(b) For each of the following sampling scenarios, determine which distribution should be used to calculate the critical value for the $90 \%$ confidence interval for the population mean.
(In the table, $Z$ refers to a standard normal distribution, and $t$ refers to a $t$ distribution.)
\begin{tabular}{|c|c|c|c|c|}
\hline Sampling scenario & $\mathbf{Z}$ & $t$ & \begin{tabular}{l}
Could use \\
either $Z$ or $t$
\end{tabular} & Unclear \\
\hline \begin{tabular}{l}
The sample has size 90 , and it is from a non-normally distributed \\
population with a known standard deviation of 0.25 .
\end{tabular} & (C) & 0 & 0 & 0 \\
\hline \begin{tabular}{l}
The sample has size 10 , and it is from a normally distributed \\
population with an unknown standard deviation.
\end{tabular} & 0 & 0 & 0 & 0 \\
\hline \begin{tabular}{l}
The sample has size 100 , and it is from a non-normally distributed \\
population.
\end{tabular} & 0 & 0 & 0 & 0 \\
\hline
\end{tabular}

Explanation
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Step 1 ：The question is asking to determine which distribution should be used to calculate the critical value for the 90% confidence interval for the population mean in each of the given sampling scenarios.

Step 2 ：The first scenario is a sample of size 90 from a non-normally distributed population with a known standard deviation of 0.25. Since the sample size is large (greater than 30) and the standard deviation is known, we can use the Z-distribution.

Step 3 ：For the first scenario, we should use the \(\boxed{Z}\) distribution.

Step 4 ：The second scenario is a sample of size 10 from a normally distributed population with an unknown standard deviation. Since the sample size is small (less than 30) and the standard deviation is unknown, we should use the t-distribution.

Step 5 ：For the second scenario, we should use the \(\boxed{t}\) distribution.

Step 6 ：The third scenario is a sample of size 100 from a non-normally distributed population. Since the sample size is large (greater than 30), we can use the Z-distribution, even though the population is not normally distributed.

Step 7 ：For the third scenario, we should use the \(\boxed{Z}\) distribution.