Questions: According to previous studies, the mean distance each visitor in Greenspan National Park hikes during their visit is 21 kilometers. The park recently introduces a shuttle system, which is used to transport hikers to many of the park's most popular hiking trails. Because of this, an administrator at the park supposes the mean distance, μ, is now less than 21 kilometers. The administrator chooses a random sample of 85 visitors. The mean distance hiked for the sample is 19.1 kilometers. Assume the population standard deviation is 9.1 kilometers. Can the administrator conclude that the mean distance hiked by each visitor is now less than 21 kilometers? Perform a hypothesis test, using the 0.10 level of significance. (a) State the null hypothesis H0 and the alternative hypothesis H1. (b) Perform a Z-test and find the p-value. Here is some information to help you do your Z-test: - The value of the test statistic is given by (x̄ - μ) / (σ / sqrt(n)), - The p-value is the area under the curve to the left of the value of the test statistic.

According to previous studies, the mean distance each visitor in Greenspan National Park hikes during their visit is 21 kilometers. The park recently introduces a shuttle system, which is used to transport hikers to many of the park's most popular hiking trails. Because of this, an administrator at the park supposes the mean distance, μ, is now less than 21 kilometers. The administrator chooses a random sample of 85 visitors. The mean distance hiked for the sample is 19.1 kilometers. Assume the population standard deviation is 9.1 kilometers. Can the administrator conclude that the mean distance hiked by each visitor is now less than 21 kilometers? Perform a hypothesis test, using the 0.10 level of significance. (a) State the null hypothesis H0 and the alternative hypothesis H1. (b) Perform a Z-test and find the p-value. Here is some information to help you do your Z-test: - The value of the test statistic is given by (x̄ - μ) / (σ / sqrt(n)), - The p-value is the area under the curve to the left of the value of the test statistic.
Transcript text: According to previous studies, the mean distance each visitor in Greenspan National Park hikes during their visit is 21 kilometers. The park recently introduces a shuttle system, which is used to transport hikers to many of the park's most popular hiking trails. Because of this, an administrator at the park supposes the mean distance, $\mu$, is now less than 21 kilometers. The administrator chooses a random sample of 85 visitors. The mean distance hiked for the sample is 19.1 kilometers. Assume the population standard deviation is 9.1 kilometers. Can the administrator conclude that the mean distance hiked by each visitor is now less than 21 kilometers? Perform a hypothesis test, using the 0.10 level of significance. (a) State the null hypothesis $H_{0}$ and the alternative hypothesis $H_{1}$. (b) Perform a Z-test and find the p-value. Here is some information to help you do your $Z$-test: - The value of the test statistic is given by $\frac{\bar{x}-\mu}{\frac{\sigma}{\sqrt{n}}}$, - The p-value is the area under the curve to the left of the value of the test statistic.
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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:

  • Null Hypothesis: \( H_0: \mu = 21 \)
  • Alternative Hypothesis: \( H_1: \mu < 21 \)
Step 2: Calculate the Standard Error

The standard error \( SE \) is calculated using the formula: \[ SE = \frac{\sigma}{\sqrt{n}} = \frac{9.1}{\sqrt{85}} \approx 0.987 \]

Step 3: Calculate the Test Statistic

The Z-test statistic is calculated using the formula: \[ Z = \frac{\bar{x} - \mu_0}{SE} = \frac{19.1 - 21}{0.987} \approx -1.925 \]

Step 4: Calculate the P-value

For a left-tailed test, the p-value is determined as: \[ P = T(z) \approx 0.0271 \]

Step 5: Make a Decision

At a significance level of \( \alpha = 0.10 \):

  • Since \( P < \alpha \) (i.e., \( 0.0271 < 0.10 \)), we reject the null hypothesis.

Final Answer

There is sufficient evidence to conclude that the mean distance hiked by each visitor is less than 21 kilometers.

\(\boxed{\text{Reject } H_0}\)

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