Points: 0 of 1 Construct the confidence interval for the population mean $\mu$. \[ c=0.95, \bar{x}=7.8, \sigma=0.5, \text { and } n=55 \] A $95 \%$ confidence interval for $\mu$ is $(\square, \square$. (Round to two decimal places as needed.)
Solution
Step 1 ：Given values are: confidence level \(c = 0.95\), sample mean \(\bar{x} = 7.8\), standard deviation \(\sigma = 0.5\), and sample size \(n = 55\).
Step 2 ：The z-score for a 95% confidence level is \(z = 1.96\).
Step 3 ：Calculate the margin of error using the formula: \(margin\_of\_error = z \times \frac{\sigma}{\sqrt{n}}\). Substituting the given values, we get \(margin\_of\_error = 0.13214317304279546\).
Step 4 ：Calculate the confidence interval using the formula: \(lower\_limit = \bar{x} - margin\_of\_error\) and \(upper\_limit = \bar{x} + margin\_of\_error\). Substituting the given values, we get \(lower\_limit = 7.6678568269572045\) and \(upper\_limit = 7.932143173042795\).
Step 5 ：Round the lower and upper limits to two decimal places. The final 95% confidence interval for \(\mu\) is \(\boxed{(7.67, 7.93)}\).
