You wish to test the following claim $\left(H_{a}\right)$ at a significance level of $\alpha=0.02$.
\[
\begin{array}{l}
H_{o}: \mu_{1}=\mu_{2} \\
H_{a}: \mu_{1} \neq \mu_{2}
\end{array}
\]
You obtain a sample of size $n_{1}=109$ with a mean of $M_{1}=65.1$ and a standard deviation of $S D_{1}=14.1$ from the first population. You obtain a sample of size $n_{2}=72$ with a mean of $M_{2}=67.1$ and a standard deviation of $S D_{2}=12.9$ from the second population.

What is the critical value for this test? For this calculation, use the conservative under-estimate for the degrees of freedom as mentioned in the textbook. (Report answer accurate to three decimal places.) critical value ?
What is the test statistic for this sample? (Report answer accurate to three decimal places.) test statistic ?
The test statistic is?
in the critical region
not in the critical region
This test statistic leads to a decision to?reject the null
accept the null
fail to reject the null
As such, the final conclusion is that?

Step 1 ：Given that we have two samples from two different populations, we want to test the claim that the means of these two populations are not equal. The null hypothesis \(H_{0}\) is that the means are equal, \(\mu_{1} = \mu_{2}\), and the alternative hypothesis \(H_{a}\) is that the means are not equal, \(\mu_{1} \neq \mu_{2}\).

Step 2 ：The sample from the first population has a size of \(n_{1} = 109\), a mean of \(M_{1} = 65.1\), and a standard deviation of \(SD_{1} = 14.1\). The sample from the second population has a size of \(n_{2} = 72\), a mean of \(M_{2} = 67.1\), and a standard deviation of \(SD_{2} = 12.9\).

Step 3 ：We are conducting this test at a significance level of \(\alpha = 0.02\). Since this is a two-tailed test, we will divide the alpha level by 2.

Step 4 ：To find the critical value for this test, we need to use the t-distribution table. The degrees of freedom is the smaller of \(n_{1} - 1\) and \(n_{2} - 1\), which is \(df = 71\).

Step 5 ：Looking up the t-distribution table with \(df = 71\) and \(\alpha/2 = 0.01\) (for a two-tailed test), we find that the critical value is approximately 2.380.

Step 6 ：\(\boxed{2.380}\) is the critical value for this test.