Questions: Determine the t-value in each of the cases. (a) Find the t-value such that the area in the right tail is 0.025 with 26 degrees of freedom. (Round to three decimal places as needed.) (b) Find the t-value such that the area in the right tail is 0.01 with 18 degrees of freedom. (Round to three decimal places as needed.) (c) Find the t-value such that the area left of the t-value is 0.15 with 29 degrees of freedom. -1.055 (Round to three decimal places as needed.) (d) Find the critical t-value that corresponds to 99% confidence. Assume 30 degrees of freedom. (Round to three decimal places as needed.)

Solution Steps

We need to find the $t$-values for different scenarios using the $t$-distribution. Each sub-question provides a specific area in the tail or left of the $t$-value and the degrees of freedom. We will use a $t$-distribution table or a calculator to find these values.

For part (a), we need to find the $t$-value such that the area in the right tail is 0.025 with 26 degrees of freedom. This is a common scenario for a two-tailed test with a 5% significance level (2.5% in each tail).

Using a $t$-distribution table or calculator, we find:

\[ t_{0.025, 26} \approx 2.056 \]

For part (b), we need to find the $t$-value such that the area in the right tail is 0.01 with 18 degrees of freedom.

Using a $t$-distribution table or calculator, we find:

\[ t_{0.01, 18} \approx 2.552 \]

For part (c), we need to find the $t$-value such that the area left of the $t$-value is 0.15 with 29 degrees of freedom. The hint suggests using symmetry, which means we are looking for a negative $t$-value.

Using a $t$-distribution table or calculator, we find:

\[ t_{0.15, 29} \approx -1.055 \]

Final Answer