Question

1. Elevated levels of plasma HDL cholesterol may be associated with lowered risk of coronary heart disease. Several studies have suggested that vigorous exercise may result in increased levels of HDL cholesterol. To investigate this theory, researchers measured HDL concentrations (mg/dL) in middle-aged (35-66 years) marathon runners, as well as inactive men. The data are summarized below. (Assume equal population variances in the two groups.)

sample size mean st. dev

Inactive men 70 43.3 14.2

Marathon runners 70 52.8 14.3

a. (4pts) Conduct a hypothesis test to see if marathon runners have significantly lower mean HDL cholesterol levels in this age group. Use α=0.05. Write your hypotheses, calculate the test statistic, and p-value, and write your conclusion. (SE = 2.41)

b. (2pts) Calculate the 90% confidence interval for the difference in mean HDL cholesterol among inactive men and marathon runners in this age group. Interpret your interval.

c. (1pt) State two ways in which the confidence interval in (i) could be made narrower.

d. (1pt) Use your interval in (b), Explain how this confidence interval corresponds to the results you found in part (a).

2. As part of a study of environmental influences on sex determination in the fish Menidia, eggs from a single mating were divided into two groups and raised in either a warm or a cold environment. The data show that 65 of 141 offspring in the warm environment and 107 of 169 offspring in the cold environment were females.

a. Test the hypothesis that the distribution of female and male fish follows a 1:1 ratio in the cold environment. Ascertain that your assumptions are met, and then do the test. (2pts)

b. What are the odds of the fish in the cold environment being female? (1pt)

c. Calculate the odds ratio of the fish being female in the cold environment compared to the warm environment. Interpret this odds ratio in the context of the question. (1pt)

Answer 1

(a)

Let sample 1 Marathon runners and sample 2 Inactive men

Hypotheses are:

The test statistics is

The p-value is

p-value = P(z<= 3.94) = 1.0000

Since p-value is greater than 0.05 so we fail to reject the null hypothesis. That is we cannot conclude that marathon runners have significantly lower mean HDL cholesterol levels in this age group.

(b)

The confidence interval is:

(c)

Either decreasing confidence interval or decreasing sample sizes will make the confidence interval narrower.

(d)

Since confidence interval contains all values are greater than zero so we cannot conclude that marathon runners have significantly lower mean HDL cholesterol levels in this age group.