To determine the upper-tail critical value $t_{\alpha/2}$, we need to use the t-distribution table. The formula for $t_{\alpha/2}$ requires:

1. $\alpha$, which is the complement of $1 - \alpha$.
2. $n$, the sample size, which is used to find the degrees of freedom $df = n - 1$.

The upper-tail critical value is typically used in the context of hypothesis testing or confidence intervals, particularly in two-tailed tests.

Part (a):

Given:

• $1 - \alpha = 0.99$, so $\alpha = 0.01$.
• $n = 27$, so degrees of freedom $df = 27 - 1 = 26$.

We need the critical value for $t_{0.005}$ (since $\alpha/2 = 0.005$ for the upper-tail critical value) with 26 degrees of freedom.

Using the t-distribution table, we look up the value for $df = 26$ and $\alpha/2 = 0.005$, which is approximately 2.779.

Part (b):

• $1 - \alpha = 0.95$, so $\alpha = 0.05$.
• $n = 27$, so degrees of freedom $df = 27 - 1 = 26$.

For $\alpha/2 = 0.025$, the critical value $t_{0.025}$ for 26 degrees of freedom is approximately 2.056.

Part (c):

• $1 - \alpha = 0.99$, so $\alpha = 0.01$.
• $n = 45$, so degrees of freedom $df = 45 - 1 = 44$.

For $\alpha/2 = 0.005$, the critical value $t_{0.005}$ for 44 degrees of freedom is approximately 2.776.

Part (d):

• $1 - \alpha = 0.99$, so $\alpha = 0.01$.
• $n = 19$, so degrees of freedom $df = 19 - 1 = 18$.

For $\alpha/2 = 0.005$, the critical value $t_{0.005}$ for 18 degrees of freedom is approximately 2.878.

Part (e):

• $1 - \alpha = 0.90$, so $\alpha = 0.10$.
• $n = 36$, so degrees of freedom $df = 36 - 1 = 35$.

For $\alpha/2 = 0.05$, the critical value $t_{0.05}$ for 35 degrees of freedom is approximately 1.690.

Summary of Results:

• Part (a): $t_{0.005} \approx 2.779$
• Part (b): $t_{0.025} \approx 2.056$
• Part (c): $t_{0.005} \approx 2.776$
• Part (d): $t_{0.005} \approx 2.878$
• Part (e): $t_{0.05} \approx 1.690$

Would you like further explanations or any specific details for these calculations? Here are some related questions you might find interesting:

1. How do you calculate the critical value for a z-distribution, and how does it differ from the t-distribution?
2. What is the relationship between degrees of freedom and the shape of the t-distribution?
3. Can the critical value $t_{\alpha/2}$ be used for both one-tailed and two-tailed tests?
4. How do you interpret the critical value in the context of hypothesis testing?
5. How do changes in the sample size affect the critical value for the t-distribution?

Tip: When looking up t-values, always ensure that you're using the correct degrees of freedom and significance level to get the precise critical value.