Question

In a survey of

607

males ages​ 18-64,

393

say they have gone to the dentist in the past year.

Construct​ 90% and​ 95% confidence intervals for the population proportion. Interpret the results and compare the widths of the confidence intervals. If​ convenient, use technology to construct the confidence intervals.

The​ 90% confidence interval for the population proportion p is

​(______,______​). ​(Round to three decimal places as​ needed.)

The​ 95% confidence interval for the population proportion p is

​(______,_______). ​(Round to three decimal places as​ needed.)

Interpret your results of both confidence intervals.

A. With the given​ confidence, it can be said that the population proportion of males ages​ 18-64 who say they have gone to the dentist in the past year is between the endpoints of the given confidence interval.

B. With the given​ confidence, it can be said that the population proportion of males ages​ 18-64 who say they have gone to the dentist in the past year is not between the endpoints of the given confidence interval.

C.With the given​ confidence, it can be said that the sample proportion of males ages​ 18-64 who say they have gone to the dentist in the past year is between the endpoints of the given confidence interval.

D. Which interval is​ wider?

(a) The​ 90% confidence interval

(b) The​ 95% confidence interval

Answer 1

Solution:

The point estimate is 393/628=0.626

90% confidence interval has a z-value of 1.645 (5% in each tail) so now use the confidence interval formula:

CI = (mean proportion) +/- z* (sample standard deviation) where z* is 1.645 for 90% confidence:

(0.594,0.658) is the 90% CI

95% confidence interval has a z-value of 1.96 so now use the confidence interval formula:

CI = (mean proportion) +/- z* (sample standard deviation)

(0.588,0.664) is the 95% CI

The correct choice is B. CI are statements about the population. We know what the sample was; we are 90% or 95% confident that the true proportion, which we don't know and never will know, is in the interval constructed. It is not a probability.

d)
The 95% interval is wider, and one needs a wider interval if one wants to be more confident. One way to remember that is one needs a 100% interval if one wants to be 100% confident.