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.) interpret your results. choose the correct answer below. a. with 95% confidence, it can be said that the population proportion of adults who believe in ufos is between the endpoints of the given confidence interval. b. the endpoints of the given confidence interval shows that 95% of adults believe in ufos. c. with 95% probability, the population proportion of adults who do not believe in ufos is between the endpoints of the given confidence interval. d. with 95% confidence, it can be said that the sample proportion of adults who believe in ufos is between the endpoints of the given confidence interval.

Explanation:

Step1: Calculate sample proportion $\hat{p}$

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: Calculate $q = 1-\hat{p}$

$q=1 - 0.323=0.677$

Step3: Find $z$-value for 95% confidence interval

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

Step4: Calculate margin of error $E$

Margin of error formula is $E = z_{\alpha/2}\sqrt{\frac{\hat{p}q}{n}}$
$E=1.96\sqrt{\frac{0.323\times0.677}{2302}}$
First calculate $\frac{0.323\times0.677}{2302}=\frac{0.218771}{2302}\approx0.000095$
$\sqrt{0.000095}\approx0.0097$
$E = 1.96\times0.0097\approx0.019$

Step5: Calculate confidence interval

Confidence interval is $\hat{p}-E$0.323- 0.019=0.304$
$0.323+0.019 = 0.342$

A confidence interval gives a range of values within which the population parameter (here, population proportion) is likely to lie. A 95% confidence interval means that if we were to take many samples and construct confidence intervals, about 95% of them would contain the true population parameter.
Option A is correct because it correctly states that with 95% confidence the population proportion lies within the interval. Option B is wrong because the confidence interval is about the population proportion, not the percentage of adults. Option C is wrong as it is about non - believers instead of believers. Option D is wrong because it refers to the sample proportion (which is a single value, $\hat{p}=0.323$ in our case) instead of the population proportion.

Answer:

A 95% confidence interval for the population proportion is $(0.304,0.342)$

For interpretation: