A doctor wants to estimate the mean HDL cholesterol of all 20- to 29-year-old females. How many subjects are needed to estimate the mean HDL cholesterol within 2 points with $99 \%$ confidence assuming $s=15.7$ based on earlier studies? Suppose the doctor would be content with $90 \%$ confidence. How does the decrease in confidence affect the sample size required? Click the icon to view a partial table of critical values. A $99 \%$ confidence level requires subjects, (Round up to the nearest subject.)

Step 1 ：The doctor wants to estimate the mean HDL cholesterol within 2 points with 99% confidence. The standard deviation is given as 15.7. We can use the formula for sample size in estimating a population mean: \(n = (Z*σ/E)^2\) where: n is the sample size, Z is the Z-score (which depends on the desired confidence level), σ is the standard deviation, and E is the desired margin of error.

Step 2 ：For a 99% confidence level, the Z-score is approximately 2.576 (from the Z-table or standard normal distribution table). The standard deviation σ is 15.7 and the desired margin of error E is 2. We can substitute these values into the formula to calculate the required sample size. Since we can't have a fraction of a subject, we'll round up to the nearest whole number.

Step 3 ：Substituting the values into the formula, we get \(n = (2.576*15.7/2)^2\).

Step 4 ：Calculating the above expression, we find that the required sample size is 409. This means that the doctor needs to sample 409 subjects to estimate the mean HDL cholesterol within 2 points with 99% confidence. This is assuming that the standard deviation is 15.7 based on earlier studies.

Step 5 ：Final Answer: The doctor needs to sample \(\boxed{409}\) subjects to estimate the mean HDL cholesterol within 2 points with 99% confidence.