Questions: Suppose a university advertises that its average class size is 34 or less. A student organization is concerned that budget cuts have led to increased class sizes and would like to test this claim. A random sample of 42 classes was selected, and the average class size was found to be 36.3 students. Assume that the standard deviation for class size at the college is 7 students. Using α=0.01, complete parts a and b below. H1 · μ>34 The z-test statistic is 2.13 (Round to two decimal places as needed.) The critical z-score(s) is(are) 2.33 (Round to two decimal places as needed. Use a comma to separate answers as needed.) Because the test statistic is less than the critical value, do not reject the null hypothesis. b. Determine the p -value for this test. The p-value is . (Round to three decimal places as needed.)

Suppose a university advertises that its average class size is 34 or less. A student organization is concerned that budget cuts have led to increased class sizes and would like to test this claim. A random sample of 42 classes was selected, and the average class size was found to be 36.3 students. Assume that the standard deviation for class size at the college is 7 students. Using α=0.01, complete parts a and b below.

H1 · μ>34

The z-test statistic is 2.13 (Round to two decimal places as needed.) The critical z-score(s) is(are) 2.33 (Round to two decimal places as needed. Use a comma to separate answers as needed.) Because the test statistic is less than the critical value, do not reject the null hypothesis. b. Determine the p -value for this test.

The p-value is . (Round to three decimal places as needed.)

Transcript text: Suppose a university advertises that its average class size is 34 or less. A student organization is concerned that budget cuts have led to increased class sizes and would like to test this claim. A random sample of 42 classes was selected, and the average class size was found to be 36.3 students. Assume that the standard deviation for class size at the college is 7 students. Using $\alpha=0.01$, complete parts a and b below. \[ H_{1} \cdot \mu>34 \] The $z$-test statistic is 2.13 (Round to two decimal places as needed.) The critical z-score(s) is(are) 2.33 (Round to two decimal places as needed. Use a comma to separate answers as needed.) Because the test statistic $\square$ is less than the critical value, do not reject the null hypothesis. b. Determine the p -value for this test. The $p$-value is $\square$ . (Round to three decimal places as needed.)

Solution

Solution Steps

Step 1: Calculate the Z-test Statistic

Given:

Sample size $ n = 42 $
Sample mean $ \bar{x} = 36.3 $
Population mean $ \mu = 34 $
Population standard deviation $ \sigma = 7 $

The standard error $ SE $ is calculated as: \[ SE = \frac{\sigma}{\sqrt{n}} = \frac{7}{\sqrt{42}} \approx 1.0801 \]

The Z-test statistic $ Z_{\text{test}} $ is: \[ Z_{\text{test}} = \frac{\bar{x} - \mu}{SE} = \frac{36.3 - 34}{1.0801} \approx 2.13 \]

Step 2: Determine the Critical Z-score

For a right-tailed test with $ \alpha = 0.01 $, the critical Z-score is: \[ Z_{\text{critical}} = 2.33 \]

Step 3: Decision Based on Z-test Statistic and Critical Z-score

Since $ Z_{\text{test}} = 2.13 $ is less than $ Z_{\text{critical}} = 2.33 $, we do not reject the null hypothesis.

Step 4: Determine the P-value

The P-value for the right-tailed test is calculated as: \[ P = 1 - \Phi(Z_{\text{test}}) = 1 - \Phi(2.13) \approx 0.0166 \]

Final Answer

The Z-test statistic is $ 2.13 $.
The critical Z-score is $ 2.33 $.
Because the test statistic $ 2.13 $ is less than the critical value $ 2.33 $, we do not reject the null hypothesis.
The P-value is $ 0.0166 $.

\[ \boxed{\text{Do not reject the null hypothesis}} \]

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