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Question 1. Which of the following best defines a “distribution‑free” statistical method? A) It assumes normality of the data. B) It does not require any assumptions about the shape of the population distribution. C) It only works with categorical data. D) It requires the data to be measured on an interval scale. Answer: B Explanation: Distribution‑free (nonparametric) methods make no assumptions about the underlying population distribution, unlike parametric tests that often require normality. Question 2. Compared with a parametric test, a nonparametric test typically has: A) Higher statistical power for all sample sizes. B) Lower power when the parametric assumptions are met. C) The same power regardless of data distribution. D) Greater sensitivity to outliers. Answer: B Explanation: When parametric assumptions hold, parametric tests are more powerful; nonparametric tests sacrifice some power to gain robustness. Question 3. Which level of measurement is required for the Wilcoxon Signed‑Rank test? A) Nominal B) Ordinal or higher C) Ratio only D) Binary only Answer: B Explanation: The Wilcoxon Signed‑Rank test uses ranks of differences, so data must be at least ordinal.
Question 4. In hypothesis testing for a median, the null hypothesis $H_0$ is usually stated as: A) The population mean equals a specific value. B) The population median equals a specific value. C) The distribution is symmetric. D) The population variance is zero. Answer: B Explanation: Nonparametric median tests (e.g., Sign test) target the population median. Question 5. The significance level ($\alpha$) of 0.05 indicates: A) A 5 % chance of committing a Type II error. B) A 5 % chance of rejecting $H_0$ when it is true. C) A 95 % confidence that $H_0$ is false. D) The probability that the p‑value will be exactly 0.05. Answer: B Explanation: $\alpha$ is the probability of a Type I error—incorrectly rejecting a true null hypothesis. Question 6. Which of the following is the correct decision rule for a two‑tailed test with $\alpha = 0.05$? A) Reject $H_0$ if $p < 0.05$. B) Reject $H_0$ if $p > 0.95$. C) Accept $H_0$ if $p < 0.05$. D) Accept $H_0$ if $p > 0.05$. Answer: A Explanation: For any significance level, reject $H_0$ when the p‑value is less than $\alpha$.
Explanation: The test assumes that the distribution of differences between observations and the hypothesized median is symmetric around zero. Question 10. The test statistic $W$ in the Wilcoxon Signed‑Rank test is: A) The sum of positive ranks only. B) The sum of negative ranks only. C) The smaller of the sum of positive and negative ranks. D) The total number of observations. Answer: C Explanation: $W$ is defined as the smaller of the two rank sums, which simplifies the reference distribution. Question 11. A Runs test is primarily used to assess: A) Equality of medians between two groups. B) Randomness (independence) in a sequence of binary observations. C) Homogeneity of variances. D) Correlation between two continuous variables. Answer: B Explanation: The Runs test counts the number of runs (consecutive identical symbols) to test for randomness. Question 12. In a Runs test, a “run” is defined as: A) A series of increasing values. B) A consecutive sequence of identical symbols (e.g., + or – ). C) The difference between the largest and smallest observation. D) The sum of all observations. Answer: B
Explanation: A run consists of one or more identical symbols that are preceded and followed by a different symbol or by the start/end of the series. Question 13. The Mann‑Whitney U test is the nonparametric analogue of which parametric test? A) Paired t‑test B) Independent‑samples t‑test C) One‑sample t‑test D) Repeated‑measures ANOVA Answer: B Explanation: Mann‑Whitney compares two independent groups, analogous to the independent‑samples t‑test. Question 14. When calculating the Mann‑Whitney U statistic, the sum of ranks for group 1 is denoted $R_1$. The formula for $U_1$ is: A) $U_1 = n_1 n_2 + \frac{n_1 (n_1+1)}{2} - R_1$ B) $U_1 = R_1 - \frac{n_1 (n_1+1)}{2}$ C) $U_1 = n_1 n_2 - R_1$ D) $U_1 = \frac{R_1}{n_1}$ Answer: B Explanation: $U_1 = R_1 - n_1(n_1+1)/2$ subtracts the minimum possible rank sum for group 1. Question 15. For large samples in the Mann‑Whitney test, the sampling distribution of $U$ can be approximated by: A) t‑distribution B) Normal distribution with mean $n_1 n_2 /2$ and variance $n_1 n_2 (n_1+n_2+1)/12$ C) Chi‑square distribution with 1 degree of freedom
