Questions: An adventure company runs two obstacle courses, Fundash and Coolsprint. The designer of the courses suspects that the mean completion time of Fundash is not equal to the mean completion time of Coolsprint. To test this, she selects 290 Fundash runners and 260 Coolsprint runners. (Consider these as random samples of the Fundash and Coolsprint runners.) The 290 Fundash runners complete the course with a mean time of 71.2 minutes and a standard deviation of 3.4 minutes. The 260 Coolsprint runners complete the course with a mean time of 71.8 minutes and a standard deviation of 3.1 minutes. Assume that the population standard deviations of the completion times can be estimated to be the sample standard deviations, since the samples that are used to compute them are quite large. At the 0.01 level of significance, is there enough evidence to support the claim that the mean completion time, μ1, of Fundash is not equal to the mean completion time, μ2, of Coolsprint? Perform a two-tailed test. Then complete the parts below. (a) State the null hypothesis H0 and the alternative hypothesis H1. - H0: - H1: (b) Determine the type of test statistic to use. (c) Find the value of the test statistic. (Round to three or more decimal places.) (d) Find the two critical values at the 0.01 level of significance. (Round to three or more decimal places.) and (e) Can we support the claim that the mean completion time of Fundash is not equal to the mean completion time of Coolsprint?

An adventure company runs two obstacle courses, Fundash and Coolsprint. The designer of the courses suspects that the mean completion time of Fundash is not equal to the mean completion time of Coolsprint. To test this, she selects 290 Fundash runners and 260 Coolsprint runners. (Consider these as random samples of the Fundash and Coolsprint runners.) The 290 Fundash runners complete the course with a mean time of 71.2 minutes and a standard deviation of 3.4 minutes. The 260 Coolsprint runners complete the course with a mean time of 71.8 minutes and a standard deviation of 3.1 minutes. Assume that the population standard deviations of the completion times can be estimated to be the sample standard deviations, since the samples that are used to compute them are quite large. At the 0.01 level of significance, is there enough evidence to support the claim that the mean completion time, μ1, of Fundash is not equal to the mean completion time, μ2, of Coolsprint? Perform a two-tailed test. Then complete the parts below. (a) State the null hypothesis H0 and the alternative hypothesis H1. - H0: - H1: (b) Determine the type of test statistic to use. (c) Find the value of the test statistic. (Round to three or more decimal places.) (d) Find the two critical values at the 0.01 level of significance. (Round to three or more decimal places.) and (e) Can we support the claim that the mean completion time of Fundash is not equal to the mean completion time of Coolsprint?
Transcript text: An adventure company runs two obstacle courses, Fundash and Coolsprint. The designer of the courses suspects that the mean completion time of Fundash is not equal to the mean completion time of Coolsprint. To test this, she selects 290 Fundash runners and 260 Coolsprint runners. (Consider these as random samples of the Fundash and Coolsprint runners.) The 290 Fundash runners complete the course with a mean time of 71.2 minutes and a standard deviation of 3.4 minutes. The 260 Coolsprint runners complete the course with a mean time of 71.8 minutes and a standard deviation of 3.1 minutes. Assume that the population standard deviations of the completion times can be estimated to be the sample standard deviations, since the samples that are used to compute them are quite large. At the 0.01 level of significance, is there enough evidence to support the claim that the mean completion time, $\mu_{1}$, of Fundash is not equal to the mean completion time, $\mu_{2}$, of Coolsprint? Perform a two-tailed test. Then complete the parts below. (a) State the null hypothesis $H_{0}$ and the alternative hypothesis $H_{1}$. \[ \begin{array}{l} H_{0}: \square \\ H_{1}: \square \end{array} \] (b) Determine the type of test statistic to use. $\square$ (c) Find the value of the test statistic. (Round to three or more decimal places.) $\square$ (d) Find the two critical values at the 0.01 level of significance. (Round to three or more decimal places.) $\square$ and $\square$ (e) Can we support the claim that the mean completion time of Fundash is not equal to the mean completion time of Coolsprint? $\square$
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Solution

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Solution Steps

Step 1: State the Hypotheses

The null hypothesis \( H_0 \) and the alternative hypothesis \( H_1 \) are defined as follows: \[ H_0: \mu_{1} = \mu_{2} \] \[ H_1: \mu_{1} \neq \mu_{2} \]

Step 2: Determine the Type of Test Statistic

Since the sample sizes are large and the population standard deviations are unknown, we use a two-sample Z-test.

Step 3: Calculate the Test Statistic

The standard error (SE) is calculated using the formula: \[ SE = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} = \sqrt{\frac{3.4^2}{290} + \frac{3.1^2}{260}} = 0.2772 \] The test statistic \( z \) is then calculated as: \[ z = \frac{\bar{x}_1 - \bar{x}_2}{SE} = \frac{71.2 - 71.8}{0.2772} = -2.1647 \]

Step 4: Determine the Critical Values

For a two-tailed test at the \( \alpha = 0.01 \) significance level, the critical values are determined using: \[ Z = \Phi^{-1}(1 - \frac{\alpha}{2}) = \Phi^{-1}(0.995) \] The critical values are: \[ -2.5758 \quad \text{and} \quad 2.5758 \]

Step 5: Make a Decision

We compare the calculated test statistic with the critical values: \[ \text{If } |z| > z_{\text{critical}}, \text{ reject } H_0 \] In this case: \[ |-2.1647| < 2.5758 \] Thus, we fail to reject the null hypothesis.

Final Answer

There is not enough evidence to support the claim that the mean completion time of Fundash is not equal to the mean completion time of Coolsprint.

\(\boxed{\text{Fail to reject } H_0}\)

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