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To answer this question, let's break it down systematically:
### Step 1: Calculate the Proportion of Clothes Receiving a Rating of 7 or Higher
For detergent A:
[tex]\[ \hat{p}_A = \frac{228}{250} = 0.912 \][/tex]
For detergent B:
[tex]\[ \hat{p}_B = \frac{210}{250} = 0.84 \][/tex]
### Step 2: Calculate the Pooled Proportion
The pooled proportion is calculated by combining the results from both groups:
[tex]\[ \hat{p}_{\text{pooled}} = \frac{228 + 210}{250 + 250} = \frac{438}{500} = 0.876 \][/tex]
The complement of the pooled proportion (i.e., the proportion of clothes not receiving a rating of 7 or higher) is:
[tex]\[ q_{\text{pooled}} = 1 - \hat{p}_{\text{pooled}} = 1 - 0.876 = 0.124 \][/tex]
### Step 3: Calculate the Standard Error
We need to calculate the standard error of the difference in proportions:
[tex]\[ \text{SE} = \sqrt{ \hat{p}_{\text{pooled}} \cdot q_{\text{pooled}} \cdot \left(\frac{1}{n_A} + \frac{1}{n_B}\right) } \][/tex]
where [tex]\( n_A \)[/tex] and [tex]\( n_B \)[/tex] are the number of observations in groups A and B, respectively:
[tex]\[ \text{SE} = \sqrt{ 0.876 \cdot 0.124 \cdot \left(\frac{1}{250} + \frac{1}{250}\right) } \][/tex]
[tex]\[ \text{SE} = \sqrt{0.876 \cdot 0.124 \cdot 0.008} \][/tex]
[tex]\[ \text{SE} = \sqrt{0.00086976} \][/tex]
[tex]\[ \text{SE} \approx 0.02947867 \][/tex]
### Step 4: Calculate the Test Statistic (z)
The test statistic (z-value) is calculated as:
[tex]\[ z = \frac{ \hat{p}_A - \hat{p}_B }{ \text{SE} } \][/tex]
[tex]\[ z = \frac{ 0.912 - 0.84 }{ 0.02947867 } \][/tex]
[tex]\[ z \approx 2.44244 \][/tex]
### Step 5: Calculate the p-value
The p-value is determined by finding the cumulative distribution function (CDF) for the calculated z-value under the standard normal distribution. For a one-tailed test:
[tex]\[ \text{p-value} = 1 - \Phi(z) \][/tex]
where [tex]\( \Phi(z) \)[/tex] is the CDF of the standard normal distribution at z.
Given [tex]\( z \approx 2.44244 \)[/tex]:
[tex]\[ \text{p-value} \approx 0.00729 \][/tex]
### Conclusion
The standardized test statistic is:
[tex]\[ z \approx 2.44244 \][/tex]
The p-value is:
[tex]\[ \text{p-value} \approx 0.00729 \][/tex]
Thus, the correct standardized test statistic and p-value for the given hypothesis test are:
[tex]\[ z = 2.44244 \][/tex]
[tex]\[ \text{p-value} = 0.00729 \][/tex]
### Step 1: Calculate the Proportion of Clothes Receiving a Rating of 7 or Higher
For detergent A:
[tex]\[ \hat{p}_A = \frac{228}{250} = 0.912 \][/tex]
For detergent B:
[tex]\[ \hat{p}_B = \frac{210}{250} = 0.84 \][/tex]
### Step 2: Calculate the Pooled Proportion
The pooled proportion is calculated by combining the results from both groups:
[tex]\[ \hat{p}_{\text{pooled}} = \frac{228 + 210}{250 + 250} = \frac{438}{500} = 0.876 \][/tex]
The complement of the pooled proportion (i.e., the proportion of clothes not receiving a rating of 7 or higher) is:
[tex]\[ q_{\text{pooled}} = 1 - \hat{p}_{\text{pooled}} = 1 - 0.876 = 0.124 \][/tex]
### Step 3: Calculate the Standard Error
We need to calculate the standard error of the difference in proportions:
[tex]\[ \text{SE} = \sqrt{ \hat{p}_{\text{pooled}} \cdot q_{\text{pooled}} \cdot \left(\frac{1}{n_A} + \frac{1}{n_B}\right) } \][/tex]
where [tex]\( n_A \)[/tex] and [tex]\( n_B \)[/tex] are the number of observations in groups A and B, respectively:
[tex]\[ \text{SE} = \sqrt{ 0.876 \cdot 0.124 \cdot \left(\frac{1}{250} + \frac{1}{250}\right) } \][/tex]
[tex]\[ \text{SE} = \sqrt{0.876 \cdot 0.124 \cdot 0.008} \][/tex]
[tex]\[ \text{SE} = \sqrt{0.00086976} \][/tex]
[tex]\[ \text{SE} \approx 0.02947867 \][/tex]
### Step 4: Calculate the Test Statistic (z)
The test statistic (z-value) is calculated as:
[tex]\[ z = \frac{ \hat{p}_A - \hat{p}_B }{ \text{SE} } \][/tex]
[tex]\[ z = \frac{ 0.912 - 0.84 }{ 0.02947867 } \][/tex]
[tex]\[ z \approx 2.44244 \][/tex]
### Step 5: Calculate the p-value
The p-value is determined by finding the cumulative distribution function (CDF) for the calculated z-value under the standard normal distribution. For a one-tailed test:
[tex]\[ \text{p-value} = 1 - \Phi(z) \][/tex]
where [tex]\( \Phi(z) \)[/tex] is the CDF of the standard normal distribution at z.
Given [tex]\( z \approx 2.44244 \)[/tex]:
[tex]\[ \text{p-value} \approx 0.00729 \][/tex]
### Conclusion
The standardized test statistic is:
[tex]\[ z \approx 2.44244 \][/tex]
The p-value is:
[tex]\[ \text{p-value} \approx 0.00729 \][/tex]
Thus, the correct standardized test statistic and p-value for the given hypothesis test are:
[tex]\[ z = 2.44244 \][/tex]
[tex]\[ \text{p-value} = 0.00729 \][/tex]
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