Questions: Fran is training for her first marathon, and she wants to know if there is a significant difference between the mean number of miles run each week by group runners and individual runners who are training for marathons. She interviews 31 randomly selected people who train in groups, and finds that they run a mean of 47.5 miles per week. Assume that the population standard deviation for group runners is known to be 4.2 miles per week. She also interviews a random sample of 37 people who train on their own and finds that they run a mean of 49.4 miles per week. Assume that the population standard deviation for people who run by themselves is 3.1 miles per week. Test the claim at the 0.02 level of significance. Let group runners training for marathons be Population 1 and let individual runners training for marathons be Population 2. State the null and alternative hypotheses for the test. Fill in the blank below. H0: μ1=μ2 Ha: μ1 ≠ μ2

• Null hypothesis ($H_0$): $\mu_1 = \mu_2$, indicating there is no significant difference between the population means.
• Alternative hypothesis ($H_a$): $\mu_1 \neq \mu_2$, indicating there is a significant difference between the population means.

Using the formula $z = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}$, where $\mu_1 - \mu_2 = 0$ under the null hypothesis, we find: $z = \frac{(5.1 - 4.9)}{\sqrt{\frac{1.2^2}{100} + \frac{1.3^2}{100}}} = 1.131$

For a level of significance of 0.05, the critical values are $\pm1.96$.

Since the absolute value of the test statistic $|1.131|$ is not greater than the critical value $\pm1.96$, we do not reject the null hypothesis.

There is not significant evidence to suggest a difference in the population means.

• Null hypothesis ($H_0$): $\mu_1 = \mu_2$, indicating there is no significant difference between the population means.
• Alternative hypothesis ($H_a$): $\mu_1 \neq \mu_2$, indicating there is a significant difference between the population means.

Using the formula $z = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}$, where $\mu_1 - \mu_2 = 0$ under the null hypothesis, we find: $z = \frac{(47.5 - 49.4)}{\sqrt{\frac{4.2^2}{31} + \frac{3.1.2}{37}}} = -2.087$

For a level of significance of 0.02, the critical values are $\pm2.3263$.

Since the absolute value of the test statistic $|-2.087|$ is not greater than the critical value $\pm2.3263$, we do not reject the null hypothesis.

There is not significant evidence to suggest a difference in the population means.