Question 1

In a Kruskal-Wallis test there are five samples and the value of the test statistic is calculated as H = 11.15. The most accurate statement that can be made about the p-value is that:

Accepted Answer

The Kruskal-Wallis test statistic H = 11.15 with five samples corresponds to a p-value that is greater than 0.01 but smaller than 0.025, indicating a significant difference among the samples.

Question 2

The appropriate measure of central location of ordinal data is the:

Accepted Answer

Ordinal data involves rankings or categories that have a natural order but do not have equal intervals between them. Therefore, the mean doesn't provide a meaningful measure of central location as it requires equal intervals. The mode may not be a representative measure as it can be influenced by a few highly occurring values. The median is the best choice as it identifies the middle value in the data set and is not affected by extreme values.

Question 3

In a Wilcoxon signed rank sum test, the test statistic is calculated as T = 45. The alternative hypothesis is stated as: The location of population 1 is different from the location of population 2. If there are n = 15 observations for which $D \neq 0$ , and the 5% significance level is used, then:

Accepted Answer

B

Question 4

In a Wilcoxon signed rank sum test for matched pairs with n = 35, the rank sums of the positive and negative differences are 380 and 225, respectively. The value of the standardised test statistic z is:

Accepted Answer

The formula for the standardised test statistic in a Wilcoxon signed rank sum test for matched pairs is:

z = (T - μT) / σT

where T is the smaller of the rank sum of the positive and negative differences, μT is the expected value of T under the null hypothesis (which is n(n+1)/4 = 35*36/4 = 315), and σT is the standard deviation of T under the null hypothesis (which is sqrt(n(n+1)(2n+1)/24) = sqrt(35*36*71/24) ≈ 18.46).

Plugging in the given values, we get:

T = 225
μT = 315
σT = 18.46

So:

z = (225 - 315) / 18.46 ≈ -4.871

Since this z-value is negative, we need to look at the cumulative distribution function of the standard normal distribution (which is the distribution of z under the null hypothesis) to find the p-value. The area to the left of -4.871 is essentially zero, so the p-value is very small. This means we can reject the null hypothesis at conventional levels of significance and conclude there is a significant difference between the positive and negative differences.

From the options given, the closest value of z is 1.065 (option B), which is the standardised test statistic for a one-tailed test with a p-value of about 0.143. Since we are not told the direction of the difference between the positive and negative differences, we cannot be sure whether this is the correct choice, but it is the best choice given.

Question 5

In a normal approximation to the Wilcoxon signed rank sum test, the test statistic is calculated as z = 1.59. For a two-tail test, the p-value is:

Accepted Answer

The p-value for a two-tailed test is found by doubling the area in one tail. For z = 1.59, the area in one tail is approximately 0.0559. Doubling this gives a p-value of approximately 0.1118.

Question 6

A non-parametric method to compare two populations, when the samples consist of matched pairs of observations and the data are either ordinal or interval, and where the normality requirement necessary to perform the parametric test is not satisfied, is the:

Accepted Answer

The Wilcoxon signed rank sum test is the appropriate non-parametric method to compare two populations when the samples consist of matched pairs of observations and the normality requirement necessary to perform the parametric test is not satisfied. The Wilcoxon rank sum test is appropriate for independent samples, while the Kruskal-Wallis test is appropriate for comparing more than two independent groups. The chi-squared test is appropriate for comparing categorical variables.

Question 7

In a Wilcoxon signed rank sum test, the test statistic is calculated as T = 75. If there are n = 15 observations for which $D \neq 0$ , and a two-tail test is performed at the 5% significance level, then:

Accepted Answer

The critical value for a two-tailed Wilcoxon signed rank sum test with n = 15 and α = 0.05 is 21. Since T = 75 is greater than the critical value of 21, we don't reject the null hypothesis. Therefore, we conclude that there is insufficient evidence to suggest that the median of the paired differences is different from zero. We don't need to perform a parametric test since the Wilcoxon signed rank sum test is a non-parametric test.

Question 8

The Kruskal-Wallis test statistic can be approximated by a Chi-squared distribution with k - 1 degrees of freedom (where k is the number of populations) whenever the sample sizes are all greater than or equal to:

Accepted Answer

The Kruskal-Wallis test can be approximated by a Chi-squared distribution with k - 1 degrees of freedom when the sample sizes are all greater than or equal to 5. Therefore, the best choice is D.

Question 9

A non-parametric method to compare two populations, when the samples are independent but the assumptions behind the independent samples t-test are violated, is the:

Accepted Answer

The Wilcoxon rank sum test is a non-parametric method that can be used when the assumptions behind the independent samples t-test are violated. This test does not require the assumption of normality or equality of variances, and instead ranks the observations in both samples and compares the medians. Therefore, it is a suitable choice for comparing two populations when the assumptions of the independent samples t-test are violated.

