Answer:
[tex]CI = (0.028636,0.071364)[/tex]
I am 95% confident that the true proportion of couples where the wife is taller than her husband is captured in the interval (.028, .071)
Step-by-step explanation:
Given
[tex]n = 400[/tex]
[tex]x = 20[/tex] --- taller wife
[tex]y = 380[/tex] --- shorter wife
Required
Determine the 95% confidence interval of taller wives
First, calculate the proportion of taller wives
[tex]\hat p = \frac{x}{n}[/tex]
[tex]\hat p = \frac{20}{400}[/tex]
[tex]\hat p = 0.05[/tex]
The z value for 95% confidence interval is:
[tex]z = 1.96[/tex]
The confidence interval is calculated as:
[tex]CI = \hat p \± z \sqrt{\frac{\hat p (1 - \hat p)}{n}}[/tex]
[tex]CI = 0.05 \± 1.96* \sqrt{\frac{0.05 (1 - 0.05)}{400}}[/tex]
[tex]CI = 0.05 \± 1.96 * \sqrt{\frac{0.0475}{400}}[/tex]
[tex]CI = 0.05 \± 1.96 * \sqrt{0.00011875}[/tex]
[tex]CI = 0.05 \± 1.96 * 0.01090[/tex]
[tex]CI = 0.05 \± 0.021364[/tex]
This gives:
[tex]CI = (0.05 - 0.021364,0.05 + 0.021364)[/tex]
[tex]CI = (0.028636,0.071364)[/tex]
