Question

3 multiple choice 1.42 points
one of the most common places where people have met their significant other is at the work place. according to a recent survey of 600 married people, 186 of them met their spouse at work. construct and interpret a 95% confidence interval for the proportion of married people that met their spouse at work. (make sure to verify the requirements before constructing the confidence interval.)
○ we are 95% confident that the proportion of married people that met their spouse at work will lie between 0.25 and 0.37.
○ we are 95% confident that the proportion of married people that met their spouse at work will lie between 0.26 and 0.36.
○ we are 95% confident that the proportion of married people that met their spouse at work will lie between 0.28 and 0.34.
○

1. we are 95% confident that the proportion of married people that met their spouse at work will lie between 0.27 and 0.35.

Explanation:

Step1: Calculate sample proportion

$\hat{p} = \frac{186}{600} = 0.31$

Step2: Verify normality conditions

$n\hat{p} = 600×0.31 = 186 ≥ 10$, $n(1-\hat{p}) = 600×0.69 = 414 ≥10$ (satisfied)

Step3: Find z-score for 95% CI

$z^* = 1.96$ (standard 95% confidence z-value)

Step4: Compute margin of error

$ME = z^*×\sqrt{\frac{\hat{p}(1-\hat{p})}{n}} = 1.96×\sqrt{\frac{0.31×0.69}{600}} ≈1.96×0.0188≈0.0368$

Step5: Construct CI

$0.31 - 0.0368 ≈0.273$, $0.31 + 0.0368≈0.3468$ (rounded to 0.27 and 0.35)

Answer:

1. We are 95% confident that the proportion of married people that met their spouse at work will lie between 0.27 and 0.35.