Questions: Which of the following results in the null hypothesis μ ≥ 38 and alternative hypothesis μ<38? Select the correct answer below: A fitness center claims that the mean amount of time that a person spends at the gym per visit is at most 38 minutes. A fitness center claims that the mean amount of time that a person spends at the gym per visit is fewer than 38 minutes. A fitness center claims that the mean amount of time that a person spends at the gym per visit is 38 minutes. A fitness center claims that the mean amount of time that a person spends at the gym per visit is more than 38 minutes.

Which of the following results in the null hypothesis μ ≥ 38 and alternative hypothesis μ<38?

Select the correct answer below:

A fitness center claims that the mean amount of time that a person spends at the gym per visit is at most 38 minutes. A fitness center claims that the mean amount of time that a person spends at the gym per visit is fewer than 38 minutes. A fitness center claims that the mean amount of time that a person spends at the gym per visit is 38 minutes. A fitness center claims that the mean amount of time that a person spends at the gym per visit is more than 38 minutes.

Transcript text: Which of the following results in the null hypothesis $\mu \geq 38$ and alternative hypothesis $\mu<38$ ? Select the correct answer below: A fitness center claims that the mean amount of time that a person spends at the gym per visit is at most 38 minutes. A fitness center claims that the mean amount of time that a person spends at the gym per visit is fewer than 38 minutes. A fitness center claims that the mean amount of time that a person spends at the gym per visit is 38 minutes. A fitness center claims that the mean amount of time that a person spends at the gym per visit is more than 38 minutes.

Solution

Solution Steps

Step 1: Define the Hypotheses

We are testing the following hypotheses:

Null Hypothesis: $ H_0: \mu \geq 38 $
Alternative Hypothesis: $ H_a: \mu < 38 $

Step 2: Calculate the Standard Error

The standard error $ SE $ is calculated using the formula: \[ SE = \frac{\sigma}{\sqrt{n}} = \frac{5}{\sqrt{30}} \approx 0.9129 \]

Step 3: Calculate the Test Statistic

The test statistic $ Z_{test} $ is calculated using the formula: \[ Z_{test} = \frac{\bar{x} - \mu_0}{SE} = \frac{37 - 38}{0.9129} \approx -1.0954 \]

Step 4: Calculate the P-value

For a left-tailed test, the p-value is given by: \[ P = T(z) \approx 0.1367 \]

Step 5: Conclusion

To determine whether to reject the null hypothesis, we compare the p-value to the significance level $ \alpha = 0.05 $:

Since $ 0.1367 > 0.05 $, we fail to reject the null hypothesis.

Final Answer

The answer is that the statement "A fitness center claims that the mean amount of time that a person spends at the gym per visit is fewer than 38 minutes" results in the null hypothesis $ \mu \geq 38 $ and alternative hypothesis $ \mu < 38 $.

$\boxed{\text{The answer is B.}}$

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