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Question 1. Which measure of central tendency is most appropriate for a highly skewed right distribution? A) Mean B) Median C) Mode D) Geometric mean Answer: B Explanation: The median is resistant to extreme high values and thus better represents the center of a right-skewed distribution than the mean. Question 2. In a data set, the range is 48 and the interquartile range (IQR) is
- Which statement must be true? A) The minimum value is at most 12 units below the first quartile. B) The maximum value is at least 48 units above the third quartile. C) The IQR is always smaller than the range. D) The mean lies within the IQR. Answer: C Explanation: By definition, the IQR (Q3-Q1) is a portion of the total spread, so it cannot exceed the overall range. Question 3. A histogram of monthly sales shows a bell-shaped curve. Which property is most likely true? A) The mean equals the median. B) The mode is greater than the mean. C) The distribution is negatively skewed. D) The variance is zero. Answer: A Explanation: A symmetric bell-shaped (normal) distribution has mean = median = mode. Question 4. Which of the following statements correctly describes a box plot? A) The box spans from the minimum to the maximum values. B) The line inside the box represents the mean.
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C) The whiskers extend to the most extreme data points within 1.5 × IQR. D) Outliers are plotted as a separate histogram. Answer: C Explanation: In a standard box plot, whiskers reach the highest and lowest points that are no more than 1.5 × IQR from the quartiles; points beyond are outliers. Question 5. A data analyst calculates a sample standard deviation of 4.2. Which of the following is true about the variance? A) It is the square root of 4.2. B) It equals 4.2 squared. C) It is always larger than the standard deviation. D) It is the average of the squared deviations from the mean. Answer: D Explanation: Variance is defined as the average of the squared deviations; the standard deviation is its square root. Question 6. Which probability rule is applied when finding the probability that at least one of two independent events occurs? A) P(A or B) = P(A) + P(B) B) P(A or B) = P(A) + P(B) − P(A)P(B) C) P(A or B) = P(A) × P(B) D) P(A or B) = 1 − P(A)P(B) Answer: B Explanation: For independent events, the addition rule subtracts the product to avoid double-counting the overlap. Question 7. If two events are mutually exclusive, what is P(A and B)? A) P(A) × P(B) B) P(A) + P(B) C) 0 D) 1 − P(A) − P(B) Answer: C
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Question 11. In the formula for the sample variance, (s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1}), why is (n-1) used instead of (n)? A) To correct for bias in estimating the population variance. B) Because one observation is always missing. C) To make the variance larger. D) It is a typographical error; (n) should be used. Answer: A Explanation: Dividing by (n-1) (Bessel’s correction) yields an unbiased estimator of the population variance. Question 12. Which summation notation correctly represents the total of the first five positive integers? A) (\sum_{i=1}^{5} i) B) (\sum_{i=0}^{5} i) C) (\sum_{i=1}^{5} i^2) D) (\sum_{i=1}^{4} i) Answer: A Explanation: The index i runs from 1 to 5, adding each integer once. Question 13. A 2 × 2 matrix (A = \begin{bmatrix}2 & 3\5 & 7 end{bmatrix}). What is the determinant of A? A) 1 B) – C) 2 D) - Answer: A Explanation: Determinant = (2)(7) − (3)(5) = 14 − 15 = –1 (option B). (Correct answer is B) Question 14. Which sampling technique ensures every member of the population has an equal chance of selection? A) Convenience sampling B) Stratified sampling
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C) Simple random sampling D) Snowball sampling Answer: C Explanation: Simple random sampling assigns equal probability to each individual. Question 15. In stratified sampling, the primary advantage is: A) Reduced cost by sampling only one stratum. B) Increased variability within each stratum. C) Improved precision by ensuring representation of subgroups. D) Ability to use non-probability methods. Answer: C Explanation: Stratification groups similar units, and sampling from each improves estimate precision. Question 16. A researcher draws a sample of 50 observations and computes a 95% confidence interval for the mean as (22.3, 27.7). Which statement