91Ó°ÊÓ

Refer again to Fig. 10.8 on page 427, where each pair of graphs shows the distributions of a variable on two populations. Suppose that, in each case, you want to perform a small-sample hypothesis test based on independent simple random samples to compare the means of the two populations. In each case, decide which, if any, of the following tests is preferable: the pooled $t$-test, the non pooled $t$-test, or the Mann-Whitney test. Explain your answers.

Tukey's Quick Test.

In this exercise, we examine an alternative method, conceived by the late Professor John Tukey, for performing a two-tailed hypothesis test for two population means based on independent random samples. To apply this procedure, one of the samples must contain the largest observation (high group) and the other sample must contain the smallest observation (low group). Here are the steps for performing Tukey's quick test.
Step I Count the number of observations in the high group that are greater than or equal to the largest observation in the low group. Count ties as $\frac{1}{2}$.

Step 2 Count the number of observations in the low group that are less than or equal to the smallest observation in the high group. Count ties as $\frac{1}{2}$.

Step 3 Add the two counts obtained in Steps 1 and 2, and denote the sum c.

Step 4 Reject the null hypothesis at the 5% significance level if and only if ³¦â‰¥7; reject it at the 1% significance level if and only if ³¦â‰¥10; and reject it at the0.1% significance level if and only
if³¦â‰¥13.
a. Can Tukey's quick test be applied to Exercise 10.48 on page 416? Explain your answer.
b. If your answer to part (a) was yes, apply Tukey's quick test and compare your result to that found in Exercise 10.48, where a t-test was used.
c. Can Tukey's quick test be applied to Exercise 10.74? Explain your answer.
d. If your answer to part (c) was yes, apply Tukey's quick test and compare your result to that found in Exercise 10.74, where a t-test was used.

State one possible advantage of using paired samples instead of independent samples.

State the two conditions required for performing a paired r-procedure. How important are those conditions?

Identify the assumption for using the two means $z$test and the two mean $z$interval procedure that renders those procedures generally impractical.

Measuring Treadwear. R. Stichler et al. compared two methods of measuring treadwear in their paper "Measurement of Treadwear of Commercial Tires" (Rubber Age, Vol. 73:2). Eleven tires were each measured for treadwear by two methods, one based on weight and the other on groove wear. The data, in thousands of miles, are as follows.

At the$5\%$ significance level, do the data provide sufficient evidence to conclude that, on average, the two measurement methods give different results?

Cooling Down. Cooling down with a cold drink before exercise in the heat is believed to help an athlete perform. Researcher 1. Dugas explored the difference between cooling down with an ice slurry (slushy) and with cold water in the article "lce Slurry Ingestion Increases Running Time in the Heat" (Clinical Journal of Sports Medicine, Vol. 21, No, 6, pp. 541-542). Ten male participants drank a flavored ice slurry and ran on a treadmill in a controlled hot and humid environment. Days later, the same participants drank cold water and ran on a treadmill in the same bot and humid environment. The following table shows the times, in minutes, it took to fatigue on the treadmill for both the ice slurry and the cold water.

At the $1\%$significance level, do the data provide sufficient evidence to conclude that, on average, cold water is less effective than ice slurry For optimizing athletic performance in the heat? (Note; The mean and standard deviation of the paired differences are $-5.9$minutes and $1.60$minutes, respectively.)

Gasoline Additive. This exercise shows what can happen when a hypothesis-testing procedure designed for use with independent samples is applied to perform a hypothesis test on a paired sample. The gas mileages, in miles per gallon (mpg), of 10 randomly selected cars, both with and without a new gasoline additive, are shown in the following table.

1. Apply the paired $t$-test to decide, at the $5\%$significance level, whether the gasoline additive is effective in increasing gas mileage.
2. Apply the pooled $t$-test to the sample data to perform the hypothesis test.
3. Why is performing the hypothesis test the way you did in part (b) inappropriate?
4. Compare your result in parts (a) and (b).

In Exercises 10.25-10.30, hypothesis tests are proposed. For each

hypothesis test,

a. identify the variable.

b. identify the two populations,

c. determine the null and alternative hypotheses.

d. classify the hypothesis test as two-tailed, left-tailed, or right-tailed.

Teaching Duties. Contingent faculty members in higher education are non-tenure track faculty, adjuncts, postdocs, lecturers, of instructors. R. Bowden and L. Gonzalez researched whether contingent faculty members are different from tenure-track faculty members with regards to teaching, research, and service in the article "The Rise of Contingent Faculty: Its Impact on the Professoriate and Higher Education" (Journal of Applied Research in Higher Education, Vol. 4, No. 1, pp. 5-22). A hypothesis test was conducted to decide whether the mean number of classes taught for credit per semester was less for contingent faculty than for tenure-track faculty.

In Exercises 10.25-10.30, hypothesis tests are proposed. For each

hypothesis test,

a. identify the variable.

b. identify the two populations,

c. determine the null and alternative hypotheses.

d. classify the hypothesis test as two-tailed, left-tailed, or right-tailed.

Wing Length. D. Cristol et al. published results of their studies of two subspecies of dark-eyed juncos in the paper "Migratory Dark-Eyed Juncos, Junco hyemalis, Have Better Spatial Memory and Denser Hippocampal Neurons Than Nonmigratory Conspecifics" (Animal Behaviour, Vol. 66, Issue 2, pp. 317-328). One of the subspecies migrates each year, and the other does not migrate. A hypothesis test is to be performed to decide whether the mean wing lengths for the two subspecies (migratory and nonmigratory) are different.