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 4040 randomly selected people who train in groups, and finds that they run a mean of 44.244.2 miles per week. Assume that the population standard deviation for group runners is known to be 1.61.6 miles per week. She also interviews a random sample of 3131 people who train on their own and finds that they run a mean of 45.345.3 miles per week. Assume that the population standard deviation for people who run by themselves is 2.12.1 miles per week. Test the claim at the 0.050.05 level of significance. Let group runners training for marathons be Population 1 and let individual runners training for marathons be Population 2.

Step 2 of 3 :  

Compute the value of the test statistic. Round your answer to two decimal places.

This problem is asking for a test to determine if there is a significant difference between the mean miles run by group runners (Population 1) and individual runners (Population 2) using a hypothesis test at the \( \alpha = 0.05 \) significance level.

### Step 1: Define the null and alternative hypotheses:

\[
H_0: \mu_1 = \mu_2 \quad \text{(no difference in the mean miles run per week)}
\]
\[
H_a: \mu_1 \neq \mu_2 \quad \text{(there is a difference in the mean miles run per week)}
\]

This will be a two-tailed test.

### Step 2: Compute the test statistic

The formula for the test statistic \( z \) for the difference in means when the population standard deviations are known is:

\[
z = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}
\]

Where:
- \( \bar{x}_1 = 44.2 \) is the mean for group runners.
- \( \bar{x}_2 = 45.3 \) is the mean for individual runners.
- \( \sigma_1 = 1.6 \) is the population standard deviation for group runners.
- \( \sigma_2 = 2.1 \) is the population standard deviation for individual runners.
- \( n_1 = 40 \) is the sample size for group runners.
- \( n_2 = 31 \) is the sample size for individual runners.

Now, plug in the values into the formula:

\[
z = \frac{44.2 - 45.3}{\sqrt{\frac{1.6^2}{40} + \frac{2.1^2}{31}}}
\]

Let's calculate the test statistic.The value of the test statistic is approximately \( z = -2.42 \) (rounded to two decimal places).

Would you like further details or have any questions?

Here are 5 related questions to expand your understanding:

1. What is the next step in determining whether to reject the null hypothesis?
2. How do we calculate the p-value based on the test statistic?
3. What is the critical value for a two-tailed test at the 0.05 significance level?
4. How would the test change if we didn't know the population standard deviations?
5. What are the assumptions behind using this test statistic?

**Tip**: For two-tailed tests, you need to account for both the lower and upper tails when checking the significance of the test statistic.

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 40 randomly selected people who train in groups, and finds that they run a mean of 44.2 miles per week. Assume that the population standard deviation for group runners is known to be 1.6 miles per week. She also interviews a random sample of 31 people who train on their own and finds that they run a mean of 45.3 miles per week. Assume that the population standard deviation for people who run by themselves is 2.1 miles per week. Test the claim at the 0.05 level of significance. Let group runners training for marathons be Population 1 and let individual runners training for marathons be Population 2. Compute the value of the test statistic. Round your answer to two decimal places.

Explore how to perform a two-sample z-test to determine if there is a significant difference between the mean number of miles run per week by group runners and individual runners training for marathons. The problem includes data such as means (44.2 and 45.3 miles) and population standard deviations (1.6 and 2.1 miles). Suitable for undergraduate statistics courses.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation