Question 1

If the sample size is large enough (n > 20), the distribution of the signed-rank sum statistic when H₀ is true is well approximated by the normal distribution with a mean of 0 and standard deviation of n(n+1)/4.

Accepted Answer

The distribution of the signed-rank sum statistic under the null hypothesis is approximated by a normal distribution with mean $ \frac{n(n+1)}{4} $ and standard deviation $ \sqrt{\frac{n(n+1)(2n+1)}{24}} $, not with a mean of 0 and standard deviation of $ \frac{n(n+1)}{4} $.

Question 2

Peter noticed that the nearest cafe, where he had previously been often, changed the signboard. He decided to find out how this place has changed using a rating from five popular sites. He took the rating of this place under the previous name and now and, after that, he conducted a signed-rank test. The result of this study is a signed-rank sum equal to 11. What can he say about a change in the rating of this place? Use α = 0.10.

Accepted Answer

The signed-rank test is a non-parametric test used for paired data when the data is not normally distributed. The null hypothesis for the signed-rank test is that there is no difference between the two groups. In this case, the two groups are the rating under the previous name and the rating under the current name.

The signed-rank sum is 11, and we need to compare it to the critical value at α = 0.10. Since there are only 5 ratings, we can use a table to find the critical value. The critical value for a sample size of 5 at α = 0.10 is 10. Therefore, the test-statistic (11) is greater than the critical value (10). This means that we cannot reject the null hypothesis and conclude that the rating has remained unchanged. So, the correct answer is C.

Question 3

A researcher wants to determine the difference in scores for three dentist clinics by the patients' responses. The scores (from 0 to 10) were taken from the official websites of each clinic for the study. These are the data of the scores.  Conduct the Kruskal-Wallis test and make a conclusion about the acceptance or rejection of the null hypothesis. Use α = 0.05.

Accepted Answer

D

Question 4

If l is not too small and H₀ is true F_r has approximately a chi-squared distribution with df = k - 1. The null hypothesis is rejected if F_r > chi-square critical value for the given level of significance α.

Accepted Answer

This statement is true according to the chi-square test for independence. If the null hypothesis is true (i.e. no association between two variables), then the test statistic has a chi-square distribution with degrees of freedom equal to (number of rows - 1) times (number of columns - 1). The critical value of chi-square depends on the chosen level of significance α and the degrees of freedom. If the calculated test statistic is greater than the critical value, then the null hypothesis is rejected in favor of the alternative hypothesis (i.e. there is evidence of a significant association between the variables).

Question 5

To conduct the Kruskal-Wallis test for comparing three different methods of treatment, one needs to state the null hypothesis as H₀: μ₁ = μ₂ = μ₃ and alternative hypothesis as Ha: μ₁ ≠ μ₂ ≠ μ₃.

Accepted Answer

The correct null hypothesis for the Kruskal-Wallis test is that the median (or mean rank) of all groups are equal, not specifically about the means (μ) being equal. The alternative hypothesis should suggest that at least one group's median (or mean rank) differs from the others, not necessarily that all group medians are different from each other.

Question 6

A sedentary lifestyle often contributes to a weight gain. An anonymous survey was conducted among the employees of a small office company asking for a change in the weight of a person during the year of working in the office. Data on those who agreed to respond to this survey are given below (in pounds).   Provide both the paired t-test and the signed-rank sum test for this study assuming the data have a normal distribution. Use α = 0.05. 
Find the test statistic and the critical value for the signed-rank sum test. 
Test statistic: ___________ 
Critical value: ___________ 
Make a conclusion for the signed-rank sum test. __________________ 
Find the P-value for the paired t-test. 
P-value: ___________ 
Make a conclusion for the paired t-test. __________________ 
Compare the results. _____________________ ​ ​

Accepted Answer

The answer of A sedentary lifestyle often contributes to a...

Question 7

There is a set of n pairs of observation. To conduct a signed-rank sum test, the null hypothesis should be stated. Choose the possible correct statement for testing hypotheses.

