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 sizes \(n_{1}=15\) and \(n_{2}=25\).

Short answer

The endpoints of the t-distribution with 2.5% beyond them in each tail when the sample sizes are \(n_{1}=15\) and \(n_{2}=25\) are approximately ±2.024.

Step-by-step solution

Degrees of Freedom

The formula for degrees of freedom for a two-sample t-test when sample sizes are different and variance is assumed unequal is given as: \[ df = {(s_{1}^2/n_{1} + s_{2}^2/n_{2})^2 \over {(s_{1}^2/n_{1})^2/(n_{1}-1) + (s_{2}^2/n_{2})^2/(n_{2}-1)} } \] Where: \n \(s_{1}^2\) and \(s_{2}^2\) are the sample variances \n \(n_{1}\) and \(n_{2}\) are the sample sizes Since we don't have standard deviations here, we will use the simpler formula to calculate degrees of freedom which is: \[ df = n_{1} + n_{2} - 2 \] Substituting with given values, we get : \[ df = 15 + 25 - 2 = 38 \]

Calculating the t-distribution endpoints

The next step is to use a t-table or a t-distribution calculator to obtain the t-distribution values, accounting for the 2.5% in each tail. The t-distribution value that leaves 2.5% in each tail for 38 degrees of freedom is approximately ±2.024.