To find the critical value for the given one-sample t-test, we will use the following information:

• Null Hypothesis ($H_0$): $\mu \leq 20$
• Significance level ($\alpha$): 0.10
• Sample size ($n$): 20
• Degrees of freedom ($df$): $n - 1 = 19$

Since the null hypothesis is $\mu \leq 20$, we are dealing with a one-tailed test (specifically a left-tailed test). We need to find the t-value that corresponds to the given level of significance ($\alpha = 0.10$) for 19 degrees of freedom.

Steps to Find the Critical Value:

1. Determine the degrees of freedom: $df = 20 - 1 = 19$.
2. Locate the critical t-value for a one-tailed test with $\alpha = 0.10$ and $df = 19$.

For a one-tailed t-test with $\alpha = 0.10$ and 19 degrees of freedom, we can use a t-distribution table or statistical software to find the critical value.

Finding the t-value:

Looking up the value in a t-distribution table, the critical value (negative since it's a left-tailed test) for $\alpha = 0.10$ and $df = 19$ is approximately:

$t_{0.10, 19} \approx -1.328$

Thus, the correct critical value is:

D. $-1.328$

Would you like more details or have any questions?

Here are some related questions to explore:

1. How do you determine whether to use a one-tailed or two-tailed t-test?
2. What is the difference between a one-sample t-test and a two-sample t-test?
3. How does the sample size affect the critical value in a t-test?
4. What are degrees of freedom, and why are they important in statistical tests?
5. How does changing the significance level $\alpha$ affect the critical value?
6. What assumptions must be met to perform a t-test?
7. How do you interpret the result of a t-test in context?
8. What is the difference between a t-test and a z-test?

Tip: When using a t-distribution table, always ensure you are using the correct tail and degrees of freedom for your specific test.