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The confidence interval is given by

[tex]\mu_s\pm Z_{\alpha_0}\sigma_s[/tex]

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

We have

[tex]\sigma_s=\dfrac6{\sqrt{25}}=1.2[/tex]

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

[tex]\begin{cases}\mu_s-1.2Z_{\alpha_0}=28\\\mu_s+1.2Z_{\alpha_0}=32\end{cases}[/tex]

for which we can solve for [tex]Z_{\alpha_0}[/tex]. We get

[tex]Z_{\alpha_0}=\dfrac53\approx1.67[/tex]

which is the critical value for a confidence level of [tex]\alpha_0\approx90.44\%[/tex].

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