Question 1

How is the alternative hypothesis to be tested always stated in all applications of the Kruskal-Wallis test?

Accepted Answer

The Kruskal-Wallis test is used to compare three or more independent samples to determine if there is a difference between the population medians. The alternative hypothesis is always that at least two population locations differ. Therefore, the best choice is D.

Question 2

In which of the following situations might the Wilcoxon signed-rank test be more appropriate than the paired-difference t test?

Accepted Answer

The Wilcoxon signed-rank test is a nonparametric alternative to the paired-difference t test and does not assume normality of the population of differences. Therefore, it is more appropriate when dealing with non-normal data.

Question 3

In a Kruskal-Wallis test, there are four samples and the value of the test statistic is calculated as H = 8.79. Under these conditions, which of the following is the most accurate statement that can be made about the p-value?

Accepted Answer

To determine the p-value, we need to compare the calculated H value to the critical values of the chi-squared distribution with degrees of freedom equal to the number of samples minus one (k-1). For this problem, we have k = 4 samples, so degrees of freedom = 4-1 = 3. Using a chi-squared distribution table, we find that the critical value for a significance level of 0.05 and degrees of freedom = 3 is 7.815. Since our calculated H value (8.79) is greater than the critical value, we can reject the null hypothesis and conclude that there is a significant difference between at least two of the samples. The p-value for this test is the probability of obtaining an H value greater than or equal to 8.79, given that the null hypothesis is true. This probability is between 0.025 and 0.05, so the most accurate statement about the p-value is that it is greater than 0.025 but smaller than 0.05, which corresponds to answer choice D.

Question 4

Which of these tests is a nonparametric method to compare two or more populations when the samples are independent and the data are either ordinal or interval but not normal?

Accepted Answer

The Kruskal-Wallis test is a nonparametric method used to compare two or more populations when the samples are independent and the data are either ordinal or interval but not normal. The Friedman test is also a nonparametric test for dependent samples, and the Wilcoxon rank sum test and the Wilcoxon signed-rank sum test are used to compare two populations.

Question 5

Which of these tests is the nonparametric counterpart of the parametric t test of  for matched pairs?

Accepted Answer

The Wilcoxon signed-rank test is the nonparametric counterpart of the parametric t test for matched pairs. It is used to test the hypothesis that the median of the differences between paired observations is zero. The Friedman test is used to compare three or more matched samples, the Kruskal-Wallis test is used to compare three or more independent samples, and the Wilcoxon rank sum test is used to compare two independent samples.

Question 6

Which of the following is a nonparametric alternative to the Kruskal-Wallis test for differences in more than two medians?

Accepted Answer

The answer of Which of the following is a nonparametric...

Question 7

Which of the following is the nonparametric counterpart of the parametric one-way analysis of variance F-test?

Accepted Answer

The Kruskal-Wallis test is the nonparametric counterpart of the parametric one-way analysis of variance F-test. It is used to test whether there are differences in the medians of two or more groups. The Friedman test is used to compare three or more related samples, while the Wilcoxon rank sum and Wilcoxon signed-rank sum tests are used to compare two groups.

Question 8

In a Kruskal-Wallis test for comparing three populations, the test statistic is calculated as H = 2.80. If the test is conducted at the 5% significance level, then what may one conclude?

Accepted Answer

The Kruskal-Wallis test statistic H = 2.80 is compared to a chi-square distribution with degrees of freedom equal to the number of groups minus one. For three groups, the critical value at the 5% significance level is approximately 5.99. Since 2.80 is less than 5.99, the null hypothesis is not rejected.

Question 9

The significance level for a Wilcoxon signed-rank test is 0.05. The alternative hypothesis is stated as: The location of population 1 is to the left of the location of population 2. Under these circumstances, what is the appropriate critical value for a sample of size 20 (i.e., the number of non-zero differences)?

Accepted Answer

The answer of The significance level for a Wilcoxon signed-rank...

Question 10

In a Wilcoxon signed-rank test for matched pairs with n = 35, the rank sums of the positive and negative differences are 380 and 225, respectively. What, then, is the value of the standardized test statistic z?

Accepted Answer

The standardized test statistic z is calculated as (W+ - W-) / sqrt(n(n+1)(2n+1)/6), where W+ and W- are the rank sums of the positive and negative differences, respectively, and n is the sample size. In this case, z = (380 - 225) / sqrt(35(35+1)(2*35+1)/6) = 1.065.

Question 11

Which of the following distributions approximates the Kruskal-Wallis test statistic H when the objective is to compare k distributions and the sample sizes are greater than or equal to 5?

Accepted Answer

The Kruskal-Wallis test statistic H follows a chi-square distribution with k - 1 degrees of freedom when the sample sizes are greater than or equal to 5. Therefore, the best choice is B.

