Question

The number of men and women among professors in Math, Physics, Chemistry, Linguistics, and English departments from a SRS of small colleges were counted, and the results are shown in the table below.

Dept. Math. Physics. Chemistry. Linguistics English

Men 58 74 36 12 25

Women. 5 7 10 11 20

Test the claim that the gender of a professor is independent of the department. Use the significance level 𝛼=0.01

(a) The test statistic is 𝜒2 =

 

(b) The critical value is 𝜒2 =

Answer 1

The hypotheses tested are,

$H_0:$ Gender of a professor is independent of the department.

$Against$

$H_1:$ gender of a professor is not independent of the department.

We first determine the expected count for each cell using the formula below,

$E_{ij}=(r_i*c_j)/n, \space i=1,2\space \& \space j=1,2,3,4,5$, where $r_i$ is the corresponding row total for each cell and $c_j$ is the corresponding column total for each cell. $n=258$ is the sample size.

The expected counts are as follows,

$E_{11}=(r_1*c_1)/n=(63*205)/258=50.06$

$E_{12}=(r_1*c_2)/n=(81*205)/258=64.36$

$E_{13}=(r_1*c_3)/n=(46*205)/258=36.55$

$E_{14}=(r_1*c_4)/n=(23*205)/258=18.28$

$E_{15}=(r_1*c_5)/n=(45*205)/258=35.76$

$E_{21}=(r_2*c_1)/n=(63*53)/258=12.94$

$E_{22}=(r_2*c_2)/n=(81*53)/258=16.64$

$E_{23}=(r_2*c_3)/n=(46*53)/258=9.45$

$E_{24}=(r_2*c_4)/n=(23*53)/258=4.72$

$E_{25}=(r_2*c_5)/n=(45*53)/258=9.24$

Next is to determine the test statistic given as,

$\chi^2_*=\displaystyle\sum^2_{i=1}\displaystyle\sum^5_{j=1}(O_{ij}-E_{ij})^2/E_{ij}$

Now,

$\chi^2_*=(58-50.06)^2/50.05+(74-64.36)^2/64.36+(36-36.55)^2/36.55+(12-18.28)^2/18.28+(25-35.76)^2/35.76+(5-12.94)^2/12.94+(7-16.64)^2/16.64+(10-9.45)^2/9.45+(11-4.72)^2/4.72+(20-9.24)^2/9.24=39.4412(4dp)$

$\chi^2_*$ is compared with the table value at $\alpha$ level of significance with $(r-1)*(c-1)=(2-1)*(5-1)=1*4=4$ degrees of freedom.

The table value is $\chi^2_{\alpha=0.01,4}=13.2767$ and the null hypothesis is rejected if, $\chi^2_*\gt\chi^2_{0.01,4}.$

Since $\chi^2_*=39.4412\gt\chi^2_{0.01,4}=13.2767,$ we reject the null hypothesis and conclude that there is no sufficient evidence to show that the gender of a professor is independent of the department at 1% significance level.