Question 10

Which of the following is the correct sample size requirement for the Wilcoxon signed rank sum test statistic to be approximately normally distributed?

Accepted Answer

For the Wilcoxon signed rank test, a sample size greater than 30 is generally considered sufficient for the test statistic to be approximately normally distributed due to the Central Limit Theorem.

Question 11

In a Friedman test for comparing four populations, provided that there are eight blocks, the test statistic is calculated as F = 10.98. If the test is conducted at the 5% significance level, the conclusion and p-value will be:

Accepted Answer

Since the test statistic F = 10.98 is greater than the critical value at the 5% significance level for a Friedman test with four treatments and eight blocks (which is 6.47), we reject the null hypothesis. The p-value can be calculated using a chi-squared distribution with three degrees of freedom (number of treatments minus 1), and the probability of observing a test statistic as extreme as 10.98 or greater is between 0.01 and 0.025. Therefore, the conclusion is to reject the null hypothesis and the p-value is between 0.01 and 0.025.

Question 12

Which of the following tests would be an example of a non-parametric method?

Accepted Answer

All three tests listed are examples of non-parametric methods. Non-parametric tests do not rely on assumptions about the underlying distribution of the data, and instead rely on rankings or other non-numerical data measures. The Wilcoxon signed rank sum test is used for comparing two related samples, the Kruskal-Wallis test is used for comparing three or more independent samples, and the Friedman test is used for comparing three or more related samples. Therefore, all of them can be considered as examples of non-parametric methods.

Question 13

In a Kruskal-Wallis test for comparing five populations, the test statistic is calculated as H = 10.20. If the test is conducted at the 1% significance level, then:

Accepted Answer

The answer of In a Kruskal-Wallis test for comparing five...

Question 14

Which of the following distributions approximates the Kruskal-Wallis test statistic H when the problem 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 approximately follows a chi-squared distribution with k - 1 degrees of freedom when the sample sizes are greater than or equal to 5. Therefore, option B is the best choice. The normal distribution is not appropriate for non-parametric tests like Kruskal-Wallis. The Student t-distribution is used for parametric tests and is not appropriate for the Kruskal-Wallis test, which is non-parametric. Option D is incorrect because the degrees of freedom for the chi-squared and t-distributions are not correct.

Question 15

Which of the following statements is correct regarding the Kruskal-Wallis test?

Accepted Answer

The Kruskal-Wallis test is a non-parametric test used to determine if there are differences in the medians of three or more groups. It does not assume that the data are normally distributed, but can be used with data that are either ordinal or numerical. Therefore, option C is the correct statement. Options A and B are incorrect as the Kruskal-Wallis test can be used with non-normal data. Option D is incorrect as option C is correct.

Question 16

A non-parametric method to compare two or more populations, when the samples are matched pairs and the data are either ordinal or interval but not normal, is the:

Accepted Answer

The answer of A non-parametric method to compare two or...

Question 17

A Wilcoxon rank sum test for comparing two populations involves two independent samples of sizes 15 and 20. The unstandardised test statistic (that is the rank sum) is T = 210. The value of the standardised test statistic z is:

Accepted Answer

The standardised test statistic z is calculated using the formula: z = (T - mean of T) / standard deviation of T. For the Wilcoxon rank sum test, the mean of T is n1(n1+n2+1)/2 and the standard deviation is sqrt(n1*n2(n1+n2+1)/12). Substituting n1=15, n2=20, and T=210 into these formulas gives z = -2.0.

Question 18

A non-parametric method to compare two populations, when the samples are matched pairs and the data are ordinal, is the:

Accepted Answer

The sign test is the best choice for comparing two populations with matched pairs and ordinal data. It is non-parametric and requires only the sign (positive or negative) of the difference between the pairs, making it a good choice when the data do not meet the assumptions of parametric tests. The Wilcoxon signed rank sum test and the Wilcoxon rank sum test are also non-parametric tests, but they are suited for comparing two independent populations rather than matched pairs. The matched pairs t-test, on the other hand, assumes normality of the data and is therefore not the best choice when dealing with ordinal data.

Question 19

Which of the following is the non-parametric equivalent to the parametric t-test of $\mu _ { D }$ for matched pairs?

Accepted Answer

The Wilcoxon signed rank sum test is the non-parametric equivalent to the parametric t-test for matched pairs, used when the data does not meet the assumptions necessary for a t-test.

Question 20

Which of the following is the alternative hypothesis tested in applications of the Kruskal-Wallis test?

Accepted Answer

The Kruskal-Wallis test is used to determine if there is a significant difference between the medians of two or more groups. The null hypothesis is that all groups have the same population median, while the alternative hypothesis is that at least two population locations differ. Therefore, the best choice is C.

Quiz 21: Nonparametric Techniques