Answered

Welcome to Westonci.ca, where curiosity meets expertise. Ask any question and receive fast, accurate answers from our knowledgeable community. Find reliable answers to your questions from a wide community of knowledgeable experts on our user-friendly Q&A platform. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.

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

Sagot :

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]
Thank you for choosing our service. We're dedicated to providing the best answers for all your questions. Visit us again. We hope this was helpful. Please come back whenever you need more information or answers to your queries. Thank you for trusting Westonci.ca. Don't forget to revisit us for more accurate and insightful answers.