For the theory section, show your work and do not round off your middle work.
Keep your final answer to four (4) decimal places, where relevant.
1. A researcher wants to investigate whether there is any difference in the average number of hours worked by post-secondary students who are at least second-generation Canadians and those who are new immigrants. Seven college students who are second-generation Canadians (Group A), and five students who are new immigrants (Group B), were examined, and the average number of hours worked during summer months were recorded and are given below:
Group A 36 38 36 29 32 34 37
Group B 30 39 31 35 33
Since it was not clear whether the assumptions for a t-test were valid, the researcher decided to employ non-parametric methods. Use the Wilcoxon rank sum test to determine whether the average number of hours worked by Group A is significantly different from those of Group B. Use alpha = 0.05.
2. Suppose you want to use the Wilcoxon rank sum test to detect a shift in Distribution 1 to the right of Distribution 2, based on samples of sizes n1 = 7 and n2 = 9.
a. Should you use T1 or T1* as the test statistic? Why?
b. What is the rejection region for the test if alpha = 0.05?
c. What is the rejection region for the test if alpha = 0.01?
3. A paired-difference experiment was conducted to compare two populations. The data are shown in the table below. Use a sign test to determine whether the population distributions are different.
Population 1 2 3 4 5 6 7
1 9.6 8.1 9.7 8.4 11.1 8.8 7.6
2 9.5 8.8 10.0 8.5 10.6 9.0 8.1
a. State the null and alternative hypotheses for the test.
b. Calculate the observed value of the test statistic.
c. Calculate the exact p value using binomial tables.
d. Do the data present sufficient evidence to indicate that Populations 1 and 2 are different?
4. A dog owner was interested in determining whether there is a difference in the time it takes a dog to complete two different obstacle courses. A random sample of 36 dogs was selected, and the time it took each dog to complete each course was recorded. In twelve cases it took the dog longer to complete Course 1. Use the normal approximation to the sign test to determine if there is a significant difference in the time it takes to complete the two obstacle courses. Use alpha = 0.05.
5. Two aptitude tests are currently being used to screen applicants for a certain position within a company. The question arose as to whether the two tests are comparable; that is, whether they yield the same results. Six applicants were selected at random to take both tests (in a random order). The following scores were recorded:
Applicant Test A Test B
1 85 87
2 93 94
3 98 92
4 68 73
5 76 73
6 83 85
Use the Wilcoxon signed-rank test to determine whether there is a difference in scores between the two tests. Use alpha = 0.10.
6. Suppose you wish to detect a difference in the locations of two population distributions based on a paired-difference experiment consisting of n = 35 pairs.
a. Give the null and alternative hypotheses for the Wilcoxon Signed-rank test.
b. Give the test statistic.
c. Give the rejection region for the test for alpha = 0.05.
d. If T+ = 339, what are your conclusions?
Note: T+ + T- = n(n+1)/2
e. Conduct the test using the large-sample z test. Compare your results with the nonparametric test results above.
7. A computer laboratory manager was interested in whether there was a difference in functioning time for three different branded battery packs for laptop. The manager took a random sample of six battery packs of each brand and tested them. The results, in hours of functioning before the need for a recharge, were recorded as follows:
Brand 1 Brand 2 Brand 3
6.75 7.80 6.25
7.30 7.65 6.54
7.60 7.72 6.20
7.50 7.85 6.35
6.90 7.45 6.39
7.25 7.00 6.95
As the manager was unsure whether the assumptions for the usual parametric analysis of variance were valid, she decided to employ nonparametric methods. Use the appropriate nonparametric procedure to determine whether the distribution of functioning time before needing to be recharged is the same for the three brands of battery packs. Use alpha = 0.05.
8. The reaction times to three stimuli were recorded for each of eight subjects. The data, recorded in seconds, are shown below. This problem uses Friedman's Fr test to determine if there is a difference among the population distributions of reaction times.
Subject Stimulus I Stimulus II Stimulus III
1 3.5 4.8 5.3
2 4.7 6.2 5.9
3 6.1 5.2 5.8
4 2.8 4.5 5.7
5 4.3 4.2 5.8
6 2.6 4.3 3.7
7 5.0 4.4 5.1
8 3.2 3.9 4.6
a. What experimental design is being used in this problem?
b. State the null and alternative hypotheses.
c. Describe what the test statistic Fr is. What is the value of Fr in this problem.
d. Find the rejection region for alpha = 0.05.
e. Is there a difference among the population distributions of reaction times?
f. What is the approximate p-value for this problem?
Note: Give your answer as a range.
9. Two psychometricians were asked to rank six designs for a new standardized provincial math exam. The ranks are given below:
Educator 1 2 3 4 5 6
1 1 3 4 2 5 6
2 4 3 2 1 6 5
Use Spearman’s rank correlation coefficient to see if there is a (positive) relationship between the educators’ rankings.
a. State the null and alternative hypotheses.
b. Why is the test statistic rs called the rank correlation coefficient. What is the value of rs in this problem?
c. Find the rejection region for alpha = 0.05.
d. Do the data present sufficient evidence to indicate a positive correlation in the rankings of the two educators? What does this result mean in the context of the problem?
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