Question

Two random samples are taken, one from among first-year students and the other from among fourth-year students at a public university. Both samples are asked if they favor modifying the student Honor Code. A summary of the sample sizes and number of each group answering "yes'' are given below:

First-Years (Pop. 1):n1=93 x2=56

Fourth-Years (Pop. 2):,n2=97 x1=62

Is there evidence, at an α=0.07 level of significance, to conclude that there is a difference in proportions between first-years and fourth-years? Carry out an appropriate hypothesis test, filling in the information requested.

A. The value of the standardized test statistic:

B. The P-value is

Answer 1

To Test :-

H0 :- P1 = P2
H1 :- P1 ≠ P2

p̂1 = 56 / 93 = 0.6022
p̂2 = 62 / 97 = 0.6392

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 = ( 56 + 62 ) / ( 93 + 97 )
p̂ = 0.6211
q̂ = 1 - p̂ = 0.3789
Z = ( 0.6022 - 0.6392) / √( 0.6211 * 0.3789 * (1/93 + 1/97) )
Z = -0.5259

Test Criteria :-
Reject null hypothesis if Z < -Z(α/2)
Z(α/2) = Z(0.07/2) = 1.81
Z > -Z(α/2) = -0.5259 > -1.81, hence we fail to reject the null hypothesis
Conclusion :- We Fail to Reject H0

Decision based on P value
P value = 2 * P ( Z < -0.5259 )
P value = 0.5990
Reject null hypothesis if P value < α = 0.07
Since P value = 0.599 > 0.07, hence we fail to reject the null hypothesis
Conclusion :- We Fail to Reject H0

There is insufficient evidence to support the claim  that there is a difference in proportions between first-years and fourth-years.