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?
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A 99\% confidence level requires 409 subjects. (Round up to the nearest subject.)
A 90\% confidence level requires 167 subjects. (Round up to the nearest subject.)
How does the decrease in confidence affect the sample size required?
A. Decreasing the confidence level increases the sample size needed.
B. Decreasing the confidence level decreases the sample size needed.
C. The sample size is the same for all levels of confidence.
Answer
\(\boxed{\text{Final Answer: B. Decreasing the confidence level decreases the sample size needed.}}\)
Step 1 ：A doctor wants to estimate the mean HDL cholesterol of all 20 - to 29 -year-old females. The doctor wants to estimate this within 2 points with a confidence level of 99% and 90%. The standard deviation, based on earlier studies, is 15.7.
Step 2 ：For a 99% confidence level, the required sample size is 409 subjects. This is rounded up to the nearest subject.
Step 3 ：For a 90% confidence level, the required sample size is 167 subjects. This is also rounded up to the nearest subject.
Step 4 ：The question asks how the sample size required changes when the confidence level decreases. The confidence level is the probability that the confidence interval contains the true population parameter.
Step 5 ：A higher confidence level means that there is a higher probability that the confidence interval contains the true population parameter, but it also means that the confidence interval is wider. Therefore, to maintain the same level of precision, a larger sample size is needed.
Step 6 ：Conversely, a lower confidence level means that the confidence interval is narrower, so a smaller sample size is needed to maintain the same level of precision.
Step 7 ：\(\boxed{\text{Final Answer: B. Decreasing the confidence level decreases the sample size needed.}}\)
