Question

In a survey of 2272 adults in a recent​ year, 1210 say they have made a New​ Year's resolution.

What is a 90% confidence intervals

what is a 95% confidence intervals for the population proportion.

Interpret the results and compare the widths of the confidence intervals.

Answer 1

Solution:

Given,

n = 2272 ....... Sample size

x = 1210 .......no. of successes in the sample

Let denotes the sample proportion.

     = x/n   = 1210/2272 = 0.5326

1)

90% confidence interval

c = 90% = 0.90

= 1- c = 1- 0.90 = 0.10

  /2 = 0.10 2 = 0.05 and 1- /2 = 0.950

= 1.645 (use z table)

Now , the margin of error is given by

E = *  

= 1.645 * [0.5326 *(1 - 0.5326)/2272]

= 0.0172

Now the confidence interval is given by

( - E)   ( + E)

(0.5326 - 0.0172)   (0.5326 + 0.0172)

0.5154 0.5498

Required 90% Confidence Interval is (0.5154 , 0.5498)

2)

95% confidence interval

c = 95% = 0.95

= 1- c = 1- 0.95 = 0.05

  /2 = 0.05 2 = 0.025 and 1- /2 = 0.9750

= 1.96 (use z table)

Now , the margin of error is given by

E = *  

= 1.96 * [0.5326 *(1 - 0.5326)/2272]

= 0.0205

Now the confidence interval is given by

( - E)   ( + E)

(0.5326 - 0.0205)   (0.5326 + 0.0205)

0.5121   0.5531

Required 95% Confidence Interval is (0.5121 , 0.5531)

3)

Width = 2 * Margin of error

Width of 90% confidence interval is less than width of 95% confidence interval

90% confidence interval is narrower than 95% confidence interval.