Questions: According to previous studies, the mean distance each visitor in Greenspan National Park hikes during their visit is 30 kilometers. The park recently closed its shuttle system, which used to transport hikers to many of the park's most popular hiking trails. Because of this, an administrator at the park suspects the mean distance, μ, is now less than 30 kilometers. The administrator chooses a random sample of 45 visitors. The mean distance hiked for the sample is 27.2 kilometers. Assume the population standard deviation is 9.9 kilometers. Can the administrator conclude that the mean distance hiked by each visitor is now less than 30 kilometers? Perform a hypothesis test, using the 0.10 level of significance. (a) State the null hypothesis H0 and the alternative hypothesis H1.

According to previous studies, the mean distance each visitor in Greenspan National Park hikes during their visit is 30 kilometers. The park recently closed its shuttle system, which used to transport hikers to many of the park's most popular hiking trails. Because of this, an administrator at the park suspects the mean distance, μ, is now less than 30 kilometers. The administrator chooses a random sample of 45 visitors. The mean distance hiked for the sample is 27.2 kilometers. Assume the population standard deviation is 9.9 kilometers.

Can the administrator conclude that the mean distance hiked by each visitor is now less than 30 kilometers? Perform a hypothesis test, using the 0.10 level of significance.
(a) State the null hypothesis H0 and the alternative hypothesis H1.

Transcript text: According to previous studies, the mean distance each visitor in Greenspan National Park hikes during their visit is 30 kilometers. The park recently closed its shuttle system, which used to transport hikers to many of the park's most popular hiking trails. Because of this, an administrator at the park suspects the mean distance, $\mu$, is now less than 30 kilometers. The administrator chooses a random sample of 45 visitors. The mean distance hiked for the sample is 27.2 kilometers. Assume the population standard deviation is 9.9 kilometers. Can the administrator conclude that the mean distance hiked by each visitor is now less than 30 kilometers? Perform a hypothesis test, using the 0.10 level of significance. (a) State the null hypothesis $H_{0}$ and the alternative hypothesis $H_{11}$. \[ \begin{array}{l} H_{0}: \square \\ H_{1}: \square \end{array} \]

Solution

Step 1: State the null and alternative hypotheses

The null hypothesis $H_0$: $\mu = 30$. The alternative hypothesis $H_1$: $\mu < 30$.

Step 2: Calculate the test statistic

Using the formula $z = \frac{\bar{x} - \mu_0}{\frac{\sigma}{\sqrt{n}}} = \frac{27.2 - 30}{\frac{9.9}{\sqrt{45}}} = -1.897$, we find that the test statistic $z$ is approximately -1.897.

Step 3: Determine the critical value

The critical value for $z$ at the significance level $\alpha = 0.1$ is approximately -1.282. This is found by looking up the cumulative probability of $1 - \alpha$ in a standard normal (Z) distribution table.

Step 4: Make a decision

Since the calculated test statistic -1.897 is less than the critical value -1.282, we reject the null hypothesis.

Step 5: Interpret the result

There is sufficient evidence at the $\alpha = 0.1$ level of significance to conclude that the population mean is less than 30.

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