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A laundry detergent company wants to determine if a new formula of detergent, A, cleans better than the original formula, B.

- Researchers randomly assign 500 pieces of similarly soiled clothes to the two detergents, with 250 pieces in each group.
- After washing the clothes, independent reviewers rate the cleanliness of the clothes on a scale of 1-10, with 10 being the cleanest.
- The researchers calculate the proportion of clothes in each group that receive a rating of 7 or higher.

Results:
- For detergent A, 228 pieces of clothing received a 7 or higher.
- For detergent B, 210 pieces of clothing received a rating of 7 or higher.

Let [tex]\( p_A \)[/tex] be the true proportion of clothes receiving a rating of 7 or higher for detergent A and [tex]\( p_B \)[/tex] be the true proportion for detergent B.

Which of the following is the correct standardized test statistic and P-value for the hypotheses [tex]\( H_0: p_A - p_B = 0 \)[/tex] and [tex]\( H_A: p_A - p_B \ \textgreater \ 0 \)[/tex]?

A. [tex]\( z = \frac{0.912 - 0.84}{\sqrt{\frac{(0.876)(0.124)}{500}}}, P\)[/tex]-value = 0.014

B. [tex]\( z = \frac{0.912 - 0.84}{\sqrt{\frac{(0.876)(0.124)}{250} + \frac{(0.876)(0.124)}{250}}}, P\)[/tex]-value = 0.007



Answer :

GinnyAnswer
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]

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