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.

Solution Steps

We are testing the following hypotheses:

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

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

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

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

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