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Sagot :
### Step-by-Step Solution
We are testing the claim that vinyl gloves have a greater virus leak rate than latex gloves using a significance level of 0.10. Let's go through the hypothesis test step-by-step.
#### Step 1: State the Hypotheses
We need to set up our null and alternative hypotheses.
- Null Hypothesis ([tex]${H_0}$[/tex]): [tex]\( p_1 = p_2 \)[/tex]
This means that the proportion of leaks in vinyl gloves is equal to the proportion of leaks in latex gloves.
- Alternative Hypothesis ([tex]${H_1}$[/tex]): [tex]\( p_1 > p_2 \)[/tex]
This means that the proportion of leaks in vinyl gloves is greater than the proportion of leaks in latex gloves.
#### Step 2: Collect and Summarize the Data
Given data:
- Number of vinyl gloves: [tex]\( n_1 = 252 \)[/tex]
- Number of latex gloves: [tex]\( n_2 = 252 \)[/tex]
- Proportion of leaks in vinyl gloves: [tex]\( \hat{p}_1 = 0.63 \)[/tex]
- Proportion of leaks in latex gloves: [tex]\( \hat{p}_2 = 0.08 \)[/tex]
#### Step 3: Calculate the Pooled Proportion
The pooled proportion ([tex]\( p_{\text{pool}} \)[/tex]) is calculated using the combined data from both groups.
[tex]\[ p_{\text{pool}} = \frac{\hat{p}_1 \cdot n_1 + \hat{p}_2 \cdot n_2}{n_1 + n_2} \][/tex]
Substituting the values:
[tex]\[ p_{\text{pool}} = \frac{0.63 \cdot 252 + 0.08 \cdot 252}{252 + 252} = 0.355 \][/tex]
#### Step 4: Calculate the Standard Error
The standard error (SE) for the difference in proportions is calculated as follows:
[tex]\[ SE = \sqrt{p_{\text{pool}} \cdot (1 - p_{\text{pool}}) \cdot \left(\frac{1}{n_1} + \frac{1}{n_2}\right)} \][/tex]
Substituting the values:
[tex]\[ SE = \sqrt{0.355 \cdot (1 - 0.355) \cdot \left(\frac{1}{252} + \frac{1}{252}\right)} = 0.0426 \][/tex]
#### Step 5: Calculate the Test Statistic
The test statistic (z) is calculated as:
[tex]\[ z = \frac{\hat{p}_1 - \hat{p}_2}{SE} \][/tex]
Substituting the values:
[tex]\[ z = \frac{0.63 - 0.08}{0.0426} \approx 12.90 \][/tex]
#### Step 6: Determine the P-value
Using the z-value from the test statistic, we can find the P-value. The P-value corresponds to the area under the normal curve to the right of the computed z-value for a one-tailed test.
Given the z-value of 12.90, the P-value is as follows:
[tex]\[ P-value \approx 0.000 \][/tex]
Since small P-values imply strong evidence against the null hypothesis, this P-value indicates extremely strong evidence that the virus leak rate for vinyl gloves is greater than that of latex gloves.
#### Step 7: Conclusion
Compare the P-value with the significance level ([tex]\(\alpha = 0.10\)[/tex]):
- The P-value (0.000) is less than our significance level of 0.10.
Therefore, we reject the null hypothesis.
### Final Answer
The P-value is less than the significance level [tex]\(\alpha\)[/tex], so we reject the null hypothesis. There is sufficient evidence to support the claim that vinyl gloves have a greater virus leak rate than latex gloves.
We are testing the claim that vinyl gloves have a greater virus leak rate than latex gloves using a significance level of 0.10. Let's go through the hypothesis test step-by-step.
#### Step 1: State the Hypotheses
We need to set up our null and alternative hypotheses.
- Null Hypothesis ([tex]${H_0}$[/tex]): [tex]\( p_1 = p_2 \)[/tex]
This means that the proportion of leaks in vinyl gloves is equal to the proportion of leaks in latex gloves.
- Alternative Hypothesis ([tex]${H_1}$[/tex]): [tex]\( p_1 > p_2 \)[/tex]
This means that the proportion of leaks in vinyl gloves is greater than the proportion of leaks in latex gloves.
#### Step 2: Collect and Summarize the Data
Given data:
- Number of vinyl gloves: [tex]\( n_1 = 252 \)[/tex]
- Number of latex gloves: [tex]\( n_2 = 252 \)[/tex]
- Proportion of leaks in vinyl gloves: [tex]\( \hat{p}_1 = 0.63 \)[/tex]
- Proportion of leaks in latex gloves: [tex]\( \hat{p}_2 = 0.08 \)[/tex]
#### Step 3: Calculate the Pooled Proportion
The pooled proportion ([tex]\( p_{\text{pool}} \)[/tex]) is calculated using the combined data from both groups.
[tex]\[ p_{\text{pool}} = \frac{\hat{p}_1 \cdot n_1 + \hat{p}_2 \cdot n_2}{n_1 + n_2} \][/tex]
Substituting the values:
[tex]\[ p_{\text{pool}} = \frac{0.63 \cdot 252 + 0.08 \cdot 252}{252 + 252} = 0.355 \][/tex]
#### Step 4: Calculate the Standard Error
The standard error (SE) for the difference in proportions is calculated as follows:
[tex]\[ SE = \sqrt{p_{\text{pool}} \cdot (1 - p_{\text{pool}}) \cdot \left(\frac{1}{n_1} + \frac{1}{n_2}\right)} \][/tex]
Substituting the values:
[tex]\[ SE = \sqrt{0.355 \cdot (1 - 0.355) \cdot \left(\frac{1}{252} + \frac{1}{252}\right)} = 0.0426 \][/tex]
#### Step 5: Calculate the Test Statistic
The test statistic (z) is calculated as:
[tex]\[ z = \frac{\hat{p}_1 - \hat{p}_2}{SE} \][/tex]
Substituting the values:
[tex]\[ z = \frac{0.63 - 0.08}{0.0426} \approx 12.90 \][/tex]
#### Step 6: Determine the P-value
Using the z-value from the test statistic, we can find the P-value. The P-value corresponds to the area under the normal curve to the right of the computed z-value for a one-tailed test.
Given the z-value of 12.90, the P-value is as follows:
[tex]\[ P-value \approx 0.000 \][/tex]
Since small P-values imply strong evidence against the null hypothesis, this P-value indicates extremely strong evidence that the virus leak rate for vinyl gloves is greater than that of latex gloves.
#### Step 7: Conclusion
Compare the P-value with the significance level ([tex]\(\alpha = 0.10\)[/tex]):
- The P-value (0.000) is less than our significance level of 0.10.
Therefore, we reject the null hypothesis.
### Final Answer
The P-value is less than the significance level [tex]\(\alpha\)[/tex], so we reject the null hypothesis. There is sufficient evidence to support the claim that vinyl gloves have a greater virus leak rate than latex gloves.
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