B) The numbers of discordant pairs in opposite directions. C) The total sample size and number of successes. D) The counts of missing data. Answer: B Explanation: $b$ and $c$ are the off‑diagonal cells representing opposite changes; only discordant pairs contribute to the statistic. Question 19. The Kruskal‑Wallis H test assumes that: A) All groups have the same shape and spread, differing only in medians. B) The data are normally distributed in each group. C) Observations are paired across groups. D) Each group contains the same number of observations. Answer: A Explanation: Kruskal‑Wallis tests for differences in central tendency under the assumption of similarly shaped distributions. Question 20. The test statistic $H$ for Kruskal‑Wallis is computed as $H = \frac{12}{N(N+1)}\sum\frac{R_i^2}{n_i} - 3(N+1)$. Which symbol represents $R_i$? A) The raw sum of observations in group $i$. B) The average rank of group $i$. C) The sum of ranks for group $i$. D) The number of ties in group $i$. Answer: C Explanation: $R_i$ is the total of the ranks assigned to observations in group $i$. Question 21. If the Kruskal‑Wallis test yields a significant result, the appropriate next step is:
A) Conclude that all group medians differ. B) Perform post‑hoc pairwise comparisons with adjusted significance levels. C. Transform the data to meet parametric assumptions. D. Accept the null hypothesis. Answer: B Explanation: A significant $H$ indicates at least one difference; post‑hoc tests (e.g., Dunn’s) identify which pairs differ while controlling Type I error. Question 22. The Friedman test is the nonparametric counterpart of which parametric test? A) One‑way ANOVA for independent groups. B) Repeated‑measures ANOVA. C) Two‑sample t‑test. D) Chi‑square test of independence. Answer: B Explanation: Friedman analyzes rankings across multiple related blocks (e.g., subjects measured under several conditions). Question 23. In the Friedman test, the test statistic $\chi^2_F$ is calculated using: A) The sum of squared rank differences between treatments. B) The total of treatment rank sums. C) The variance of raw scores. D) The number of ties only. Answer: B Explanation: $\chi^2_F = \frac{12}{k n(k+1)}\sum_{j=1}^{k} R_j^2 - 3n(k+1)$, where $R_j$ are the rank sums for each treatment.
Question 27. For a chi‑square test of independence with a $2 \times 3$ table, the degrees of freedom are: A) $(2-1)(3-1) = 2$ B) $2+3-1 = 4$ C) $(2)(3) = 6$ D) $(2-1)+(3-1) = 3$ Answer: A Explanation: Degrees of freedom = $(r-1)(c-1)$ for an $r \times c$ contingency table. Question 28. In a chi‑square test, the expected frequency for a cell is calculated as: A) Row total × Column total ÷ Grand total B) Row total + Column total C) Grand total ÷ Number of cells D) Row total ÷ Column total Answer: A Explanation: Expected count = (row total × column total) / overall total. Question 29. Spearman’s rank correlation coefficient $\rho$ is appropriate when: A) Both variables are measured on a nominal scale. B) Data are normally distributed and measured on an interval scale. C) Variables are ordinal or not normally distributed. D) One variable is binary and the other is continuous. Answer: C Explanation: $\rho$ uses ranks, making it suitable for ordinal data or non‑normal continuous data.
Question 30. The formula $\rho = 1 - \frac{6\sum d_i^2}{n(n^2-1)}$ requires that there be: A) No tied ranks. B) At least one tie. C) Only nominal data. D) Exactly three observations. Answer: A Explanation: The simple formula assumes no ties; ties require a correction factor. Question 31. Kendall’s Tau $\tau$ is based on: A) The difference between rank sums. B) The number of concordant and discordant pairs. C) The sum of squared deviations from the mean. D) The product of raw scores. Answer: B Explanation: $\tau = (C - D) / \frac{1}{2}n(n-1)$ where $C$ and $D$ are concordant and discordant pairs. Question 32. Which of the following statements about Kendall’s Tau is true? A) It is always larger in magnitude than Spearman’s $\rho$. B) It is less affected by ties than Spearman’s $\rho$. C) It can only be used for binary data. D) It requires the data to be normally distributed. Answer: B Explanation: Kendall’s Tau handles ties more gracefully; its calculation directly accounts for tied pairs. Question 33. The one‑sample Kolmogorov‑Smirnov (K‑S) test compares:
A) Increase the original sample size. B) Approximate the sampling distribution of a statistic by resampling with replacement. C) Convert ordinal data to interval data. D. Eliminate the need for hypothesis testing. Answer: B Explanation: Bootstrapping generates many resamples to estimate the distribution of a statistic without relying on parametric assumptions. Question 37. In a bootstrap procedure, the term “percentile interval” refers to: A) An interval based on the standard error of the bootstrap mean. B) An interval obtained by taking the 2.5th and 97.5th percentiles of the bootstrap distribution. C) The range between the minimum and maximum bootstrap estimates. D) The confidence interval assuming normality of the bootstrap estimates. Answer: B Explanation: The percentile method uses the empirical quantiles of the bootstrap replicates to form a confidence interval. Question 38. Which of the following software packages can directly produce a Mann‑Whitney U test output? A) SPSS B) Microsoft Word C) Adobe Photoshop D) AutoCAD Answer: A Explanation: SPSS, R, SAS, and similar statistical packages include procedures for the Mann‑Whitney test.
