For the provided sample mean, sample size, and population standard deviation, complete parts (a) through (c) below Assume that $\mathrm{x}$ is normally distributed. \[ \bar{x}=22, n=25, \sigma=5 \] a. Find a $95 \%$ confidence interval for the population mean. The $95 \%$ confidence interval is from $\square$ to $\square$. (Round to two decimal places as needed.)
Solution
Step 1 ：Given the sample mean (\(\bar{x}\)) is 22, the sample size (n) is 25, and the population standard deviation (\(\sigma\)) is 5.
Step 2 ：We are asked to find a 95% confidence interval for the population mean. The Z-score for a 95% confidence interval is 1.96.
Step 3 ：First, calculate the margin of error using the formula: \(Z \times \frac{\sigma}{\sqrt{n}}\). Substituting the given values, we get: \(1.96 \times \frac{5}{\sqrt{25}} = 1.96\).
Step 4 ：Next, calculate the lower limit of the confidence interval using the formula: \(\bar{x} - \text{margin of error}\). Substituting the given values, we get: \(22 - 1.96 = 20.04\).
Step 5 ：Then, calculate the upper limit of the confidence interval using the formula: \(\bar{x} + \text{margin of error}\). Substituting the given values, we get: \(22 + 1.96 = 23.96\).
Step 6 ：Finally, the 95% confidence interval for the population mean is from 20.04 to 23.96.
