Probability Distributions for HDL Cholesterol Levels Studies have shown that vigorous exercise may increase HDL cholesterol levels in blood, which may have health benefits such as lowering the risk of coronary heart disease. Suppose that the HDL cholesterol levels in non-joggers have a mean of 43.2 mg/dl and a standard deviation of 14.2 mg/dl, and the levels in joggers have a mean of 58.0 mg/dl and a standard deviation of 17.7 mg/dl. Assume normal distributions for the HDL cholesterol levels in both groups.
Part 1 If a large, random group of joggers are tested, what fraction of them would be expected to have an HDL cholesterol level of 58 mg/dl or more? What fraction of non-joggers have these levels?
Likewise, if a large, random group of non-joggers were tested, what fraction of them would be expected to have an HDL cholesterol level of 43.2 mg/dl or less? What fraction of joggers have these levels?
If a level of 58 mg/dl or more is considered to be an indication of good heart health and 43.2 mg/dl or less an indication of poor heart health, what conclusions can you draw, if any, about the value of vigorous exercise based on your numerical results?
Part 2 Notice that although normal distributions were assumed for both populations (joggers and non-joggers), in truth the distributions cannot be perfectly normal because they cannot be perfectly symmetrical. A true normal distribution has tails that extend without limit in either direction (from negative infinity to positive infinity). In both of our cases, we ignored the part of the left-hand tail below 0 mg/dl. There actually should be a cutoff at 0 mg/dl since HDL cholesterol levels cannot be negative.
Fortunately, the magnitude of the distribution at 0 mg/dl is very small and the percent of the area in the tails below 0 is correspondingly very small. Ignoring it should not result in a significant error. If one wanted to be a stickler for accuracy, though, the distribution would have to fall to zero at or below 0 mg/dl since there cannot be a level less than 0 mg/dl in real life.
Describe in a few sentences (and, if you wish, by drawings or graphics) how you might alter the normal distributions which extend below 0 to arrive at more realistic ones that might not be normal, would not extend below 0 mg/dl but would be expected to be more accurate. Note: Remember that the total area under a probability distribution function is always 1.