Question

A study was designed to compare two energy drink commercials. Each participant was shown the commercials in random order and asked to select the better one. Commercial A was selected by 54 out of 115 women and 83 out of 145 men. Give an estimate of the difference in gender proportions that favored commercial A.

Conduct a test for checking that the proportions of women and men that liked commercial A are the same versus the two-sided alternative at the 5% level.

What would be the test's result?

Answer 1

Estimate of difference = 0.4696 - 0.5724 = - 0.1028

To Test :-

H0 :- P1 = P2
H1 :- P1 ≠ P2

p̂1 = 54 / 115 = 0.4696
p̂2 = 83 / 145 = 0.5724

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 = ( 54 + 83 ) / ( 115 + 145 )
p̂ = 0.5269
q̂ = 1 - p̂ = 0.4731
Z = ( 0.4696 - 0.5724) / √( 0.5269 * 0.4731 * (1/115 + 1/145) )
Z = -1.6497

Test Criteria :-
Reject null hypothesis if Z < -Z(α/2)
Z(α/2) = Z(0.05/2) = 1.96
Z > -Z(α/2) = -1.6497 > -1.96, hence we fail to reject the null hypothesis
Conclusion :- We Fail to Reject H0

The proportions of women and men that liked commercial A are the same.