Question

A researcher selects 100 subjects at random from a population, observes 50 successes, and calculates three confidence intervals. The confidence levels are 90%, 95%, and 99%, and the intervals are (0.402, 0.598), (0.371, 0.629), and (0.418, 0.582). Match each interval with its confidence level.

Answer 1

Solution :

Given that,

Point estimate = sample proportion = = x / n = 50 / 100 = 0.50

1 - = 1 - 0.50 = 0.50

Z/2 = Z0.05 = 1.645

Margin of error = E = Z / 2 * (( * (1 - )) / n)

= 1.645 (((0.50 * 0.50) / 100)

= 0.082

A 90% confidence interval for population proportion p is ,

± E

= 0.50   ± 0.082

= ( 0.418, 0.582 )

Z/2 = Z0.025 = 1.96

Margin of error = E = Z / 2 * (( * (1 - )) / n)

= 1.96 (((0.50 * 0.50) / 100)

= 0.098

A 95% confidence interval for population proportion p is ,

± E

= 0.50   ± 0.098

= ( 0.402, 0.598 )

Z/2 = Z0.005 = 2.576  

Margin of error = E = Z / 2 * (( * (1 - )) / n)

= 2.576 (((0.50 * 0.50) / 100)

= 0.129

A 99% confidence interval for population proportion p is ,

± E

= 0.50   ± 0.129

= ( 0.371, 0.629 )