Question

5. A professor analyzed the responses of two randomly selected samples of students. The first sample had 50 students from a population of Biology freshmen. The second sample had 40 students from a population of Chemistry freshmen. The professor found that 20 of the 50 biology freshmen surveyed had already taken a math course, and 20 of the 40 chemistry freshmen surveyed had already taken a math course. The professor believes that the proportion of the population of Biology freshmen who had already taken a math course is different than the proportion of the population of Chemistry freshmen who had already taken a math course. At the 5% level of significance, test the professor’s belief.

Answer 1

To Test :-

H0 :- P1 = P2
H1 :- P1 ≠ P2

p̂1 = 20 / 50 = 0.4
p̂2 = 20 / 40 = 0.5

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 = ( 20 + 20 ) / ( 50 + 40 )
p̂ = 0.4444
q̂ = 1 - p̂ = 0.5556
Z = ( 0.4 - 0.5) / √( 0.4444 * 0.5556 * (1/50 + 1/40) )
Z = -0.9487

Test Criteria :-
Reject null hypothesis if Z < -Z(α/2)
Z(α/2) = Z(0.05/2) = 1.96
Z > -Z(α/2) = -0.9487 > -1.96, hence we fail to reject the null hypothesis
Conclusion :- We Fail to Reject H0

There is not sufficient evidence to support the claim that  the proportion of the population of Biology freshmen who had already taken a math course is different than the proportion of the population of Chemistry freshmen who had already taken a math course.