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].
