To test the claim that vinyl gloves have a higher virus leak rate than latex gloves at a [tex]\( \alpha = 0.10 \)[/tex] significance level, we will set up the hypotheses and conduct a hypothesis test for the difference in proportions.
### Step-by-Step Solution:
1. Formulate the Hypotheses:
Null Hypothesis [tex]\( H_0 \)[/tex]: The proportion of virus leaks in vinyl gloves ([tex]\( P_1 \)[/tex]) is equal to the proportion of virus leaks in latex gloves ([tex]\( P_2 \)[/tex]).
[tex]\[
H_0: P_1 = P_2
\][/tex]
Alternative Hypothesis [tex]\( H_1 \)[/tex]: The proportion of virus leaks in vinyl gloves ([tex]\( P_1 \)[/tex]) is greater than the proportion of virus leaks in latex gloves ([tex]\( P_2 \)[/tex]).
[tex]\[
H_1: P_1 > P_2
\][/tex]
Based on the problem, the correct hypotheses are:
[tex]\[
\text{D:} \quad H_0: P_1 = P_2 \quad \text{and} \quad H_1: P_1 > P_2
\][/tex]
2. Identify the Test Statistic:
The test statistic for comparing two proportions is calculated using the z-score formula for proportions. The provided test statistic (z-score) is:
[tex]\[
z = 8.2173 \quad \text{(rounded to four decimal places)}
\][/tex]
When rounded to two decimal places:
[tex]\[
z \approx 8.22
\][/tex]
3. Identify the P-value:
The p-value is determined based on the calculated z-score. For a one-tailed test to the right (since [tex]\( H_1: P_1 > P_2 \)[/tex]):
[tex]\[
p\text{-value} = P(Z > 8.2173) = 0.0
\][/tex]
When rounded to three decimal places:
[tex]\[
p\text{-value} = 0.000
\][/tex]
### Summary:
- Null and Alternative Hypotheses:
[tex]\[
\text{D:} \quad H_0: P_1 = P_2 \quad \text{and} \quad H_1: P_1 > P_2
\][/tex]
- Test Statistic:
[tex]\[
z \approx 8.22 \quad \text{(rounded to two decimal places)}
\][/tex]
- P-value:
[tex]\[
p\text{-value} = 0.000 \quad \text{(rounded to three decimal places)}
\][/tex]
Given the extremely low p-value (0.0), which is less than any reasonable significance level, including [tex]\( \alpha = 0.10 \)[/tex], we reject the null hypothesis. This strongly indicates that vinyl gloves have a significantly higher virus leak rate compared to latex gloves.
