Conduct the following test at the $\alpha=0.01$ 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 $p_{1} \neq p_{2}$. Sample data are $x_{1}=28, n_{1}=254, x_{2}=38$ and $n_{2}=301$.
(a) Determine the null and 2 ternative hypotheses. Choose the correct answer below.
$H_{0}: p_{1}=p_{2}$ versus $H_{1}: p_{1}>p_{2}$
$H_{0}: p_{1}=p_{2}$ versus $H_{1}: p_{1} \neq p_{2}$
$H_{0}: p_{1}=p_{2}$ versus $H_{1}: p_{1}
Solution
Step 1 ：First, we need to establish the null and alternative hypotheses. Given that we are testing whether \(p_{1} \neq p_{2}\), the null hypothesis is \(H_{0}: p_{1}=p_{2}\) and the alternative hypothesis is \(H_{1}: p_{1} \neq p_{2}\).
Step 2 ：Next, we calculate the test statistic using the formula: \[z = \frac{(\hat{p}_{1} - \hat{p}_{2}) - 0}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_{1}} + \frac{1}{n_{2}})}}\] where \(\hat{p}_{1}\) and \(\hat{p}_{2}\) are the sample proportions, \(n_{1}\) and \(n_{2}\) are the sample sizes, and \(\hat{p}\) is the pooled sample proportion, given by the formula: \[\hat{p} = \frac{x_{1} + x_{2}}{n_{1} + n_{2}}\]
Step 3 ：Then, we determine the critical value. The critical value for a two-tailed test at the \(\alpha=0.01\) level of significance is given by the z-score that corresponds to the upper \(\alpha/2\) percentile of the standard normal distribution. This can be found using a standard normal distribution table or a statistical calculator.
Step 4 ：Finally, we compare the test statistic and the critical value. If the absolute value of the test statistic is greater than the critical value, we reject the null hypothesis. Otherwise, we do not reject the null hypothesis.
Step 5 ：Given that x1 = 28, n1 = 254, x2 = 38, n2 = 301, and alpha = 0.01, we calculate \(\hat{p}_{1}\) = 0.11023622047244094, \(\hat{p}_{2}\) = 0.12624584717607973, and \(\hat{p}\) = 0.11891891891891893.
Step 6 ：Substituting these values into the formula for the test statistic, we find that z = -0.5804989064291958.
Step 7 ：The critical value for a two-tailed test at the \(\alpha=0.01\) level of significance is \(\pm 2.5758293035489004\).
Step 8 ：Since the absolute value of the test statistic (-0.58) is less than the critical value (2.58), we do not reject the null hypothesis.
Step 9 ：Final Answer: (a) The null and alternative hypotheses are \(H_{0}: p_{1}=p_{2}\) versus \(H_{1}: p_{1} \neq p_{2}\). (b) The test statistic \(z_{0}\) is \(\boxed{-0.58}\). (c) The critical values are \(\pm \boxed{2.58}\). We do not reject the null hypothesis.
From Solvely APP
Solution
Step 1 ：First, we need to establish the null and alternative hypotheses. Given that we are testing whether \(p_{1} \neq p_{2}\), the null hypothesis is \(H_{0}: p_{1}=p_{2}\) and the alternative hypothesis is \(H_{1}: p_{1} \neq p_{2}\).
Step 2 ：Next, we calculate the test statistic using the formula: \[z = \frac{(\hat{p}_{1} - \hat{p}_{2}) - 0}{\sqrt{\hat{p}(1-\hat{p})(\frac{1}{n_{1}} + \frac{1}{n_{2}})}}\] where \(\hat{p}_{1}\) and \(\hat{p}_{2}\) are the sample proportions, \(n_{1}\) and \(n_{2}\) are the sample sizes, and \(\hat{p}\) is the pooled sample proportion, given by the formula: \[\hat{p} = \frac{x_{1} + x_{2}}{n_{1} + n_{2}}\]
Step 3 ：Then, we determine the critical value. The critical value for a two-tailed test at the \(\alpha=0.01\) level of significance is given by the z-score that corresponds to the upper \(\alpha/2\) percentile of the standard normal distribution. This can be found using a standard normal distribution table or a statistical calculator.
Step 4 ：Finally, we compare the test statistic and the critical value. If the absolute value of the test statistic is greater than the critical value, we reject the null hypothesis. Otherwise, we do not reject the null hypothesis.
Step 5 ：Given that x1 = 28, n1 = 254, x2 = 38, n2 = 301, and alpha = 0.01, we calculate \(\hat{p}_{1}\) = 0.11023622047244094, \(\hat{p}_{2}\) = 0.12624584717607973, and \(\hat{p}\) = 0.11891891891891893.
Step 6 ：Substituting these values into the formula for the test statistic, we find that z = -0.5804989064291958.
Step 7 ：The critical value for a two-tailed test at the \(\alpha=0.01\) level of significance is \(\pm 2.5758293035489004\).
Step 8 ：Since the absolute value of the test statistic (-0.58) is less than the critical value (2.58), we do not reject the null hypothesis.
Step 9 ：Final Answer: (a) The null and alternative hypotheses are \(H_{0}: p_{1}=p_{2}\) versus \(H_{1}: p_{1} \neq p_{2}\). (b) The test statistic \(z_{0}\) is \(\boxed{-0.58}\). (c) The critical values are \(\pm \boxed{2.58}\). We do not reject the null hypothesis.
