Problem
urrent Attempt in Progress Use a t-distribution to answer this question. Assume the samples are random samples from distributions that are reasonably normally distributed, and that a t-statistic will be used for inference about the difference in sample means. State the degrees of freedom used. Find the endpoints of the t-distribution with $2.5 \%$ beyond them in each tail if the samples have sizès $n_{1}=17$ and $n_{2}=25$. Enter the exact answer for the degrees of freedom and round your answer for the endpoints to two decimal places. \[ \begin{array}{l} \text { degrees of freedom }=\bar{i} \\ \text { endpoints }= \pm \text { i } \end{array} \]
Solution
Step 1 ：Calculate the degrees of freedom using the formula: \(df = n1 + n2 - 2\)
Step 2 ：Substitute the given values into the formula: \(df = 17 + 25 - 2\)
Step 3 ：Simplify the equation to find the degrees of freedom: \(df = 40\)
Step 4 ：Find the endpoints of the t-distribution with 2.5% beyond them in each tail. For a t-distribution with 40 degrees of freedom, the t-value that leaves 2.5% in each tail (or 5% total in both tails) is approximately ±2.021
Step 5 ：So, the final answer is: degrees of freedom = \(\boxed{40}\), endpoints = \(\boxed{\pm2.02}\)
