Question

in a survey of 2302 adults, 743 say they believe in ufos. construct a 95% confidence interval for the population proportion of adults who believe in ufos. a 95% confidence interval for the population proportion is (□,□) (round to three decimal places as needed.)

Explanation:

Step1: Calculate sample proportion

The sample proportion $\hat{p}=\frac{x}{n}$, where $x = 743$ (number of successes) and $n=2302$ (sample size).
$\hat{p}=\frac{743}{2302}\approx0.323$

Step2: Find $z$-score

For a 95% confidence interval, the $z$-score $z_{\alpha/2}=1.96$

Step3: Calculate margin of error

The margin of error $E = z_{\alpha/2}\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}}$
Substitute $\hat{p}=0.323$, $n = 2302$, and $z_{\alpha/2}=1.96$
$E=1.96\sqrt{\frac{0.323\times(1 - 0.323)}{2302}}$
$E=1.96\sqrt{\frac{0.323\times0.677}{2302}}$
$E=1.96\sqrt{\frac{0.218671}{2302}}$
$E=1.96\sqrt{0.00009499}$
$E=1.96\times0.009746$
$E\approx0.019$

Step4: Calculate confidence interval

The confidence interval is $\hat{p}-E$0.323- 0.019$0.304

Answer:

$(0.304,0.342)$