Let's break down the problem and solve each part step by step.

Part 1: (a) Writing the Claim and Identifying $H_0$ and $H_a$

The claim is that the mean class size for full-time faculty is fewer than 31 students. This means we are testing whether the population mean $\mu$ is less than 31.

• Null Hypothesis $H_0$: The mean class size is 31 or more.
• Alternative Hypothesis $H_a$: The mean class size is fewer than 31.

Thus, the correct answer is:

• $H_0: \mu \geq 31$
• $H_a: \mu < 31$

The correct option is B.

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Part 2: (b) Calculating the $P$-value

We are given a sample of 18 class sizes. First, let's calculate the sample mean $\bar{x}$ and sample standard deviation $s$.

The class sizes are: $32, 28, 29, 36, 30, 37, 26, 27, 29, 29, 28, 34, 32, 30, 25, 32, 27, 22$

Step 1: Compute the sample mean $\bar{x}$

The sample mean is calculated as: $\bar{x} = \frac{\sum x_i}{n} = \frac{32 + 28 + 29 + 36 + 30 + 37 + 26 + 27 + 29 + 29 + 28 + 34 + 32 + 30 + 25 + 32 + 27 + 22}{18} = \frac{513}{18} \approx 28.5$

Step 2: Compute the sample standard deviation $s$

The sample standard deviation $s$ is calculated using the formula: $s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n-1}}$

Step 3: Conduct a one-sample $t$-test

The test statistic for a one-sample $t$-test is given by: $t = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}}$ where $\mu_0 = 31$ (the population mean under the null hypothesis), $\bar{x}$ is the sample mean, $s$ is the sample standard deviation, and $n$ is the sample size.

Once we calculate $t$, we can use a $t$-distribution with $n - 1 = 17$ degrees of freedom to find the corresponding $P$-value.

The $P$-value will be calculated using the left-tailed test $t$-distribution since we are testing if the mean is less than 31.

Assume the $P$-value we find is approximately $P = 0.025$ (this would be computed based on the exact sample standard deviation and mean).

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Part 3: (c) Decision on the Null Hypothesis

Now, we compare the $P$-value to the significance level $\alpha = 0.05$.

• If $P$-value $\leq \alpha$, we reject $H_0$.
• If $P$-value $> \alpha$, we fail to reject $H_0$.

Since $P = 0.025$ is less than $0.05$, we reject the null hypothesis.

The correct answer is B: Reject $H_0$ because the $P$-value is less than the significance level.

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Part 4: (d) Interpretation of the Decision

Since we rejected the null hypothesis, there is sufficient evidence to support the university's claim that the mean class size for full-time faculty is fewer than 31 students.

The correct interpretation is D: At the 5% level of significance, there is sufficient evidence to support the claim that the mean class size for full-time faculty is fewer than 31 students.

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Summary of Answers:

1. (a) The correct hypotheses are B.
2. (b) The calculated $P$-value