(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 variance
Inactive men 70 43.3 201.64
Marathon runners 70 52.8 204.49
Given: pooled sample variance = 203.065
(i) 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. (2 points)
(ii) State two ways in which the confidence interval in (i) could be made narrower. (1 point)
(iii) Based on your interval in (i), can you conclude that there is a significant difference in mean HDL cholesterol among inactive men and marathon runners in this age group? State your conclusion, along with a reason for your decision. (2 points)
(iv) Conduct a hypothesis test to see if inactive men 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 state your conclusion. (4 points)
(v) Suppose you obtain the histograms for HDL cholesterol levels in each group, and you find that the distribution in the marathon runners group is strongly left skewed, while the distribution for the inactive men is symmetric. Discuss how you would proceed. (1 point)