To construct a 94% confidence interval for a population mean with a sample size of 15, the appropriate critical value is determined using the t-distribution because the sample size is small ($n < 30$) and the population standard deviation is unknown.

Steps:

1. Determine the degrees of freedom (df): $\text{df} = n - 1 = 15 - 1 = 14$

2. Find the significance level ($\alpha$):

   • Confidence level = 94%, so $\alpha = 1 - 0.94 = 0.06$.
   • Divide $\alpha$ by 2 for a two-tailed test: $\frac{\alpha}{2} = 0.03$.

3. Look up the critical value ($t^*$) in the t-distribution table:

   • For $\text{df} = 14$ and $\frac{\alpha}{2} = 0.03$, the critical value can be found using a t-table or a calculator.

   Using a t-distribution table or software: $t^* \approx 2.14$

Final Answer:

The critical value for a 94% confidence interval with a sample size of 15 is: $\boxed{2.14}$

Would you like a detailed explanation of how to use the t-distribution table or compute this value?

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Here are five follow-up questions you might explore:

1. How does the critical value change if the confidence level increases to 95%?
2. What happens to the critical value as the sample size increases?
3. How does the t-distribution compare to the z-distribution?
4. Why is the t-distribution used instead of the z-distribution for small samples?
5. What assumptions are necessary for using the t-distribution?

Tip: Always check that the data approximately follows a normal distribution when using the t-distribution for small samples!