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Question 1. Which statement best describes the role of deductive reasoning in the scientific method for hypothesis testing? A) It generates new data from observations. B) It derives specific predictions from a general theory. C) It evaluates the probability of random error. D) It selects the most convenient statistical test. Answer: B Explanation: Deductive reasoning moves from a general hypothesis to specific, testable predictions that can be evaluated with data. Question 2. The null hypothesis (H₀) typically represents: A) The researcher’s preferred outcome. B) The presence of a meaningful effect. C) The status quo or “no effect” condition. D) The alternative explanation for the data. Answer: C Explanation: H₀ is formulated as “no effect” or “no difference,” serving as the baseline for testing. Question 3. A one-tailed alternative hypothesis (H₁) is appropriate when the researcher: A) Is interested in any difference, regardless of direction. B) Predicts the effect could be positive or negative. C) Has a specific directional expectation. D) Wants to increase statistical power without a directional claim. Answer: C Explanation: One-tailed tests allocate all α to one direction, matching a directional hypothesis. Question 4. In hypothesis testing, we “reject” the null hypothesis rather than “prove” it because: A) The null is always false. B) Statistical tests can only assess evidence against H₀.
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C) Proving H₀ would require infinite sample size. D) Both B and C are correct. Answer: D Explanation: Tests evaluate whether data provide sufficient evidence to reject H₀; proving H₀ would need certainty that no effect exists, which is unattainable. Question 5. Setting α = 0.01 instead of α = 0.05 will: A) Increase the probability of a Type I error. B) Decrease the probability of a Type I error. C) Increase the probability of a Type II error. D) Both B and C. Answer: D Explanation: A smaller α reduces Type I error risk but raises the chance of failing to detect a true effect (Type II error). Question 6. Which of the following is a consequence of committing a Type I error? A) Accepting a false null hypothesis. B) Rejecting a true null hypothesis. C) Failing to detect a real difference. D) Decreasing statistical power. Answer: B Explanation: Type I error occurs when we incorrectly reject a true H₀, producing a false positive. Question 7. Power (1 − β) of a test is most directly increased by: A) Reducing the sample size. B) Lowering the effect size. C) Raising the significance level α. D) Using a non-parametric test when assumptions are met. Answer: C
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A) Whether a variable follows a Poisson distribution. B) Equality of variances (homogeneity) across groups. C) Independence of two categorical variables. D) The linearity of a regression model. Answer: B Explanation: Levene’s test evaluates the null hypothesis that group variances are equal, a key ANOVA assumption. Question 12. When the assumption of equal variances is violated, the appropriate test for comparing two independent means is: A) Pooled-variance t-test. B) Welch’s t-test. C) Paired-samples t-test. D) One-sample Z-test. Answer: B Explanation: Welch’s t-test adjusts degrees of freedom and does not require homogeneity of variance. Question 13. Parametric tests differ from non-parametric tests primarily because parametric tests: A) Require categorical data. B) Assume a specific distribution (often normal). C) Are always more powerful. D) Do not need any assumptions. Answer: B Explanation: Parametric methods rely on distributional assumptions (e.g., normality) whereas non-parametric methods are distribution-free. Question 14. The standard error (SE) of the mean measures: A) The variability of individual observations. B) The spread of the sampling distribution of the mean. C) The average distance between data points and the median. D) The total variance in the population.
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Answer: B Explanation: SE = σ/√n (or s/√n) quantifies how much sample means would vary across repeated samples. Question 15. In a large sample (n > 30) with known population variance, the appropriate test for a hypothesis about a population mean is: A) One-sample t-test. B) One-sample Z-test. C) Paired-samples t-test. D) Chi-square goodness-of-fit test. Answer: B Explanation: When σ is known and n is large, the Z-test is appropriate for testing a mean. Question 16. The degrees of freedom for a one-sample t-test with n = 25 are: A) 24 B) 25 C) 23 D) 26 Answer: A Explanation: df = n − 1 for a one-sample t-test, so 25 − 1 = 24. Question 17. In an independent-samples t-test, the pooled variance estimator is used when: A) Sample sizes are unequal and variances differ. B) Sample sizes are equal but variances differ. C) Variances are assumed equal (homogeneity). D) The data are ordinal. Answer: C Explanation: Pooled variance combines the two sample variances under the assumption of equal population variances. Question 18. A paired-samples t-test is most appropriate for:
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Answer: B Explanation: A significant F suggests that the null hypothesis of equal population means is unlikely, implying some difference exists. Question 22. Which post-hoc procedure controls the family-wise error rate while comparing all pairwise group means? A) Tukey’s Honest Significant Difference (HSD). B) Pearson correlation. C) Levene’s test. D) Shapiro-Wilk test. Answer: A Explanation: Tukey HSD adjusts for multiple comparisons, maintaining the overall Type I error probability. Question 23. The Bonferroni correction adjusts the significance level by: A) Multiplying α by the number of comparisons. B) Dividing α by the number of comparisons. C) Adding α to each p-value. D) Subtracting α from each p-value. Answer: B Explanation: Bonferroni sets α_adj = α / k, where k is the number of tests, to keep the family-wise error rate at α. Question 24. The chi-square test of independence examines: A) Whether an observed frequency distribution matches an expected one. B) Whether two categorical variables are associated. C) The equality of two population means. D) The correlation between two continuous variables. Answer: B Explanation: Independence test evaluates if the joint distribution differs from what would be expected under independence.
