1. SA3202, Solution for Tutorial 5
The University Admission Data
For each department, we compute the log odds ratio, and then compute the z-test statistic based on the log odds ratio:
Male Female Total
A Yes theta=(512*19)/(313*89)=.3492, phi=-1.052
No s.e.(phi|H0)=.21374
Total z3=phi/s.e(phi|H0)=
The computation of z allows us to see if the independence test is significant or not. For Department A it seems that there is strong evidence to support p1<p2. That is, more females are admitted in this department. Similarly, we compute z for other departments, and summarize the results as follows:
Department A B C D E F z z z
The test H0:p1=p2 for all other departments are not significant. That is, overall, the admission is not sex discrimination against female. This is inconsistent with the conclusion obtained in Problem 1, Tutorial 4 using just part of the University Admission Data. Thus, use of part of the data may lead to wrong conclusions.
12/2/2018
SA3202, Solution for Tutorial 5

2. SA3202, Solution for Tutorial 5
The Death Penalty Data
Effect of Defendant’s Race
For victim White , we compute the log odds ratio, and then compute the z-test statistic based on the log odds ratio:
Victim Def. W Def. B
White Death
No Death z1=
Black Death
No Death z1= (z2 and z3 not computable)
Note that the z values are both negative (Use of the partial data z was positive as in Problem 1, Tutorial 4). That is, p1<p2: the death penalty rates are higher for Black defendants than for the White defendants. However, the difference is not statistically significant. The main point is to note the change in the direction.
In fact, “Race” does have a significant effect, but through victim’s race as showed below.
12/2/2018
SA3202, Solution for Tutorial 5

3. SA3202, Solution for Tutorial 5
The Death Penalty Data
Effect of Victim’s Race
For defendant White , we compute the log odds ratio, and then compute the z-test statistic based on the log odds ratio:
Victim
Defendant White Black
White Death
No Death z1= (z2 and z3 not computable)
Black Death
No Death z1=2.3994
So if the victim is white, there is a higher chance of getting death penalty, especially when the defendant is Black.
12/2/2018
SA3202, Solution for Tutorial 5

4. SA3202, Solution for Tutorial 5
The Chi-square test is not appropriate if the alternative is one-sided since it does not take into account the direction of the difference. The following data sets that we can’t use chi-square tests:
Vitamin C data: one sided test, to test if Vitamin C is helpful to reduce “cold”
Seat Belt data: one-sided test, to test if the seat belt is helpful to avoid injury
Death Penalty data: one-sided test, to test if there is a race bias against black
University Admission data: one-sided test, to test if there is a sex discrimination against women
12/2/2018
SA3202, Solution for Tutorial 5

5. SA3202, Solution for Tutorial 5
Proof Note that
Response G G2 Total
Positive a b a+b
Negative c d c+d
Total a+c b+d a+b+c+d
Thus, the estimated Expected Frequency Table is
Positive A B
Negative C D
Total
Where
A=(a+c)(a+b)/(a+b+c+d), B=(b+d)(a+b)/(a+b+c+d)
C=(a+c)(c+d)/(a+b+c+d), D=(b+d)(c+d)/(a+b+c+d)
Therefore
(a-A)=a-(a+c)(a+b)/(a+b+c+d)=(ad-bc)/(a+b+c+d)
(b-B)=(bc-ad)/(a+b+c+d), (c-C)=(bc-ad)/(a+b+c+d), (d-D)=(ad-bc)/(a+b+c+d)
Thus
T=(a-A)^2/A+(b-B)^2/B+(c-C)^2/C+(d-D)^2/D=(a+b+c+d)(ad-bc)^2/[(a+b)(c+d)(a+c)(b+d)]
That is, T=Z^2 as desired.
12/2/2018
SA3202, Solution for Tutorial 5