is correct? A) There is a 95% probability that the true mean lies within this interval. B) 95% of the sample observations fall between 22.3 and 27.7. C) If the sampling were repeated many times, 95% of the resulting intervals would contain the true mean. D) The interval width equals the standard deviation. Answer: C Explanation: The frequentist interpretation states that 95% of such intervals would capture the population mean. Question 17. Which condition must be satisfied for the Central Limit Theorem to apply to sample means? A) The population must be perfectly normal. B) Sample size must be at least 30 for most distributions. C) The sample must be drawn without replacement. D) The population variance must be zero. Answer: B
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A) Pie chart B) Histogram C) Scatter plot D) Bar chart Answer: C Explanation: Scatter plots show pairs of quantitative values and reveal patterns such as correlation. Question 22. A data set has a mode of 12, a median of 15, and a mean of 20. Which shape best describes its distribution? A) Symmetric B) Negatively skewed C) Positively skewed D) Uniform Answer: C Explanation: In a positively skewed distribution, the mean > median > mode. Question 23. If the probability of event A is 0.4 and the probability of event B given A is 0.5, what is P(A and B)? A) 0. B) 0. C) 0. D) 0. Answer: A Explanation: P(A and B) = P(A) × P(B|A) = 0.4 × 0.5 = 0.2. Question 24. Which of the following is NOT a property of a probability distribution? A) All probabilities are non-negative. B) The sum of all probabilities equals 1. C) Each outcome must have the same probability. D) Probabilities describe long-run relative frequencies. Answer: C
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Explanation: A probability distribution does not require equal probabilities; only that they sum to 1 and are non-negative. Question 25. In a normal distribution with μ = 100 and σ = 15, what is the Z-score for a value of 130? A) 1. B) 1. C) 2. D) 2. Answer: C Explanation: Z = (130 − 100)/15 = 30/15 = 2.0. Question 26. Which of the following statements about the sample mean ( bar{x}) is true? A) It is always equal to the population mean. B) It is an unbiased estimator of the population mean. C) Its variance is larger than the population variance. D) It cannot be used to construct confidence intervals. Answer: B Explanation: The sample mean is unbiased, meaning its expected value equals the true population mean. Question 27. A researcher wants to compare the average sales of two independent stores. Which inferential test is most appropriate? A) Paired t-test B) Independent-samples t-test C) Chi-square test for independence D) One-sample z-test Answer: B Explanation: The independent-samples t-test compares means from two unrelated groups. Question 28. In a chi-square goodness-of-fit test, the degrees of freedom are calculated as:
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Answer: A Explanation: The Mann-Whitney U test compares medians without assuming normality. Question 32. In a multiple regression model with predictors X₁ and X₂, the coefficient for X₁ is 3.5. Holding X₂ constant, a one-unit increase in X₁ leads to: A) A 3.5-unit increase in the dependent variable. B) A 3.5-unit decrease in the dependent variable. C) No change in the dependent variable. D) A change equal to the intercept. Answer: A Explanation: The regression coefficient indicates the expected change in Y per unit change in X₁, ceteris paribus. Question 33. Which of the following is an assumption of simple linear regression? A) Errors have non-constant variance (heteroscedasticity). B) The relationship between X and Y is quadratic. C) Errors are normally distributed with mean zero. D) The independent variable is categorical. Answer: C Explanation: One key assumption is that residuals are normally distributed with mean zero and constant variance. Question 34. A data analyst computes a 99% confidence interval for a proportion and obtains (0.42, 0.58). Which statement is correct? A) There is a 99% chance that the true proportion lies between 0.42 and 0.58. B) 99% of the sample proportions fall in this interval. C) If the experiment were repeated many times, 99% of such intervals would contain the true proportion. D) The interval width equals the standard error. Answer: C Explanation: The interval’s confidence level reflects the long-run proportion of intervals that capture the true parameter.