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Question 25. In a chi-square goodness-of-fit test, the degrees of freedom are calculated as: A) (Number of categories − 1). B) (Number of categories − 2). C) (Number of observations − 1). D) (Number of observations − Number of categories). Answer: A Explanation: df = k − 1 for goodness-of-fit, where k is the number of mutually exclusive categories. Question 26. Pearson’s r measures: A) Linear association between two continuous variables. B) Rank-order association between ordinal variables. C) Difference between two proportions. D) Equality of variances across groups. Answer: A Explanation: Pearson correlation quantifies the strength and direction of a linear relationship. Question 27. Spearman’s rho is preferred over Pearson’s r when: A) Both variables are nominal. B) The relationship is non-linear but monotonic. C) Data are normally distributed. D) Sample size exceeds 1000. Answer: B Explanation: Spearman’s rho uses ranks and captures monotonic relationships without requiring linearity or normality. Question 28. In simple linear regression, the null hypothesis for the slope coefficient (β₁) is: A) H₀: β₁ = 0 B) H₀: β₁ ≠ 0 C) H₀: β₁ > 0
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Question 32. An 95 % confidence interval for a population mean that does not include the null value (e.g., 0) implies: A) The test is non-significant at α = 0.05. B) The result is statistically significant at α = 0.05. C) The sample size is too small. D) The data are non-normal. Answer: B Explanation: If the null value lies outside the CI, the corresponding two-tailed test would reject H₀ at the same confidence level. Question 33. Which of the following best reflects the ethical issue of “p-hacking”? A) Reporting all performed analyses regardless of outcome. B) Selecting only the statistically significant results after many unplanned tests. C) Using a larger α level to achieve significance. D) Conducting a power analysis before data collection. Answer: B Explanation: P-hacking involves data-driven manipulation (e.g., cherry-picking) to obtain significant p-values, inflating Type I error. Question 34. When reporting a t-test result, the conventional format includes: A) t, degrees of freedom, p-value, and effect size. B) t, sample size, confidence interval, and R². C) F, degrees of freedom, and η². D) χ², df, and standardized residuals. Answer: A Explanation: Standard reporting provides the test statistic, df, p-value, and often an effect-size measure like Cohen’s d. Question 35. In a two-sample proportion test, the pooled proportion is calculated because: A) It provides a more accurate estimate of the common proportion under H₀.
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B) It eliminates the need for a continuity correction. C) It increases the test’s power. D) It is only used when sample sizes are equal. Answer: A Explanation: Under H₀ the two population proportions are equal; pooling yields the best estimate of that common value. Question 36. The effect of increasing sample size on the standard error of the mean is to: A) Increase it linearly. B) Decrease it proportionally to 1/√n. C) Keep it constant. D) Increase it exponentially. Answer: B Explanation: SE = σ/√n; as n grows, SE shrinks at the rate of the square root of n. Question 37. Which test is appropriate for comparing the median of a single sample to a hypothesized value when data are heavily skewed? A) One-sample t-test. B) One-sample Z-test. C) One-sample Wilcoxon signed-rank test. D) Chi-square test. Answer: C Explanation: The Wilcoxon signed-rank test is a non-parametric alternative to the one-sample t when normality is violated. Question 38. For an ANOVA with unequal group sizes, which assumption is most critical to check? A) Normality of each group. B) Homogeneity of variances. C) Independence of observations. D) Linear relationship between variables.
