Question

This is AP Statistics Question. Can you help me with this?

Answer 1

The confidence interval is given by

$\mu_s\pm Z_{\alpha_0}\sigma_s$

where $\mu_s$ is the sample mean and $\sigma_s$ is the standard error of the mean. In turn, the standard error of the mean is $\sigma_s=\dfrac\sigma{\sqrt n}$ where $n$ is sample size.

We have

$\sigma_s=\dfrac6{\sqrt{25}}=1.2$

The endpoints of the confidence interval correspond to the finite endpoints of the rejection region. That is,

$\begin{cases}\mu_s-1.2Z_{\alpha_0}=28\\\mu_s+1.2Z_{\alpha_0}=32\end{cases}$

for which we can solve for $Z_{\alpha_0}$. We get

$Z_{\alpha_0}=\dfrac53\approx1.67$

which is the critical value for a confidence level of $\alpha_0\approx90.44\%$.