In: Statistics and Probability
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.
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
= Z0.05 = 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
= Z0.025 = 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.