Here are summary statistics for randomly selected weights of newborn girls: $n=214, \bar{x}=30.9 \mathrm{hg}, \mathrm{s}=6.9$ hg. Construct a confidence interval estimate of the mean. Use a $99 \%$ confidence level. Are these results very different from the confidence interval $28.6 \mathrm{hg}< \mu< 32.6 \mathrm{hg}$ with only 15 sample values, $\bar{x}=30.6 \mathrm{hg}$, and $\mathrm{s}=2.6 \mathrm{hg}$ ?
What is the confidence interval for the population mean $\mu$ ?
$\mathrm{hg}< \mu< \square$ hg (Round to one decimal place as needed.)
Answer
Final Answer: The confidence interval for the population mean \(\mu\) is approximately \(29.7 \mathrm{hg}<\mu<32.1 \mathrm{hg}\). So, the answer is \(\boxed{32.1}\) hg.
Step 1 ：Given that the sample size \(n=214\), the sample mean \(\bar{x}=30.9\) hg, and the standard deviation \(s=6.9\) hg, we are asked to construct a confidence interval estimate of the mean using a 99% confidence level.
Step 2 ：The formula for the confidence interval of a mean is \(\bar{x} \pm z \frac{s}{\sqrt{n}}\), where \(\bar{x}\) is the sample mean, \(z\) is the z-score corresponding to the desired confidence level, \(s\) is the standard deviation, and \(n\) is the sample size.
Step 3 ：The z-score for a 99% confidence level is approximately 2.576.
Step 4 ：Substituting the given values into the formula, we get \(30.9 \pm 2.576 \frac{6.9}{\sqrt{214}}\).
Step 5 ：Solving this gives us a margin of error of approximately 1.215.
Step 6 ：Subtracting this margin of error from the sample mean gives us the lower bound of the confidence interval, which is approximately 29.7 hg.
Step 7 ：Adding the margin of error to the sample mean gives us the upper bound of the confidence interval, which is approximately 32.1 hg.
Step 8 ：Therefore, the confidence interval for the population mean \(\mu\) is approximately \(29.7 \mathrm{hg}<\mu<32.1 \mathrm{hg}\).
Step 9 ：Comparing this with the given confidence interval of \(28.6 \mathrm{hg}<\mu<32.6 \mathrm{hg}\), we see that our interval is slightly wider, which is expected given the larger standard deviation and sample size in our calculation.
Step 10 ：Final Answer: The confidence interval for the population mean \(\mu\) is approximately \(29.7 \mathrm{hg}<\mu<32.1 \mathrm{hg}\). So, the answer is \(\boxed{32.1}\) hg.
