Question

In 1991, 48% of of 674 Americans surveyed favored admitting Hawaii as a state, while in 1946, 60% of 739 Americans surveyed did.

(a) Test the claim that more Americans favored admitting Hawaii as a state in 1946 than in 1941 at a significance level of .05.

(b) Test the claim that the proportion of Americans who favored admitting Hawaii as a state was at least 8 percentage points higher in 1946 than it was in 1941 at a significance level of .05.

Answer 1

Part a)

To Test :-

H0 :- P1 = P2
H1 :- P1 < P2

Test Statistic :-
Z = ( p̂1 - p̂2 ) / √(p̂ * q̂ * (1/n1 + 1/n2) ) )
p̂ is the pooled estimate of the proportion P
p̂ = ( x1 + x2) / ( n1 + n2)
p̂ = ( 323.52 + 443.4 ) / ( 674 + 739 )
p̂ = 0.5428
q̂ = 1 - p̂ = 0.4572
Z = ( 0.48 - 0.6) / √( 0.5428 * 0.4572 * (1/674 + 1/739) )
Z = -4.5226

Test Criteria :-
Reject null hypothesis if Z < -Z(α)
Z(α) = Z(0.05) = 1.645
Z < -Z(α) = -4.5226 < -1.645, hence we reject the null hypothesis
Conclusion :- We Reject H0

There is sufficient evidence to support the claim that  more Americans favored admitting Hawaii as a state in 1946 than in 1941 at a significance level of .05.

Part b)

To Test :-

H0 :- P1 - P2 >= 0.08
H1 :- P1 - P2 < 0.08

Test Statistic :-
Z = ( p̂1 - p̂2 ) / √(p̂ * q̂ * (1/n1 + 1/n2) ) )

Z = -7.59

Test Criteria :-
Reject null hypothesis if Z < -Z(α)
Z(α) = Z(0.05) = 1.645
Z < -Z(α) = -7.59 < -1.645, hence we reject the null hypothesis
Conclusion :- We Reject H0

There is sufficient evidence to support the claim that  more Americans favored admitting Hawaii as a state in 1946 than in 1941 at a significance level of .05.