To address the problem, we'll go through the process of hypothesis testing in a step-by-step manner. Here is a detailed approach:
1. Formulate the Hypotheses:
- Null Hypothesis [tex]\( H_0 \)[/tex]: This is the hypothesis that there is no difference in the leak rates between the vinyl and latex gloves. Thus, we set [tex]\( P_1 = P_2 \)[/tex].
- Alternative Hypothesis [tex]\( H_a \)[/tex]: This is the hypothesis that the leak rate of vinyl gloves is greater than that of latex gloves, denoted as [tex]\( P_1 > P_2 \)[/tex].
Given the choices in the problem, the correct hypotheses are:
- [tex]\( H_0: P_1 = P_2 \)[/tex]
- [tex]\( H_a: P_1 > P_2 \)[/tex]
Therefore, the correct answer is:
- E. [tex]\( H_0: P_1 = P_2 \)[/tex], [tex]\( H_1: P_1 > P_2 \)[/tex]
2. Identify the Test Statistic:
To determine if the proportion of leaked vinyl gloves is significantly greater than that of latex gloves, we use a Z-test for proportions.
From the information provided:
- Number of vinyl gloves ([tex]\(n_1\)[/tex]) = 222
- Number of latex gloves ([tex]\(n_2\)[/tex]) = 222
- Proportion of vinyl gloves that leaked ([tex]\(\hat{p}_1\)[/tex]) = 0.66
- Proportion of latex gloves that leaked ([tex]\(\hat{p}_2\)[/tex]) = 0.05
- Significance level ([tex]\(\alpha\)[/tex]) = 0.10
Next, we calculate the pooled proportion ([tex]\( p_{pool} \)[/tex]):
[tex]\[ p_{pool} = \frac{(\hat{p}_1 \times n_1) + (\hat{p}_2 \times n_2)}{n_1 + n_2} \][/tex]
The calculation for the standard error ([tex]\( SE \)[/tex]) is as follows:
[tex]\[ SE = \sqrt{ p_{pool} \times (1 - p_{pool}) \times \left( \frac{1}{n_1} + \frac{1}{n_2} \right)} \][/tex]
The test statistic (Z-score) can then be calculated by:
[tex]\[ Z = \frac{\hat{p}_1 - \hat{p}_2}{SE} \][/tex]
After computing these values, the test statistic (Z-score) is derived. According to the results, the calculated Z-score is:
[tex]\[ Z \approx 13.43 \][/tex]
Therefore, the test statistic rounded to two decimal places is:
[tex]\[ 13.43 \][/tex]
This calculation indicates a very high Z-score, which we would next compare to critical Z values for the significance level ([tex]\(\alpha = 0.10\)[/tex]) to make a decision about the hypothesis. However, since the problem only asks for the identification of the null and alternative hypotheses and the computation of the test statistic, we conclude with the finalized values:
- Null Hypothesis [tex]\( H_0 \)[/tex]: [tex]\( P_1 = P_2 \)[/tex]
- Alternative Hypothesis [tex]\( H_a \)[/tex]: [tex]\( P_1 > P_2 \)[/tex]
- Test Statistic [tex]\( Z \)[/tex]: 13.43 (rounded to two decimal places)
This approach provides a step-by-step explanation that outlines the necessary parts of hypothesis formulation and test statistic calculation.
