Math 207—Large-Sample Significance Test of a Population Mean

Example 1 (Motivating Example)

For the general population of women between the ages 30 and 34, the mean diastolic blood pressure is 74.4 mm Hg. A researcher believes that diabetic women between the ages 30 and 34 have a higher mean diastolic blood pressure than the general population.

1. Let be the mean diastolic blood pressure for the diabetic women. List the null and alternative hypotheses.

2. Suppose the researcher suspected that diabetic women simply have a different mean blood pressure (not necessarily a higher mean blood pressure). List the null and alternative hypotheses.

3. The researcher takes a random sample of 30 diabetic women between the ages 30 and 34 and determines their blood pressures. The sample mean diastolic blood pressure is 79.1 mm Hg (for now, make the unrealistic assumption that we know the population standard deviation of the blood pressures of diabetic women is 9.8 mm Hg). We want to test the researcher’s original hypothesis (listed in part a).

Assuming that the mean for diabetic women is the same as the mean for the general population, what is the probability of observing a sample mean of 79.1 mm Hg or greater?

4. Suppose the researcher wants to conduct the test at the = 0.01 level. Are the results statistically significant?

Example 2

The mean yield of corn in the United States last year was about 120 bushels per acre. A farmer is interested in if the mean yield is different this year. A random sample of 40 farmers is taken and their yields are recorded. In this sample, bushels/acre and bushels/acre. Carry out the significance test (state the hypotheses, check any assumptions of the test, calculate the test statistic, calculate and interpret the p-value).