You wish to test the following claim ( $$ H_{a} $$ ) at a significance level of $$ \alpha=0.002 $$. $$ \begin{array}{l}H_{0}: \mu_{1}=\mu_{2} \ H_{a}: \mu_{1}\mu_{2}\end{array} $$ 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 ( $$ H_{a} $$ ) at a significance level of $$ \alpha=0.002 $$. $$ \begin{array}{l}H_{0}: \mu_{1}=\mu_{2} \ H_{a}: \mu_{1}>\mu_{2}\end{array} $$ 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.

Wilson Wood · Accepted Answer

To test the hypothesis $$ H_0: \mu_1 = \mu_2 $$ against $$ H_a: \mu_1 > \mu_2 $$ with $$ \alpha = 0.002 $$, calculate the t-statistic and compare it to the critical t-value. If the t-statistic is greater than the critical value, reject $$ H_0 $$. In this case, the t-statistic is 3.71, and the critical value is 3.50, so we reject $$ H_0 $$, supporting the claim that $$ \mu_1 > \mu_2 $$.

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