You wish to test the following claim $$ \left(H_{a}\right) $$ at a significance level of $$ \alpha=0.002 $$. $$ \begin{array}{l}H_{a}: \mu_{1}=\mu_{2} \ H_{a}: \mu_{1}\mu_{2}\end{array} $$ You believe both populations are normally distributed, but you do not know the standard deviations for either. However, you also have no reason to believe the variances of the two populations are not equal. You obtain a sample of size $$ n_{1}=20 $$ with a mean of $$ M_{1}=51.3 $$ and a standard deviation of $$ S D_{1}=13.2 $$ from the first population. You obtain a sample of size $$ n_{2}=23 $$ with a mean of $$ M_{2}=38.8 $$ and a standard deviation of $$ S D_{2}=8.7 $$ from the second population.

Question

You wish to test the following claim $$ \left(H_{a}\right) $$ at a significance level of $$ \alpha=0.002 $$. $$ \begin{array}{l}H_{a}: \mu_{1}=\mu_{2} \ H_{a}: \mu_{1}>\mu_{2}\end{array} $$ You believe both populations are normally distributed, but you do not know the standard deviations for either. However, you also have no reason to believe the variances of the two populations are not equal. You obtain a sample of size $$ n_{1}=20 $$ with a mean of $$ M_{1}=51.3 $$ and a standard deviation of $$ S D_{1}=13.2 $$ from the first population. You obtain a sample of size $$ n_{2}=23 $$ with a mean of $$ M_{2}=38.8 $$ and a standard deviation of $$ S D_{2}=8.7 $$ from the second population.

Garrett Campos · Accepted Answer

To test the claim $$ H_{a}: \mu_{1}>\mu_{2} $$ at $$ \alpha=0.002 $$, we used a two-sample t-test with equal variances. The test statistic was 3.608, and the critical value was 2.423. Since the test statistic is greater than the critical value, we reject the null hypothesis, concluding that the mean of the first population is greater than the mean of the second population.

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