Unit 2 Section 2 Exercises

1. If two independent random variables, y₁ and y₂, are normally distributed with means and variances (m₁, s²₁) and (m₂,s²₂) respectively, the difference between the random variables has what distribution (be specific - give mean and variance).
2. The sampling distribution of the difference between two sample means has an approximate normal distribution in large samples with mean equal to what?
3. The standard error of the sampling distribution of the difference between two sample means has what value?
4. The following estimate for the standard error of the difference between two means is used when what assumption about the two population variances can be made?
5. What assumption about the two population variances is made when the following standard error estimate is used?
6. Most of the time in two-population tests, we test the hypothesis H₀: m₁-m₂ =0. What does it mean to test the hypothesis H₀: m₁-m₂ = D₀ where D₀ does not equal to zero?
7. Why is the Wilcoxon Rank Sum test use the words "Rank Sum" in its name?
8. Use Table 5 in the Appendix to find the critical value of the Wilcoxon rank sum test for independent samples when n₁=7 and n₂ = 6 and the alternative hypothesis is "Population 1 is shifted to the right of Population 2" with Type I error probability of a=0.05.
9. How is the critical value for the Wilcoxon rank sum test found if one or both of the sample sizes are greater than 10?
10. What do we mean when we talk about "Paired Data"?
11. Is a two-sample Paired Data t-test equivalent to a one-sample t-test performed on the differences in values for each sample unit?
12. Which of the two test is for testing the difference in means from samples of two independent populations? The Wilcoxon Rank Sum Test or the Wilcoxon Signed Rank Test? So what does the other test?
13. The equation for estimating sample sizes for the two-sided hypothesis test of differences of means is given by:. Can you define each of the terms in this equation?
14. Do problem 6.83 as a paired samples t-test and using the Wilcoxon Signed Rank test, both with a Type I error probability of 0.05. Do you get different results?
15. Again, using the scenario of problem 6.83 (page 334), how many water samples would we need if we wanted to be certain that the two Analysts did not differ by more than 2 ppm with Type I error probability of 0.05 and Power of 0.90 assuming the underlying variance in the differences were 1.0?

Review the Key Formulas on pages 317-318.

1. Do problem 6.75 on page 332.
2. Using the data from problem 6.75, perform the Wilcoxon rank sum test. Do the t-test and Wilcoxon test give different results?
3. What is the 95% confidence interval for the difference between the two means?
4. How many dairy cows would be needed for each group if we wanted to know the average difference between the two groups to within plus or minus .5 kg with 95% confidence (Hint see page 314). Use the pooled variance estimate from the t-test as if it were the true variance.

1. Do problem 6.37 on page 319.
2. Using the data from problem 6.37, perform the Wilcoxon rank sum test. Do the t-test and Wilcoxon test give different results?
3. What is the 95% confidence interval for the difference between the two means?
4. How many beams would be needed for each group if we wanted to estimate the difference between the average loading of the two groups to be within plus or minus 0.5 tons with 95% confidence (Hint see page 314).Use the pooled variance estimate from the t-test as if it were the true variance.

Bailer and Oris (1993, Env Tox and Chemistry, 12, pp787-91) report data from a study of toxic reproductive response in the aquatic organism Ceriodaphnia dubia to the herbicide nitrofen. A measure of reproductive stress in C. Dubia after exposure to the chemical is offspring counts from exposed females. A decrease in mean response in the exposure group suggests a toxic response to the chemical. Data from 20 independent animals were recorded as total number of offspring for three broods per animal.

Control                27 32 34 33 36 34 33 30 24 31
Nitrofen(160mg/l) 29 29 23 27 30 31 30 26 29 29
 

1. Perform a t-test to compare the means of these two groups.
2. Perform the Wilcoxon rank sum test to these data. Do the two test provide the same results?
3. What is the 95% confidence interval for the difference between the two means?
4. If we wanted to know the difference between the average offspring count to within plus or minus 1 with 95% confidence, how many females should we use for each group. Assume the pooled variance estimated for the t-test is the true variance.

Note that these are count data. Counts are typically not very normally distributed. Typically the analysis would be performed on transformed data, either the square root of the counts or the natural logarithm of the counts. Transformation of the response could change the results of the two-sample t-test (try if for fun and see how much). The Wilcoxon rank sum test results would not change. Do you know why?

1. Do problem 6.63 on page 327.
2. Using the data from problem 6.63, perform the Wilcoxon rank sum test. Do the t-test and Wilcoxon test give different results?
3. What is the 95% confidence interval for the difference between the two means?
4. How many candidates of each gender would be needed if we wanted to estimate the difference between the average expenditures of the two groups to be within plus or minus $5 with 95% confidence (Hint see page 314).Use the pooled variance estimate from the t-test as if it were the true variance.