Question

In a survey of 2536 ​adults, 1477 say they have started paying bills online in the last year. Construct a​ 99% confidence interval for the population proportion. Interpret the results. A​ 99% confidence interval for the population proportion is left parenthesis nothing comma nothing right parenthesis . ​(Round to three decimal places as​ needed.)

Answer 1

Solution:

Given,

n = 2536 ....... Sample size

x = 1477 .......no. of successes in the sample

Let denotes the sample proportion.

     = x/n   = 1477 / 2536 = 0.5824

Our aim is to construct 99% confidence interval.

c = 0.99

= 1 - c = 1- 0.99 = 0.01

  /2 = 0.005 and 1- /2 = 0.995

Search the probability 0.995 in the Z table and see corresponding z value

= 2.576   

Now , the margin of error is given by

E = /2 *  

= 2.576 * [0.5824 *(1 - 0.5824)/2536]

= 0.025

Now the confidence interval is given by

( - E)   ( + E)

(0.5824 - 0.025)   (0.5824 + 0.025)

0.557    0.607

(0.557 , 0.607)

A​ 99% confidence interval for the population proportion is (0.557 , 0.607)

Interpretation: There is 95% chance that the population proportion will be in the interval  (0.557 , 0.607)