Question

Construct a​ 95% confidence interval to estimate the population proportion using the data below.    

x equals 23, n equals 80, N equals 500

The​ 95% 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 that,

N = 500

n = 80

x = 23

Point estimate = sample proportion = = x / n = 23 / 80 = 0.288

1 - = 1 - 0.288 = 0.712

n(0.05) N and np(1-p) > 10

80(0.05) 500 = 4 500 and 80*0.288*0.712 > 10 = 16.40 > 10

The conditions are approximately normal.

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 (((0.288 * 0.712) / 80 )

= 0.099

A 95% confidence interval for population proportion p is ,

± E

= 0.288  ± 0.099

= ( 0.189, 0.387 )