Questions: Question 18 3 points The sample mean is the point estimator of (A) μ (B) x̄ (C) 0 (D) P̄ Question 19 3 Points The school's newspaper reported that the proportion of students majoring in business is at least 30%. You plan on taking a sample to test the newspaper's claim. The correct set of hypotheses is (A) H0: p>0.30 H2: p ≤ 0.30 (B) H0: p ≤ 0.30 H3: p>0.30 (C) H0: p<0.30 H3: p ≥ 0.30 (D) H0: p ≥ 0.30 H3: p<0.30

Solution Steps

We are testing the claim that the proportion of students majoring in business is at least \(30\%\). Therefore, we set up our hypotheses as follows:

• Null Hypothesis (\(H_0\)): \(p \leq 0.30\)
• Alternative Hypothesis (\(H_1\)): \(p > 0.30\)

The test statistic for the proportion is calculated using the formula:

\[ Z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1 - p_0)}{n}}} \]

Substituting the values:

• \(\hat{p} = 0.32\) (sample proportion)
• \(p_0 = 0.30\) (hypothesized population proportion)
• \(n = 100\) (sample size)

We find:

\[ Z = \frac{0.32 - 0.30}{\sqrt{\frac{0.30(1 - 0.30)}{100}}} = 0.4364 \]

The P-value associated with the test statistic \(Z = 0.4364\) is calculated to be:

\[ \text{P-value} = 0.3313 \]

For a significance level of \(\alpha = 0.05\) in a one-tailed test, the critical value is:

\[ Z_{\text{critical}} = 1.6449 \]

The critical region is defined as:

\[ Z > 1.6449 \]

We compare the test statistic with the critical value:

• Test Statistic: \(Z = 0.4364\)
• Critical Value: \(Z_{\text{critical}} = 1.6449\)

Since \(0.4364 < 1.6449\), we fail to reject the null hypothesis.

Final Answer