Accepted Answer

The null hypothesis for a signed-rank sum test typically states that there is no difference between the pairs of observations, which means the median difference is zero. This is correctly represented in option A, where the difference (μ1 - μ2) is stated to be equal to 0.

Question 8

A signed-rank sum test is conducted for paired samples. Usually, in such cases, two samples of the same parameter are studied for the same subjects. For example, a change in temperature in a specific patient is considered as a pair of observations. Why in these cases the rank-sum test cannot be provided?

Accepted Answer

The answer of A signed-rank sum test is conducted for...

Question 9

For the first testing of a new drug against hypercalcemia, a group of 12 patients was selected. The level of calcium in blood for each patient was measured before and after taking the treatment. The results of the study are presented in the table below (blood calcium level in mg/dL).  Use a level 0.05 signed-rank sum test to determine whether the treatment decreases the calcium level in the patients with hypercalcemia. The test statistic value is ____________ The critical value: ______________ Make a conclusion about rejection or acceptance of the null hypothesis. ____________

Accepted Answer

The answer of For the first testing of a new...

Question 10

What is the difference between conventional (parametric) statistics and non-parametric statistics?

Accepted Answer

The answer of What is the difference between conventional (parametric)...

Question 11

After the opening of a new store in a city district, experts believed that the average price of a number of marketable goods would decrease. To check if this is the case, a researcher collected data from 23 district shops. The researcher conducts a signed-rank sum test and obtains, that the signed-rank statistic is 88. Assuming that the sample is large enough, make a conclusion about the probability of a decrease in the price.

Accepted Answer

Since the sample size is greater than 20, we can use the normal approximation to the signed-rank test.

The signed-rank statistic is 88.

The corresponding z-score is:

z = (88 - 138)/15.492 = -3.2181

Looking at the standard normal table, the probability of getting a z-score of -3.2181 or lower is approximately 0.00065. Therefore, the probability of getting a z-score of 1.34 or higher is approximately:

P(z ≥ 1.34) ≈ 1 - 0.0901 ≈ 0.9099

So the best choice is A, which states that the probability of the price not decreasing is about 0.0901, which is the complement of the probability of getting a z-score of 1.34 or higher.

Question 12

In a signed-rank sum test for a set of n pairs of observations, the differences in pairs are calculated, these differences are ranked from smallest to largest, and the sum of ranks is calculated.

Accepted Answer

The answer of In a signed-rank sum test for a...

Question 13

The signed-rank sum test is a non-parametric analog of the paired t-test. Describe the difference between these two methods: when we cannot use the paired t-test and we are forced to apply the signed-rank test.

Accepted Answer

The answer of The signed-rank sum test is a non-parametric...

Question 14

In the Friedman's test for a randomized block design with k = 5 and l = 8, the test statistic is F_r = 9.87. Use a significance level of 0.05 to determine if the null hypothesis should be rejected or accepted.

Accepted Answer

The test statistic of 9.87 is more than the critical value of 9.49 for a significance level of 0.05, so we reject the null hypothesis.

Question 15

The Kruskal-Wallis test has less basic assumptions than the F-test for a completely randomize design. Determine the conditions under which we will use the Kruskal-Wallis test.

Accepted Answer

The Kruskal-Wallis test is a non-parametric method used when the k populations or treatment distributions have the same shape and variability, but not necessarily normal distributions. It is used when the assumptions of the F-test (normality and equal variances) are not met.

Quiz 16: Nonparametric Distribution-Free Statistical Methods

If the sample size is large enough (n > 20), the distribution of the signed-rank sum statistic when H0 is true is well approximated by the normal distribution with a mean of 0 and standard deviation of n(n+1)/4.

If l is not too small and H0 is true Fr has approximately a chi-squared distribution with df = k - 1. The null hypothesis is rejected if Fr > chi-square critical value for the given level of significance α.

To conduct the Kruskal-Wallis test for comparing three different methods of treatment, one needs to state the null hypothesis as H0: μ1 = μ2 = μ3 and alternative hypothesis as Ha: μ1 ≠ μ2 ≠ μ3.