Question

In a survey of 3939 ​adults, 705 say they have seen a ghost. Construct a​ 99% confidence interval for the population proportion. Interpret the results.

A​ 99% confidence interval for the population proportion is ​( ​, ​). ​(Round to three decimal places as​ needed.) Interpret your results.

Choose the correct answer below.

A. With​ 99% confidence, it can be said that the sample proportion of adults who say they have seen a ghost is between the endpoints of the given confidence interval.

B. With​ 99% confidence, it can be said that the population proportion of adults who say they have seen a ghost is between the endpoints of the given confidence interval.

C. With​ 99% probability, the population proportion of adults who say they have not seen a ghost is between the endpoints of the given confidence interval.

D. The endpoints of the given confidence interval show that​ 99% of adults have seen a ghost.

Answer 1

Solution:

Here, we have to construct the 99% confidence interval for the population proportion.

Confidence interval for Population Proportion

Confidence Interval = P ± Z* sqrt(P*(1 – P)/n)

Where, P is the sample proportion, Z is critical value, and n is sample size.

We are given n = 3939, X = 705, P = X/n = 705/3939 = 0.178979436

Confidence level = 99%

Critical Z value = 2.5758 (by using z-table)

Confidence Interval = P ± Z* sqrt(P*(1 – P)/n)

Confidence Interval = 0.178979436 ± 2.5758* sqrt(0.178979436*(1 – 0.178979436)/3939)

Confidence Interval = 0.178979436 ± 0.0157

Lower limit = 0.178979436 - 0.0157 = 0.163

Upper limit = 0.178979436 + 0.0157 = 0.195

Confidence interval = (0.163, 0.195)

A​ 99% confidence interval for the population proportion is (0.163, 0.195).

Interpretation:

B. With​ 99% confidence, it can be said that the population proportion of adults who say they have seen a ghost is between the endpoints of the given confidence interval.

[Note that, we find out the confidence interval for population proportion and not for a sample proportion.]