Question

Find the endpoints of the t-distribution with \(5 \%\) beyond them in each tail if the samples have sizes \(n_{1}=8\) and \(n_{2}=10\)

Short answer

The endpoints of the t-distribution for the given sample sizes \(n_{1}=8\) and \(n_{2}=10\) and with 5% beyond them in each tail are approximately \(-2.365\) and \(2.365\).

Step-by-step solution

Identify degrees of freedom

Degrees of freedom (df) for each group is calculated as the sample size minus 1. Therefore, \(df_{1} = n_{1}-1 = 8 - 1 = 7\) and similarly \(df_{2} = n_{2}-1 = 10 - 1 = 9\). When dealing with two independent groups, we should consider the smaller degrees of freedom to be more conservative. Therefore, we use 7 (smaller among 7 and 9).

Find the critical t-value

Look up in a standard t-distribution table or use a software to find the critical t-value corresponding to 5% area in each tail and 7 degrees of freedom. From the table, we find that the critical t-value is approximately \(\pm 2.365\). Note that the t-value is negative for the left tail and positive for the right tail.

Identify the endpoints

The endpoints of the t-distribution are these critical t-values. Thus, the endpoints are \(-2.365\) and \(2.365\). These values mark the boundaries for the middle 95% of the distribution.