Question

In studying his campaign plans, Mr. Singleton wishes to estimate the difference between men's and women's views regarding his appeal as a candidate. He asks his campaign manager to take two random independent samples and find the 90%90% confidence interval for the difference. A random sample of 703703 male voters and 688688 female voters was taken. 363363 men and 400400 women favored Mr. Singleton as a candidate. Find this confidence interval.

Step 2 of 4:

Find the critical value that should be used in constructing the confidence interval.

Answer 1

p̂1 = 363 / 703 = 0.5164
p̂2 = 400 / 688 = 0.5814

(p̂1 - p̂2) ± Z(α/2) * √( ((p̂1 * q̂1)/ n1) + ((p̂2 * q̂2)/ n2) )
Z(α/2) = Z(0.1 /2) = 1.645
Lower Limit = ( 0.5164 - 0.5814 )- Z(0.1/2) * √(((0.5164 * 0.4836 )/ 703 ) + ((0.5814 * 0.4186 )/ 688 ) = -0.1088
upper Limit = ( 0.5164 - 0.5814 )+ Z(0.1/2) * √(((0.5164 * 0.4836 )/ 703 ) + ((0.5814 * 0.4186 )/ 688 )) = -0.0212
90% Confidence interval is ( -0.1088 , -0.0212 )
( -0.1088 < ( P1 - P2 ) < -0.0212 )

Critical value   Z(α/2) = Z(0.1 /2) = 1.645 ( From standard normal table )