Questions: Conduct the following test at the α=0.01 level of significance by determining (a) the null and alternative hypotheses, (b) the test statistic, and (c) the critical value. Assume that the samples were obtained independently using simple random sampling. Test whether p1 ≠ p2. Sample data are x1=28, n1=254, x2=36 and n2=301. (a) Determine the null and alternative hypotheses. Choose the correct answer below. H0: p1=p2 versus H1: p1<p2 H0: p1=p2 versus H1: p1>p2 H0: p1=p2 versus H1: p1 ≠ p2 (b) The test statistic z0 is (Round to two decimal places as needed.) (c) The critical values are ± (Round to three decimal places as needed.) Test the null hypothesis. Choose the correct conclusion below. Do not reject the null hypothesis. Reject the null hypothesis.

Solution Steps

The null and alternative hypotheses for the test are defined as follows:

• Null hypothesis: \( H_{0}: p_{1} = p_{2} \)
• Alternative hypothesis: \( H_{1}: p_{1} \neq p_{2} \)

The test statistic \( z_{0} \) is calculated using the formula for the difference in proportions. The values obtained are:

• Sample proportion for group 1: \( \hat{p}_{1} = \frac{28}{254} \)
• Sample proportion for group 2: \( \hat{p}_{2} = \frac{36}{301} \)
• Combined sample proportion: \( \hat{p} = \frac{28 + 36}{254 + 301} \)

The calculated test statistic is: \[ z_{0} = -0.34 \]

The critical values for a two-tailed test at the significance level \( \alpha = 0.01 \) are determined using the Z critical value formula: \[ Z = \Phi^{-1}\left(1 - \frac{\alpha}{2}\right) \] The critical values are: \[ \pm 2.576 \]

To determine whether to reject the null hypothesis, we compare the absolute value of the test statistic with the critical value: \[ |z_{0}| = 0.34 < 2.576 \] Since the absolute value of the test statistic is less than the critical value, we do not reject the null hypothesis.

Final Answer