Question

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 0.28 and 0.34.
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 0.27 and 0.35.
we are 95% confident that the proportion of married people that met their spouse at work will lie between 0.25 and 0.37.

Explanation:

Step1: Calculate sample proportion

The sample proportion $\hat{p}=\frac{x}{n}$, where $x = 186$ (number of successes) and $n=600$ (sample size). So, $\hat{p}=\frac{186}{600}=0.31$.

Step2: Check requirements

For a confidence interval of proportion, we need $n\hat{p}\geq5$ and $n(1 - \hat{p})\geq5$.
$n\hat{p}=600\times0.31 = 186\geq5$ and $n(1-\hat{p})=600\times(1 - 0.31)=600\times0.69 = 414\geq5$.

Step3: Find critical value

For a 95% confidence interval, the critical value $z_{\alpha/2}$ is 1.96.

Step4: Calculate margin of error

The margin of error $E=z_{\alpha/2}\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$.
Substitute $\hat{p}=0.31$, $n = 600$, and $z_{\alpha/2}=1.96$.
$E=1.96\sqrt{\frac{0.31\times(1 - 0.31)}{600}}=1.96\sqrt{\frac{0.31\times0.69}{600}}=1.96\sqrt{\frac{0.2139}{600}}\approx1.96\times0.0189\approx0.037$.

Step5: Calculate confidence interval

The confidence interval is $\hat{p}-ESubstitute $\hat{p}=0.31$ and $E\approx0.037$.
$0.31- 0.037=0.273\approx0.27$ and $0.31 + 0.037=0.347\approx0.35$.

Answer:

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