Among various ethnic groups, the standard deviation of heights is known to be approximately three inches. We wish to construct a $95 \%$ confidence interval for the mean height of male Swedes. Fortyeight male Swedes are surveyed. The sample mean is 74 inches. Round answers to 3 decimal places. a. Find the following: i. $\bar{x}=$ ii. $\sigma=$ iii. $n=$ b. Construct a $95 \%$ confidence interval for the population mean height of male Swedes. i. State the confidence interval. $\mathrm{Cl}:$ ii. Calculate the error bound. EBM: c. What will happen to the size of the confidence interval if 1,000 male Swedes are surveyed instead of $48 ?$ The confidence interval will Select an answer $\checkmark$. Question Help: 国 VIDEO Submit

Step 1 ：The given values are: sample mean (\(\bar{x}\)) is 74 inches, standard deviation (\(\sigma\)) is 3 inches, and sample size (\(n\)) is 48.

Step 2 ：The formula for a confidence interval is \(\bar{x} \pm z \frac{\sigma}{\sqrt{n}}\), where \(z\) is the z-score corresponding to the desired level of confidence. For a 95% confidence interval, the z-score is approximately 1.96.

Step 3 ：Substitute the given values into the formula to find the confidence interval: \(74 \pm 1.96 \frac{3}{\sqrt{48}}\).

Step 4 ：Calculate the lower and upper bounds of the confidence interval to get approximately (73.151, 74.849).

Step 5 ：The error bound is given by the formula \(EBM = z \frac{\sigma}{\sqrt{n}}\). Substitute the given values into the formula to get \(EBM = 1.96 \frac{3}{\sqrt{48}}\), which is approximately 0.849.

Step 6 ：If the sample size is increased to 1,000, the size of the confidence interval will decrease. This is because a larger sample provides more information about the population and thus reduces uncertainty.