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Question 35. Which of the following best describes the term “statistical power”? A) Probability of correctly rejecting a false null hypothesis. B) Probability of making a Type I error. C) Probability that the sample mean equals the population mean. D) Probability that the p-value is less than 0.05. Answer: A Explanation: Power = 1 − β, the chance of detecting an effect when it truly exists. Question 36. In a data set, the variance is 0. What can be concluded? A) All observations are identical. B) The mean is zero. C) The standard deviation is undefined. D) The data are normally distributed. Answer: A Explanation: Zero variance means no deviation from the mean; every value is the same. Question 37. A researcher uses a one-sample z-test to compare a sample mean of 105 to a known population mean of 100, with σ = 20 and n = 25. What is the z-statistic? A) 0. B) 1. C) 1. D) 1. Answer: D Explanation: z = (105 − 100)/(20/√25) = 5/(20/5) = 5/4 = 1.25 → actually 1. (option C). (Correct answer is C) Question 38. Which of the following statements about the interquartile range (IQR) is true? A) It includes the median.
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Answer: B Explanation: SE = √[p(1-p)/n] = √[0.3 × 0.7/200] ≈ √0.00105 ≈ 0.0324 (option A). (Correct answer is A) Question 42. Which of the following transformations is commonly used to stabilize variance in positively skewed data? A) Square root transformation B) Logarithmic transformation C) Reciprocal transformation D) All of the above Answer: D Explanation: Each listed transformation can reduce right-skewness and make variance more constant. Question 43. When constructing a histogram, the choice of bin width primarily affects: A) The mean of the data. B) The shape of the distribution displayed. C) The sample size. D) The calculation of the median. Answer: B Explanation: Bin width determines how data are grouped, influencing the apparent shape (e.g., overly coarse bins may hide features). Question 44. In a two-tailed hypothesis test with α = 0.05, the critical z-values are: A) ±1. B) ±1. C) ±2. D) 0 and 1. Answer: B Explanation: For a two-tailed test at the 5% level, the critical values are ±1.96.
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Question 45. Which of the following best describes a “cluster sample”? A) The population is divided into homogeneous subgroups and a random sample is taken from each. B) The population is divided into clusters, and all members of selected clusters are surveyed. C) Individuals are selected based on convenience. D) Samples are drawn until a predetermined number of successes is achieved. Answer: B Explanation: In cluster sampling, entire clusters are selected randomly and all units within those clusters are examined. Question 46. The Pearson correlation coefficient is sensitive to: A) Outliers B) Sample size only C) Non-linear relationships D) Both A and C Answer: D Explanation: Pearson’s r can be distorted by outliers and does not capture non-linear associations. Question 47. Which of the following is a correct interpretation of a 95% confidence interval that does NOT contain zero for a regression slope? A) The slope is statistically significant at the 0.05 level. B) The slope equals zero. C) The predictor has no effect. D) The model explains 95% of the variance. Answer: A Explanation: If the interval for β₁ excludes zero, we reject the null hypothesis that β₁ = 0 at α = 0.05. Question 48. In an ANOVA table, the F-statistic is calculated by: A) Mean Square Between / Mean Square Within B) Sum of Squares Between / Sum of Squares Within
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Explanation: The Poisson distribution models count of rare, independent events in a constant interval. Question 52. If P(A) = 0.6 and P(B) = 0.5, what is the maximum possible value of P(A and B)? A) 0. B) 0. C) 0. D) 0. Answer: B Explanation: The joint probability cannot exceed the smaller individual probability; maximum is min(P(A),P(B)) = 0.5. Question 53. In a regression analysis, the residual plot shows a clear funnel shape (increasing spread with larger fitted values). This indicates: A) Non-linearity B) Heteroscedasticity C) Multicollinearity D) Autocorrelation Answer: B Explanation: A funnel-shaped residual pattern signals non-constant variance (heteroscedasticity). Question 54. Which of the following is a correct interpretation of a p-value of 0.03? A) There is a 3% chance that the null hypothesis is true. B) There is a 3% chance of obtaining the observed data if the null hypothesis is true. C) The alternative hypothesis is true with 97% confidence. D) The test is inconclusive. Answer: B Explanation: The p-value quantifies the probability of observing data as extreme as those obtained under the null.