Question 12

Suppose there is interest in comparing the median response time for three independent groups learning a new specific task. In this case, what is the appropriate nonparametric procedure to use?

Accepted Answer

The Kruskal-Wallis test is the appropriate nonparametric test to use when comparing the medians of three or more independent groups. The Wilcoxon rank sum test (option A) and the Wilcoxon signed-rank test (option B) are appropriate for comparing the medians of two independent and paired samples, respectively. Therefore, the Kruskal-Wallis test (option C) is the best choice for comparing the medians of three independent groups.

Question 13

In a Wilcoxon signed-rank test for matched pairs with n = 32, the rank sums of the positive and negative differences are 367.5 and 160.5, respectively. Given this information, what is the value of the standardized test statistic z?

Accepted Answer

The formula for the standardized test statistic z in a Wilcoxon signed-rank test is:

z = (T - μT) / σT

Where:
T = smaller of the absolute values of the rank sums (in this case, T = 160.5)
μT = expected value of T (in this case, μT = n(n + 1) / 4 = 528)
σT = standard deviation of T (in this case, σT = √[n(n + 1)(2n + 1) / 24] = 44.386)

Plugging in the values, we get:

z = (160.5 - 528) / 44.386 = -9.967

Since the Wilcoxon signed-rank test is a two-tailed test, we take the absolute value of z and compare it to the critical value for alpha = 0.05/2 = 0.025 and degrees of freedom (df) = n - 1 = 31. Looking up the critical value in a standard normal distribution table or using a calculator, we get the critical value as 2.042. Since |z| > 2.042, we reject the null hypothesis and conclude that there is a significant difference between the two groups.

The closest option to our calculated z is B) 1.935.

Question 14

The significance level for a Wilcoxon signed-rank sum test is 0.05. The alternative hypothesis is stated as: The location of population 1 is different from the location of population 2. Given this information, what is the appropriate critical value for a sample of size 20 ?

Accepted Answer

The answer of The significance level for a Wilcoxon signed-rank...

Question 15

In a Wilcoxon signed-rank test, the test statistic is calculated as T = 75. If there are n = 15 observations for which the difference is not 0, and a two-tailed test is performed at the 5% significance level, what is an appropriate conclusion?

Accepted Answer

The critical value for a two-tailed Wilcoxon signed-rank test with n = 15 at the 5% significance level is typically higher than 75. Therefore, we do not reject the null hypothesis.

Question 16

Which of these tests is a nonparametric method to compare two populations when the samples are matched pairs and the data are interval, and where the normality requirement necessary to perform the parametric test is unsatisfied?

Accepted Answer

The Wilcoxon signed-rank test is a nonparametric method used to compare two populations when the samples are matched pairs and the data are interval, and where the normality requirement necessary to perform the parametric test is unsatisfied. The Wilcoxon rank sum test and the sign test are nonparametric tests, but they are used when the samples are independent rather than matched pairs. The matched pairs t test is a parametric test that is used when the normality assumption is satisfied. Therefore, the best choice in this scenario is the Wilcoxon signed-rank test.

Question 17

A paired sample, two-sample t test corresponds to which of the following tests?

Accepted Answer

A paired sample, two-sample t test is used to compare the means of two related samples, whereas a Wilcoxon signed-rank test is used to compare the medians of two related samples. Both tests are nonparametric and are appropriate when the data do not meet the assumptions of normality and equal variances for the t-test. Therefore, the Wilcoxon signed-rank test is the best choice.

Question 18

In a normal approximation to the Wilcoxon signed-rank test, the test statistic is calculated as z = 1.36. For a two-tailed test, what is the p-value?

Accepted Answer

The p-value for a two-tailed test can be found by doubling the probability of obtaining the observed test statistic or a more extreme value. Using a standard normal distribution table, we find that the probability of obtaining a z-score of 1.36 or higher is 0.0869. Doubling this gives us a p-value of 0.1738. Therefore, the correct answer is B.

Question 19

Which of the following is a characteristic of the Kruskal-Wallis test?

Accepted Answer

The answer of Which of the following is a characteristic...

Question 20

The Wilcoxon signed-rank test statistic is approximately normally distributed whenever the sample sizes are larger than or equal to which of the following values?

Accepted Answer

The Wilcoxon signed-rank test statistic is approximately normally distributed when the sample size is at least 25. This is due to the central limit theorem, which states that as the sample size increases, the distribution of sample means (or in this case, sample test statistics) approaches a normal distribution. Therefore, option C is the best choice as it satisfies the minimum sample size requirement for the Wilcoxon signed-rank test statistic to be approximately normally distributed.

Quiz 15: A:nonparametric Statistics