In a study of 1541 men in their twenties, the men’s HDL cholesterol levels are approximately normally distributed with mean 47 mg/dl and standard deviation 12.5 mg/dl. In a study of 823 men in their fifties, the men’s HDL cholesterol levels are approximately normally distributed with mean 44 mg/dl and standard deviation 13.9 mg/dl
a. Suppose one of the men in their twenties has an HDL reading of 42 mg/dl. Find the z-score of the reading relative to the men in their twenties. What does it mean in this situation?
b. Suppose one of the men in their fifties has an HDL reading of 40 mg/dl. Find the z-score of the reading relative to the men in their fifties. What does it mean in this situation?
c. HDL is the “good” cholesterol. The higher a person’s HDL level, the lower the person’s risk of heart attack and stroke will be. Doctors sometimes judge a characteristic of a person’s health by comparing it to others of the same age. Limiting such a comparison to just those in the study, determine which of the two men described in parts (a) and (b) has the relatively better HDL reading.
d. How is it possible that of the two men described in parts (a) and (b), the man who has the higher HDL reading has the lower z-score?