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Question 1. Which measure of central tendency is most appropriate for a highly right‑skewed distribution of salaries? A) Mean B) Median C) Mode D) Harmonic mean Answer: B Explanation: In right‑skewed data the mean is pulled toward the long tail, while the median remains at the 50th percentile and better represents the typical value. Question 2. The interquartile range (IQR) is calculated as: A) Q3 – Q B) Q1 – Q C) Q2 – Q D) Q3 – Q Answer: A Explanation: IQR = third quartile (Q3) minus first quartile (Q1); it measures the spread of the middle 50 % of data. Question 3. A histogram with a symmetric bell shape most likely represents which distribution? A) Uniform B) Normal C) Exponential D) Bimodal Answer: B
Explanation: The normal distribution is symmetric and bell‑shaped; histograms of normally distributed data exhibit this pattern. Question 4. Which of the following statements about the standard deviation is true? A) It is always larger than the range. B) It is expressed in the same units as the original data. C) It can be negative. D) It is unaffected by outliers. Answer: B Explanation: Standard deviation is the square root of variance, keeping the original measurement units; it can be smaller or larger than the range and is sensitive to outliers. Question 5. In a box plot, the line inside the box represents: A) Mean B) Median C) Mode D) Minimum value Answer: B Explanation: The central line in a box plot marks the median (50th percentile) of the data. Question 6. Which chart is best for showing the proportion of market share among four companies? A) Histogram B) Scatter plot C) Pie chart D Line graph
D) Summarizes five‑number summary. Answer: B Explanation: Scatter plots display paired quantitative data points, allowing visual assessment of correlation. Question 10. When constructing a histogram, the choice of bin width primarily affects: A) The sample size. B) The shape of the distribution displayed. C) The mean of the data. D) The median of the data. Answer: B Explanation: Bin width determines how data are grouped; too wide hides detail, too narrow creates noise, altering the apparent shape. Question 11. Which of the following is NOT a measure of dispersion? A) Variance B) Standard deviation C) Mode D) Interquartile range Answer: C Explanation: Mode is a measure of central tendency, not variability. Question 12. A data set has a mean of 20 and a standard deviation of 0. Which statement is correct? A) All values equal 20. B) The data are normally distributed.
C) The data contain extreme outliers. D) The variance is 20. Answer: A Explanation: Zero standard deviation means no variation; every observation equals the mean. Question 13. Which distribution is characterized by the probability density function f(x)=λe^{-λx} for x ≥ 0? A) Normal B) Binomial C) Poisson D) Exponential Answer: D Explanation: The exponential distribution has that PDF, describing time between independent events. Question 14. The probability of event A occurring is 0.4 and of event B is 0.3. If A and B are mutually exclusive, P(A or B) equals: A) 0. B) 0. C) 0. D) 0. Answer: B Explanation: For mutually exclusive events, P(A∪B)=P(A)+P(B)=0.4+0.3=0.7. Question 15. If two events are independent, which of the following is always true? A) P(A and B)=P(A)+P(B)
Question 18. The probability of drawing an ace from a standard deck of 52 cards, then a king without replacement, is: A) (4/52)·(4/51) B) (4/52)+(4/51) C) (4/52)·(4/52) D) (4/52)·(48/51) Answer: A Explanation: First draw ace: 4/52. After removal, 51 cards remain, 4 kings left, so 4/51. Multiply for sequential independent draws without replacement. Question 19. In a normal distribution, approximately what percentage of observations lie within two standard deviations of the mean? A) 68 % B) 95 % C) 99.7 % D) 50 % Answer: B Explanation: The empirical rule states ~95 % of data fall within μ ± 2σ. Question 20. A Z‑score of – 1.5 indicates that the data point is: A) 1.5 units above the mean. B) 1.5 standard deviations below the mean. C) 1.5 standard deviations above the mean. D) Exactly at the mean. Answer: B
