unexplained.
D) 0.85; 85% of the data points are linear.
39. If the p-value for a test statistic is 0.03 and the significance level α is 0.05, what
is the correct conclusion?
A) Fail to reject the null hypothesis.
B) Reject the null hypothesis.
C) Increase the sample size.
D) There is not enough evidence to make a conclusion.
40. In a Binomial distribution with n=10 and p=0.4, what is the probability of
observing exactly 4 successes?
A) 0.2508
B) 0.2114
C) 0.1216
D) 0.0331

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ANSWER KEY
1. D) Standard Deviation
2. C) Bar chart
3. A) Mean > Median > Mode
4. B) Spread or variability
5. B) Poisson distribution
6. C) Conditional probability
7. C) Approaches a normal distribution
8. C) Pooled-variance t-test
9. A) Type I error
10. C) More than two independent groups
11. B) The mean, median, and mode are all equal.
12. B) Interquartile range
13. C) 0
14. C) The probability of observing a test statistic as extreme as, or more extreme
than, the one calculated, assuming the null hypothesis is true.
15. C) Chi-square test of independence
16. B) The proportion of the variance in the dependent variable that is predictable
from the independent variable(s).
17. C) The samples are dependent on each other.
18. B) High correlation between independent variables in a multiple regression model.
19. C) Stratified sampling
20. B) Estimate a population parameter with a specified level of confidence.
21. B) There is a strong linear relationship, but not necessarily causality.
22. B) Reject the null hypothesis.

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23. C) It applies to both the sample mean and sample proportion.
24. B) Conducting multiple tests without adjustments.
25. B) F-test
26. C) We are 95% confident that the population mean lies between 14 and 20.
27. B) The margin of error decreases.
28. B) The slope is equal to zero.
29. B) Adjusted R2 accounts for the number of predictors in the model and can
decrease when irrelevant predictors are added.
30. B) Durbin-Watson test
31. A) (42.77, 47.23)
32. C) -1.83
33. B) -2.5
34. B) 68%
35. B) -1.95
36. C) 25
37. A) (0.64, 0.76)
38. A) 0.7225; 72.25% of the variation in the dependent variable is explained by the
39. B) Reject the null hypothesis.
40. A) 0.2508

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