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.