Deck 16: Nonparametric Distribution-Free Statistical Methods

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.

Answer 1

False

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.

A) The test-statistic is 13, the critical value is 11. Therefore, he cannot reject the null hypothesis and conclude that the rating has remained unchanged.
B) The test-statistic is 11, the critical value is 13. Therefore, he can reject the null hypothesis and conclude that the rating has changed.
C) The test-statistic is 11, the critical value is 13. Therefore, he cannot reject the null hypothesis and conclude that the rating has remained unchanged.
D) The test-statistic is 13, the critical value is 11. Therefore, he can reject the null hypothesis and conclude that the rating has changed.

Answer 2

The test-statistic is 11, the critical value is 13. Therefore, he cannot reject the null hypothesis and conclude that the rating has remained unchanged.

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.

A) The test statistic is KW = 3.68, it is less than the critical value forα = 0.05 and therefore we fail to reject the null hypothesis that all clinics have the same scores.
B) The test statistic is KW = 8.19 , it is less than the critical value for α = 0.05 and therefore we fail to reject the null hypothesis that all clinics have the same scores.
C) The test statistic is KW = 3.68, it is more than the critical value for α = 0.05 and therefore we reject the null hypothesis that all clinics have the same scores.
D) The test statistic is KW = 8.19, it is more than the critical value for α = 0.05and therefore we reject the null hypothesis that all clinics have the same scores.

Answer 3

The test statistic is KW = 8.19, it is more than the critical value for α = 0.05and therefore we reject the null hypothesis that all clinics have the same scores.

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.

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.

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. _____________________ ​ ​

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.

A) H₀: μ_d = μ₁ - μ₂ = 0
B) H₀: μ_d = μ₁ - μ₂ > 0
C) H₀: μ_d = μ₁ - μ₂ < 0
D) H₀: μ_d = μ₁ - μ₂ ≠ 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?

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. ____________

Question 10

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

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.

A) The normal approximation of this study gives z = 1.34 and the probability that the price has not decreased is about P = 0.0901.
B) The normal approximation of this study gives z = 1.34 and the probability that the price has not decreased is about P = 0.9099.
C) The normal approximation of this study gives z = 0.05 and the probability that the price has not decreased is about P = 0.4801.
D) The normal approximation of this study gives z = 0.05 and the probability that the price has not decreased is about P = 0.5199.

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.

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.

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.

A) The test statistic of 9.87 is less than the critical value of 11.07; therefore, we do not reject the null hypothesis.
B) The test statistic of 9.87 is more than the critical value of 9.49; therefore, we reject the null hypothesis.
C) The test statistic of 9.87 is less than the critical value of 14.07; therefore, we do not reject the null hypothesis.
D) The test statistic of 9.87 is more than the critical value of 7.78; therefore, 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.

A) The k treatments are all drown from normal distributions with the same variance.
B) The k populations or treatment distributions all have a different shape and variability.
C) The k populations or treatment distributions all have the same shape and variability.
D) The k populations or treatment distributions all have a symmetric shape and different variability.