Section 27.1

Exercise Set A: 2, 6

Section 27.2

Exercise Set B: 2, 4, 7, 8

Section 27.3

Exercise Set C: 2, 4

And:

1. In an experiment to determine the effect of a drug on the blood glucose concentration of diabetic rats, six rats were given the drug while five others were used as controls. The blood glucose levels of the treated group (in mg/ml) were 2.02, 1.71, 2.04, 1.50, 1.83, and 1.64; for the controls, the levels were 2.15, 1.92, 1.78, 2.04, and 2.22. Assuming these samples come from normal distributions with the same standard deviation, test the hypothesis that the treatment has no effect on mean blood glucose concentration.
2. The file http://dananne.org/fu/courses/math-241/R/body-temperatures.txt contains data from measuring the body temperatures of a random sample of adult males and females. The first column, labeled Temperature, contains the data and the second column, labeled Sex, identifies the sex as Male or Female for the given observation. Test the null hypothesis that the average body temperature is the same for both men and women. Is there any difference between performing a z test or t test in this case?
3. The file http://dananne.org/fu/courses/math-241/R/heart-rates.txt contains data from measuring the heart rates from a random sample of adult males and females. The first column, labeled Rate, contains the data and the second column, labeled Sex, identifies the sex as Male or Female for the given observation. Test the null hypothesis that the average heart rate is the same for both men and women. Is there any difference between performing a z test or t test in this case?

Answers:

1. Let $\mu_x$ be the true average blood glucose level for rats using the drug and let $\mu_y$ be the true average blood glucose level for rats not using the drug. We want to test $H_0 : \mu_x = \mu_y$ against $H_1 : \mu_x \ne \mu_y$. Using a two sample t-test, the test statistics is $t = 1.9294$, which, using a t distribution with $9$ degrees of freedom, gives a p-value of $0.08575$. So, although there is some evidence that the drug reduces blood glucose level, the evidence is not statistically significant.
2. Let $\mu_x$ be the true average body temperature of the males and $\mu_y$ be the true average body temperature of the females. We want to test $H_0 : \mu_x = \mu_y$ against $H_1 : \mu_x \ne \mu_y$. The test statistic is $2.2854$ using either a two-sample t-test or a two-sample z-test, with p-values of $0.02393$ and $0.02229$, respectively. Hence there is little difference between the tests, and there is strong evidence against the null hypothesis and for the conclusion that there is a difference between the average body temperatures.
3. Let $\mu_x$ be the true average heart rate of the males and $\mu_y$ be the true average heart rate of the females. We want to test $H_0 : \mu_x = \mu_y$ against $H_1 : \mu_x \ne \mu_y$. The test statistic is $0.6319$ using either a two-sample t-test or a two-sample z-test, with p-values of $0.5286$ and $0.0.5275$, respectively. Hence there is little difference between the tests, and there is no evidence against the null hypothesis, and hence no evidence that there is a difference in the average heart rates.