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, putting 250 pieces in each group. After washing the clothes, independent reviewers determine 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. 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. Assuming the conditions for inference are met, what is the confidence interval for the difference in proportions of clothes that receive a rating of 7 or higher for the two detergents?

A. [tex]$(0.91 - 0.84) \pm 1.96 \sqrt{\frac{0.91(1 - 0.91)}{250} + \frac{0.84(1 - 0.84)}{250}}$[/tex]
B. [tex]$(0.91 - 0.84) \pm 1.65 \sqrt{\frac{0.91(1 - 0.91)}{250} + \frac{0.84(1 - 0.84)}{250}}$[/tex]
C. [tex]$(0.09 - 0.16) \pm 1.96 \sqrt{\frac{0.09(1 - 0.09)}{250} + \frac{0.16(1 - 0.16)}{250}}$[/tex]
D. [tex]$(0.09 - 0.16) \pm 1.65 \sqrt{\frac{0.09(1 - 0.09)}{500} + \frac{0.16(1 - 0.16)}{500}}$[/tex]

To find the confidence interval for the difference in proportions of clothes that receive a rating of 7 or higher for the two detergents, we can follow these steps:

1. Calculate Proportions:
- For detergent [tex]$ A $[/tex]: There are 228 out of 250 pieces of clothes that received a rating of 7 or higher.
[tex]\[ p_A = \frac{228}{250} = 0.912 \][/tex]
- For detergent [tex]$ B $[/tex]: There are 192 out of 250 pieces of clothes that received a rating of 7 or higher.
[tex]\[ p_B = \frac{192}{250} = 0.768 \][/tex]

2. Difference in Proportions:
- The difference in proportions between the two detergents:
[tex]\[ \text{Difference in proportions} = p_A - p_B = 0.912 - 0.768 = 0.144 \][/tex]

3. Standard Error Calculation:
- The standard error for the difference in proportions can be calculated using the formula:
[tex]\[ SE = \sqrt{\frac{p_A (1 - p_A)}{n_A} + \frac{p_B (1 - p_B)}{n_B}} \][/tex]
- Here, [tex]$ n_A = n_B = 250 $[/tex]:
[tex]\[ SE = \sqrt{\frac{0.912 \times (1 - 0.912)}{250} + \frac{0.768 \times (1 - 0.768)}{250}} = \sqrt{\frac{0.080736}{250} + \frac{0.177536}{250}} = \sqrt{0.000322944 + 0.000710144} = \sqrt{0.001033088} \][/tex]
[tex]\[ SE = 0.03215164070463589 \][/tex]

4. Margin of Error (MOE):
- For a 95% confidence interval, the z-value is 1.96:
[tex]\[ MOE = z \times SE = 1.96 \times 0.03215164070463589 = 0.06301721578108635 \][/tex]

5. Confidence Interval:
- Finally, the confidence interval for the difference in proportions is calculated as follows:
[tex]\[ \text{Lower bound} = \text{Difference in proportions} - MOE = 0.144 - 0.06301721578108635 = 0.08098278421891367 \][/tex]
[tex]\[ \text{Upper bound} = \text{Difference in proportions} + MOE = 0.144 + 0.06301721578108635 = 0.20701721578108637 \][/tex]

Therefore, the 95% confidence interval for the difference in proportions of clothes that receive a rating of 7 or higher for the two detergents is [tex]$(0.08098278421891367, 0.20701721578108637)$[/tex].