Question

In a survey of 624 males ages​ 18-64, 397 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.

(a) The​ 90% confidence interval for the population proportion p is (___,___)
(Round to three decimal places as​ needed.)

(b) The​ 95% confidence interval for the population proportion p is (____,____)
​(Round to three decimal places as​ needed.)

(c) Interpret your results of both confidence intervals:

1. 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.

2. 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.

3. 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.

(d) Which interval is​ wider?

The​ 90% confidence interval

or

The​ 95% confidence interval

Answer 1

Solution :

Given that,

n = 624

x = 397

Point estimate = sample proportion = = x / n = 397 / 624 = 0.636

1 - = 1 - 0.636 = 0.364

a) At 90% confidence level

= 1 - 90%

=1 - 0.90 =0.10

/2 = 0.05

Z/2 = Z_{0.05 = 1.645}

Margin of error = E = Z _{/ 2} * (( * (1 - )) / n)

= 1.645 (((0.636 * 0.364) / 624)

= 0.032

A 90% confidence interval for population proportion p is ,

± E

= 0.636  ± 0.032

= ( 0.604, 0.668 )

b) At 95% confidence level

= 1 - 95%

=1 - 0.95 =0.05

/2 = 0.025

Z/2 = Z_{0.025 = 1.960}

Margin of error = E = Z _{/ 2} * (( * (1 - )) / n)

= 1.96 (((0.636 * 0.364) / 624)

= 0.038

A 95% confidence interval for population proportion p is ,

± E

= 0.636  ± 0.038

= ( 0.598, 0.674 )

c) 1. 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.

d) The​ 95% confidence interval is wider