Question

Question 1. In 2012 the Centers for Disease Control and Prevention reported that in a sample of 4,349 African Americans 31% were Vitamin D deficient. A 90% confidence interval based on this sample is (0.30, 0.32). It is believed that among the general population of Americans 8% suffer from Vitamin D deficiency.

a. Define the appropriate parameter and state the appropriate hypotheses for testing the claim that, among African Americans, Vitamin D deficiency occurs at a rate other than 8%.

b. Does this confidence interval provide evidence that among African Americans Vitamin D deficiency occurs at a rate other than 8%? What significance level is being used to make this decision? Briefly justify your answer.

c. Using the definition of a p-value, explain why the area in the tail of a randomization distribution is used to compute a p-value.

d. In a test of the hypotheses Ho : μ1 = μ2 vs. Ha :μ1 1 μ2, the observed sample results in a p- value of 0.0256. Would you expect a 95% confidence interval for μ1 - μ2 based on this sample to contain 0? Briefly explain why or why not.

Answer 1

a.

The appropriate parameter (p) is true population proportion of African Americans with Vitamin D deficiency.

Null Hypothesis H0: p = 0.08

Alternative Hypothesis Ha: p 0.08

b.

The 90% confidence interval based on the sample is (0.30, 0.32) and does not contain the value 0.08. Thus, there is significant evidence that among African Americans Vitamin D deficiency occurs at a rate other than 8%.

Significance level = 1 - 0.9 = 0.10

c.

The P value is the probability of finding the observed, or more extreme, results when the null hypothesis (H0) of a study question is true. The area in the tail of a randomization distribution are more extreme values and thus are used to compute a p-value.

d.

p-value = 0.0256

For 95% confidence interval , Significance level = 1 - 0.95 = 0.05

Since, p-value is less than 0.05 significance level, we reject null hypothesis H0 and conclude that there is significant evidence that or .

Thus, we do not expect that a 95% confidence interval for μ1 - μ2 based on this sample to contain 0.