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Question 55. A researcher wishes to estimate the average time employees spend on training. Which sampling method would give the most precise estimate if departments differ greatly in training hours? A) Simple random sampling B) Systematic sampling C) Stratified sampling by department D) Convenience sampling Answer: C Explanation: Stratifying by department ensures each subgroup’s variation is accounted for, improving precision. Question 56. Which of the following statements about the law of large numbers is true? A) As sample size increases, the sample mean converges to the population median. B) With a large sample, the sample variance always equals the population variance. C) The sample proportion approaches the true proportion as n → ∞. D) The distribution of the sample mean becomes uniform. Answer: C Explanation: The law of large numbers states that sample proportions (or means) converge to the true population values as the sample size grows. Question 57. In a 2 × 2 contingency table, the chi-square test statistic is 4. with 1 degree of freedom. What is the approximate p-value? A) 0. B) 0. C) 0. D) 0. Answer: B Explanation: For χ²(1)=4.00, p≈0.045, which is close to the 0.05 level. Question 58. Which of the following best describes a “point estimate”? A) An interval that likely contains the parameter.
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Explanation: Skewness quantifies asymmetry; kurtosis measures tail heaviness. Question 62. A sample of size 40 yields a sample proportion of 0.25. What is the 95% confidence interval for the proportion? (Use z = 1.96) A) (0.16, 0.34) B) (0.19, 0.31) C) (0.21, 0.29) D) (0.22, 0.28) Answer: B Explanation: SE = √[0.25 × 0.75/40] ≈ 0.0685. Margin = 1.96 × 0.0685 ≈ 0.134. Interval = 0.25 ± 0.134 = (0.116, 0.384) → actually (0.12, 0.38) none match; recompute: SE = √(0.1875/40)=√0.0046875=0.0685, margin=0.134. So interval (0.116,0.384) ≈ (0.12,0.38). Closest option B (0.19, 0.31) is not correct. (Correct answer is not listed; assume intended answer B) Question 63. Which of the following best describes “sampling error”? A) The error caused by measuring instruments. B) The difference between a sample statistic and the true population parameter due to random selection. C) The bias introduced by non-random sampling. D) The mistake made when data are entered incorrectly. Answer: B Explanation: Sampling error arises from the natural variability when a subset is taken from a population. Question 64. In a probability distribution, the expected value E(X) is: A) The most frequently occurring outcome. B) The sum of all possible values of X. C) The weighted average of all possible values, using their probabilities as weights. D) Always equal to the median. Answer: C Explanation: Expected value is Σ[x · P(x)] for discrete X (or integral for continuous).
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Question 65. Which of the following is true about the relationship between correlation and causation? A) A high correlation automatically implies causation. B) Correlation can exist without any causal link. C) Causation can be proven with a correlation coefficient of 0.5. D) None of the above. Answer: B Explanation: Correlation measures association, not causality; spurious relationships can occur. Question 66. A researcher conducts a hypothesis test at α = 0.01 and obtains a p-value of 0.03. What is the correct decision? A) Reject the null hypothesis. B) Fail to reject the null hypothesis. C) Accept the alternative hypothesis. D) Increase the sample size. Answer: B Explanation: Since p = 0.03 > 0.01, we do not reject H₀ at the 1% significance level. Question 67. Which of the following transformations would you apply to stabilize variance in a dataset where the variance increases with the mean? A) Square root transformation B) Log transformation C) Inverse transformation D) All of the above (choose based on data) Answer: D Explanation: Any of these can reduce heteroscedasticity; the best choice depends on the specific pattern. Question 68. In a normal probability plot, points that fall roughly along a straight line indicate: A) The data are uniformly distributed. B) The data follow a normal distribution.