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Question 42. Which of the following statements about Type II error (β) is true? A) It is the probability of rejecting a false null hypothesis. B) It is reduced by increasing α. C) It is unaffected by sample size. D) It can be directly observed from the data. Answer: B Explanation: Raising α makes it easier to reject H₀, thereby decreasing β (increasing power). Question 43. When the sample size is small (n < 30) and the population variance is unknown, the appropriate test for a population mean is: A) Z-test. B) One-sample t-test. C) Chi-square test. D) Fisher’s exact test. Answer: B Explanation: The t-test accounts for extra uncertainty from estimating σ with s when n is small. Question 44. The term “degrees of freedom” in the context of a t-test refers to: A) The number of independent pieces of information used to estimate variance. B. The number of groups being compared. C. The number of observations in the sample. D. The number of parameters estimated in the model. Answer: A Explanation: df = n − 1 (one-sample) reflects how many values are free to vary after estimating the mean. Question 45. In a 2 × 3 contingency table, the degrees of freedom for the chi-square test of independence are: A) (2 − 1) × (3 − 1) = 2 B) (2 + 3) − 2 = 3
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C) (2 × 3) − 1 = 5
D) (2 − 1) + (3 − 1) = 3
Answer: A Explanation: df = (rows − 1)(columns − 1) = 1 × 2 = 2. Question 46. The null hypothesis for a chi-square goodness-of-fit test with three categories is that: A) All observed frequencies are equal. B) Observed frequencies match the expected proportions. C) The categories are independent. D) The sample size is large enough. Answer: B Explanation: Goodness-of-fit compares observed counts to expected counts based on a hypothesized distribution. Question 47. Which of the following is NOT a requirement for the validity of a one-sample t-test? A) Random sampling. B) Normality of the population or large n. C) Known population variance. D) Independence of observations. Answer: C Explanation: The t-test is used precisely when σ is unknown; known σ would lead to a Z-test. Question 48. The term “family-wise error rate” (FWER) refers to: A) The probability of making at least one Type I error across a set of simultaneous tests. B) The average Type II error across tests. C) The overall power of a study. D) The proportion of significant results in a meta-analysis. Answer: A
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Question 52. The term “effect size” for ANOVA is commonly expressed as: A) R². B) η² (eta-squared) or ω² (omega-squared). C) d. D) t. Answer: B Explanation: η² and ω² quantify the proportion of total variance accounted for by the factor in ANOVA. Question 53. If a 99 % confidence interval for a mean difference is (-2.1, 0.3), what can be concluded at α = 0.01? A) The difference is statistically significant. B) The difference is not statistically significant. C) The test is inconclusive because the interval includes zero. D) The interval suggests a large effect. Answer: B Explanation: Since zero lies within the CI, we fail to reject H₀ at the 1 % level. Question 54. Which of the following best describes “sampling distribution of the proportion”? A) Distribution of individual binary outcomes. B) Distribution of sample proportions across repeated samples. C) Distribution of the population proportion. D) Distribution of the difference between two proportions. Answer: B Explanation: It describes how the proportion estimator varies from sample to sample. Question 55. The “rule of thumb” for the chi-square test of independence regarding expected cell frequencies is: A) All expected counts must be > 5. B) No more than 20 % of expected counts can be < 5, and none < 1. C) Expected counts must equal observed counts.
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D) Expected counts must be normally distributed. Answer: B Explanation: This guideline ensures the chi-square approximation is reliable. Question 56. When performing a two-sample t-test with unequal variances, the degrees of freedom are calculated using: A) The smaller of (n₁ − 1) and (n₂ − 1). B) The Welch-Satterthwaite equation. C) The total sample size minus two. D) The pooled variance estimator. Answer: B Explanation: Welch’s method provides an approximate df that accounts for heteroscedasticity. Question 57. In a study with three treatment groups, the researcher wants to control the Type I error across all pairwise comparisons. Which method is most efficient? A) Bonferroni correction. B) Tukey HSD. C) Fisher’s LSD. D) Scheffé’s method. Answer: B Explanation: Tukey HSD is designed for all pairwise comparisons while maintaining the family-wise error rate, often more powerful than Bonferroni for equal-size groups. Question 58. The null hypothesis for testing the equality of variances across k groups is: A) All group means are equal. B) All group variances are equal. C) All group medians are equal. D) All group sample sizes are equal. Answer: B
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Question 62. The “effect size” for a chi-square test of independence is often reported as: A) Cohen’s d. B) Cramér’s V. C) η². D) r. Answer: B Explanation: Cramér’s V standardizes the chi-square statistic to a 0-1 scale, indicating effect magnitude. Question 63. A researcher conducts a two-tailed test at α = 0.05 and obtains a p-value of exactly 0.05. The correct interpretation is: A) Reject H₀ because p ≤ α. B) Fail to reject H₀ because p ≥ α. C) The result is ambiguous; additional data are needed. D) The test is invalid because p cannot equal α. Answer: A Explanation: When p equals the significance level, the null is conventionally rejected (though some fields adopt a stricter rule). Question 64. In a regression model with multiple predictors, multicollinearity can be detected by: A) High R². B) Large residuals. C) Variance Inflation Factor (VIF) values > 10. D) Low standard errors. Answer: C Explanation: VIF quantifies how much variance of a coefficient is inflated due to correlation with other predictors. Question 65. The “standard error of the estimate” in simple linear regression refers to: A) SE of the slope. B) SE of the intercept.
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C) The root mean square error of predictions (σgₑ). D) The standard deviation of the predictor variable. Answer: C Explanation: It estimates the typical distance between observed values and the regression line. Question 66. Which of the following is a correct statement about the relationship between α, β, and power? A) Power = α + β. B) Power = 1 − β. C) Power = 1 − α. D) Power = α × β. Answer: B Explanation: Power is the probability of correctly rejecting a false null, equal to 1 minus the Type II error rate. Question 67. When the sample size is large, the sampling distribution of a proportion can be approximated by: A) t-distribution. B) Normal distribution. C) Chi-square distribution. D) Exponential distribution. Answer: B Explanation: By the CLT, the proportion’s sampling distribution approaches normality when np and n(1-p) are both ≥ 5. Question 68. The “degrees of freedom” for the residuals in a simple linear regression with n = 30 observations is: A) 28 B) 29 C) 30 D) 31 Answer: A
