A simple random sample of size $n$ is drawn from a population that is normally distributed. The sample mean, $\bar{x}$, is found to be 18.7, and the sample standard deviation, $s$, is found to be 5.1 .
(a) Construct a $95 \%$ confidence interval about $\mu$ if the sample size, $n$, is 39 .
(b) Construct a $95 \%$ confidence interval about $\mu$ if the sample size, $n$, is 56 . How does increasing the sample size affect the margin of error, E?
(c) Construct a $99 \%$ confidence interval about $\mu$ if the sample size, $n$, is 39 . How does increasing the level of confidence affect the size of the margin of error, E?
(d) If the sample size is 19 , what conditions must be satisfied to compute the confidence interval?
(a) Construct a $95 \%$ confidence interval about $\mu$ if the sample size, $\mathrm{n}$, is 39 .
Lower bound: 17.05'; Upper bound: 20.35 '
(Round to two decimal places as needed.)
(b) Construct a $95 \%$ confidence interval about $\mu$ if the sample size, $n$, is 56 .
Lower bound: ; Upper bound:
(Round to two decimal places as needed.)
Answer
The 95% confidence interval for the population mean, when the sample size is 39, is approximately \(\boxed{[17.10, 20.30]}\).
Step 1 ：The problem provides the sample mean \(\bar{x} = 18.7\), the sample standard deviation \(s = 5.1\), and the sample size \(n = 39\). It asks for a 95% confidence interval for the population mean \(\mu\).
Step 2 ：The formula for a confidence interval is \(\bar{x} \pm z \frac{s}{\sqrt{n}}\), where \(z\) is the z-score corresponding to the desired level of confidence.
Step 3 ：For a 95% confidence interval, the z-score is approximately 1.96.
Step 4 ：Substitute the given values into the formula to calculate the margin of error, E: \(E = z \frac{s}{\sqrt{n}} = 1.96 \frac{5.1}{\sqrt{39}}\).
Step 5 ：Calculate the lower and upper bounds of the confidence interval: Lower bound = \(\bar{x} - E = 18.7 - E\), Upper bound = \(\bar{x} + E = 18.7 + E\).
Step 6 ：The 95% confidence interval for the population mean, when the sample size is 39, is approximately \(\boxed{[17.10, 20.30]}\).