Explanation: Negative Z‑scores denote positions below the mean; magnitude equals number of standard deviations. Question 21. Which of the following is the correct formula for the variance of a sample? A) Σ(x_i – μ)² / N B) Σ(x_i – \bar{x})² / (n‑1) C) Σ(x_i)² / n D) (Σx_i)² / n Answer: B Explanation: Sample variance uses the squared deviations from the sample mean divided by (n‑1) for an unbiased estimator. Question 22. The Central Limit Theorem states that: A) The sample mean equals the population mean for any sample size. B) The distribution of sample means approaches normal as sample size increases, regardless of population shape. C) All populations are normally distributed. D) The variance of the sample mean equals the population variance. Answer: B Explanation: CLT guarantees that with sufficiently large n, the sampling distribution of \bar{x} is approximately normal. Question 23. Which sampling technique ensures each member of the population has an equal chance of selection? A) Convenience sampling B) Stratified sampling C) Simple random sampling
C) √[p(1‑p)/n] D) z* p(1‑p)/n Answer: A Explanation: Margin of error = critical value (z*) times the standard error of a proportion, √[p(1‑p)/n]. Question 27. Which of the following is a requirement for using a t‑test instead of a z‑test? A) Known population variance. B) Large sample size (n > 30). C) Unknown population variance and small sample size. D) Binary response variable. Answer: C Explanation: The t‑test is appropriate when σ is unknown and the sample size is small, using the sample standard deviation. Question 28. The Pearson correlation coefficient r = – 0.85 indicates: A) A strong positive linear relationship. B) A weak negative linear relationship. C) A strong negative linear relationship. D) No linear relationship. Answer: C Explanation: Correlation magnitude near 1 denotes a strong linear association; negative sign shows inverse direction. Question 29. Which of the following statements about the coefficient of determination (R²) is true? A) R² can be negative.
B) R² equals the square of the correlation coefficient for simple linear regression. C) R² measures the slope of the regression line. D) R² is always exactly 1 for any regression model. Answer: B Explanation: In simple linear regression, R² = r², representing the proportion of variance explained by the model. Question 30. In simple linear regression, the least‑squares slope (b) is calculated as: A) Σ(x_i – \bar{x}) · (y_i – \bar{y}) / Σ(x_i – \bar{x})² B) Σ(x_i – \bar{x}) / Σ(y_i – \bar{y}) C) Σy_i / Σx_i D) (\bar{y} – \bar{x}) / (n‑1) Answer: A Explanation: The slope formula is the covariance of X and Y divided by the variance of X. Question 31. Which residual diagnostic helps assess homoscedasticity in a regression model? A) Plot of residuals versus fitted values. B) Histogram of residuals. C) Q‑Q plot of residuals. D) Autocorrelation function of residuals. Answer: A Explanation: A residual‑versus‑fitted plot reveals patterns indicating non‑constant variance (heteroscedasticity). Question 32. If the p‑value for a hypothesis test is 0.03 and the significance level α is 0.05, the correct decision is:
Question 35. Which of the following best describes a Type II error? A) Rejecting a true null hypothesis. B) Failing to reject a false null hypothesis. C) Accepting the alternative hypothesis when it is false. D) Concluding significance when p > α. Answer: B Explanation: A Type II error occurs when we do not reject H₀ even though H₀ is false. Question 36. In a binomial experiment with n = 10 trials and probability of success p = 0.2, the probability of exactly 2 successes is: A) C(10,2)(0.2)²(0.8)⁸ B) (0.2)²(0.8)⁸ C) C(10,2)(0.2)⁸(0.8)² D) 0.2² + 0.8⁸ Answer: A Explanation: Binomial probability = C(n,k) p^k (1‑p)^{n‑k}. Question 37. The expected value of a discrete random variable X is defined as: A) Σx_i · P(X = x_i) B) Σx_i / n C) √(Σx_i² · P(X = x_i)) D) Median of X Answer: A Explanation: Expectation is the weighted average of possible values, using their probabilities.
Question 38. The variance of a Bernoulli(p) random variable is: A) p(1‑p) B) p² C) (1‑p)² D) √[p(1‑p)] Answer: A Explanation: For a Bernoulli trial, Var(X)=p(1‑p). Question 39. Which of the following best describes a Poisson process? A) Fixed number of trials with two outcomes. B) Continuous outcomes with normal distribution. C) Count of events occurring independently in a fixed interval. D) Proportion of successes in a sample. Answer: C Explanation: Poisson processes model the number of rare, independent events in a given time/space interval. Question 40. If two variables have a covariance of 0, what can be concluded? A) They are independent. B) They have no linear relationship. C) Their correlation is 1. D) Their means are equal. Answer: B Explanation: Zero covariance indicates no linear association, but variables may still be non‑linearly related or dependent.
