Question

In a survey of 2508 2508 adults in a recent​ year, 1395 1395 say they have made a New​ Year's resolution. Construct​ 90% and​ 95% confidence intervals for the population proportion. Interpret the results and compare the widths of the confidence intervals. The​ 90% confidence interval for the population proportion p is left parenthesis nothing comma nothing right parenthesis , . ​(Round to three decimal places as​ needed.) The​ 95% confidence interval for the population proportion p is left parenthesis nothing comma nothing right parenthesis , . ​(Round to three decimal places as​ needed.) With the given​ confidence, it can be said that the ▼ of adults who say they have made a New​ Year's resolution is ▼ between the endpoints less than the upper endpoint not between the endpoints greater than the lower endpoint of the given confidence interval. Compare the widths of the confidence intervals. Choose the correct answer below. A. The​ 90% confidence interval is wider. B. The​ 95% confidence interval is wider. C. The confidence intervals cannot be compared. D. The confidence intervals are the same width.

Answer 1

Solution :

Given that,

n = 2508

x = 1395

Point estimate = sample proportion = = x / n = 1395 / 2508 = 0.556

1 - = 1 - 0.556 = 0.444

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 (((0.556 * 0.444) / 2508)

= 0.016

A 90% confidence interval for population proportion p is ,

± E

= 0.556 ± 0.016

= (0.540, 0.572 )

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 (((0.556 * 0.444) / 2508)

= 0.019

A 95% confidence interval for population proportion p is ,

± E

= 0.556 ± 0.019

= (0.537, 0.575 )

With the given​ confidence, it can be said that the population proportion of adults who say they have made a New​ Year's resolution is between the endpoints of the given confidence intervals.

B. The​ 95% confidence interval is wider.