Question 39. In R, the function wilcox.test(x, y, paired = FALSE) performs which test? A) Mann‑Whitney U (Wilcoxon rank‑sum) test for independent samples. B) Wilcoxon Signed‑Rank test for paired samples. C) Kruskal‑Wallis test for multiple groups. D) Chi‑square test of independence. Answer: A Explanation: With paired = FALSE, wilcox.test implements the rank‑sum (Mann‑Whitney) test. Question 40. When interpreting a Kruskal‑Wallis output in SPSS, the “Asymptotic Sig.” value corresponds to: A) The exact p‑value based on permutations. B) The p‑value derived from the chi‑square approximation. C) The confidence interval for the median. D) The effect size. Answer: B Explanation: SPSS reports the asymptotic (large‑sample) chi‑square p‑value for the H statistic. Question 41. A nonparametric test is preferred over a parametric test when the data: A) Are normally distributed with equal variances. B) Contain extreme outliers that cannot be remedied by transformation. C) Have a large sample size (> 1000). D) Are measured on a ratio scale. Answer: B Explanation: Nonparametric methods are robust to outliers and non‑normality, making them suitable when such issues exist. Question 42. In the context of the Sign test, a “positive difference” means:
Question 45. If a researcher wishes to compare three related treatments measured on an ordinal scale, the most appropriate test is: A) Kruskal‑Wallis H test B) Friedman test C) One‑way ANOVA D) McNemar test Answer: B Explanation: Friedman handles repeated measures (related samples) with ordinal data. Question 46. In the Friedman test, the null hypothesis states that: A) All treatment medians are equal. B) All treatment rank sums are equal. C) The data are normally distributed. D) The variances of treatments are equal. Answer: B Explanation: Friedman tests whether the distributions of ranks across treatments are the same. Question 47. When performing a Wilcoxon Signed‑Rank test, zero differences are: A) Assigned the average rank of the tied values. B) Excluded from the analysis. C) Counted as positive differences. D. Counted as negative differences. Answer: B Explanation: Zero differences provide no information about direction and are omitted. Question 48. The “effect size” for a Mann‑Whitney U test can be estimated by:
A) $r = Z / \sqrt{N}$ where $Z$ is the standardized statistic and $N$ the total sample size. B) $\eta^2 = U / (n_1 n_2)$. C) $d = (M_1 - M_2) / SD_{pooled}$. D) $R^2$ from a regression model. Answer: A Explanation: $r = Z/\sqrt{N}$ provides a standardized effect size analogous to Pearson’s $r$. Question 49. In a $2 \times 2$ contingency table, the odds ratio is calculated as: A) $(a \times d) / (b \times c)$ where $a,b,c,d$ are cell frequencies. B) $(a + b) / (c + d)$. C) $(a - b) / (c - d)$. D) $(a + d) / (b + c)$. Answer: A Explanation: Odds ratio = (exposed & outcome)/(exposed & no outcome) divided by (unexposed & outcome)/(unexposed & no outcome). Question 50. Which of the following is a key assumption of the McNemar test? A) The two binary outcomes are independent. B) The discordant cells are sufficiently large (generally $b + c \ge 10$). C) The data are normally distributed. D) The sample sizes of the two groups are equal. Answer: B Explanation: The chi‑square approximation for McNemar works well when the sum of discordant counts is at least about 10. Question 51. When the data contain many tied ranks, the appropriate version of the Wilcoxon Signed‑Rank test:
B. Summing raw scores for each group. C. Computing the difference of means. D. Ranking all observations together. Answer: A Explanation: Mood’s Median test categorizes each observation as above or below the overall median and then applies chi‑square. Question 55. Which of the following statements about the Kolmogorov‑Smirnov test is FALSE? A) It is sensitive to differences in both location and shape of distributions. B) It requires the data to be continuous. C) It can be used for discrete distributions without modification. D. The test statistic is based on the maximum vertical distance between CDFs. Answer: C Explanation: The K‑S test assumes continuous data; with discrete data, the distribution of $D$ is altered and adjustments are needed. Question 56. When performing a bootstrap for the median, the resampling distribution will be: A) Exactly normal for any sample size. B) Skewed if the original data are skewed. C. Symmetric regardless of original data. D. Identical to the original sample distribution. Answer: B Explanation: Bootstrap replicates reflect the shape of the original data; a skewed original sample yields a skewed bootstrap distribution.
Question 57. In SPSS, the “Exact” option for the Mann‑Whitney test provides: A) Monte‑Carlo simulated p‑values. B) Exact p‑values based on the exact distribution of U. C. Asymptotic chi‑square p‑values. D. Confidence intervals for the mean. Answer: B Explanation: Selecting “Exact” computes the exact permutation distribution of U, useful for small samples. Question 58. The term “paired nominal data” is best illustrated by: A) Pre‑ and post‑treatment scores on a Likert scale. B) Blood type before and after a transfusion. C. Success/failure status of the same subject at two time points. D. Height measurements of two independent groups. Answer: C Explanation: Paired nominal data involve two categorical outcomes measured on the same subjects. Question 59. In the context of the Runs test for randomness, a significantly low number of runs suggests: A) The sequence is perfectly random. B) Positive autocorrelation (clustering). C) Negative autocorrelation (alternation). D. No conclusion can be drawn. Answer: B Explanation: Fewer runs than expected indicate clustering of similar symbols, violating randomness.