Explanation: The F‑ratio compares variability due to the treatment (between groups) to residual variability (within groups). Question 44. Which post‑hoc test is commonly used after a significant one‑way ANOVA to control Type I error? A) Tukey’s HSD B) Pearson correlation C) Paired t‑test D) Wilcoxon signed‑rank test Answer: A Explanation: Tukey’s Honestly Significant Difference test adjusts for multiple comparisons while identifying which means differ. Question 45. A 90 % confidence interval for a proportion is (0.42, 0.58). Which of the following is true? A) The sample proportion is 0.5. B) The margin of error is 0.08. C) The interval will contain the true proportion 90 % of the time. D) The interval width is 0.16. Answer: D Explanation: Width = upper – lower = 0.58 – 0.42 = 0.16. Question 46. Which of the following best describes the law of large numbers? A) Sample means converge to the population mean as sample size increases. B) Sample variances equal population variance for any n. C) The distribution of sample means is always normal.
D) Probabilities become more uncertain with larger samples. Answer: A Explanation: The law of large numbers states that as n → ∞, the sample average approaches the expected value. Question 47. In a contingency table, the expected count for cell (i,j) under independence is calculated as: A) (Row total_i · Column total_j) / Grand total B) Row total_i + Column total_j C) (Row total_i · Column total_j) / n² D) Grand total / (Row total_i · Column total_j) Answer: A Explanation: Expected frequency under independence = (row total × column total) / overall total. Question 48. The chi‑square test for goodness‑of‑fit compares observed frequencies to: A) Expected frequencies from a hypothesized distribution. B) Frequencies of another variable. C) The sample mean. D) The median of observed data. Answer: A Explanation: Goodness‑of‑fit assesses how well observed counts match those expected under a specified theoretical distribution. Question 49. Which of the following is a non‑parametric alternative to the independent‑samples t‑test? A) Mann‑Whitney U test
A) It is approximately normal. B) It follows a t‑distribution. C) It is skewed regardless of n. D) It equals the binomial distribution. Answer: A Explanation: With sufficient sample size and both np and n(1‑p) ≥ 5, the sampling distribution of (\hat{p}) can be approximated by a normal distribution. Question 53. A data set has a mean of 50 and a standard deviation of 5. What is the Z‑score for a value of 60? A) 1 B) 2 C) 0. D) 10 Answer: B Explanation: Z = (60 – 50)/5 = 10/5 = 2. Question 54. Which of the following best describes a Type I error? A) Failing to reject a false null hypothesis. B) Rejecting a true null hypothesis. C) Accepting the alternative hypothesis when it is false. D) Concluding no effect when there is one. Answer: B Explanation: A Type I error occurs when we incorrectly reject a true H₀; its probability is the significance level α.
Question 55. In a simple linear regression output, the intercept term is statistically significant (p < 0.01). This indicates: A) The slope is zero. B) The predicted value of Y when X = 0 is reliably different from zero. C) The model explains all variability in Y. D) Multicollinearity is present. Answer: B Explanation: A significant intercept suggests that the estimated Y at X = 0 differs from zero in a statistically meaningful way. Question 56. Which of the following is the correct interpretation of a 99 % confidence interval? A) There is a 99 % chance that the interval contains the true parameter. B) 99 % of future observations will fall inside the interval. C) If we repeated the experiment many times, 99 % of the constructed intervals would capture the true parameter. D) The interval width is 99 % of the sample mean. Answer: C Explanation: Confidence intervals refer to the long‑run proportion of intervals that would contain the parameter. Question 57. In a regression model, the Durbin‑Watson statistic close to 2 suggests: A) Strong positive autocorrelation. B) Strong negative autocorrelation. C) No autocorrelation among residuals. D) Heteroscedasticity. Answer: C
