Question

The 90 students in a statistics class are categorized by gender and by the year in school. The numbers are listed in the following table:

Year in School	Freshman	Sophmore	Junior	Senior
Gender
Male	1	4	8	17
Female	23	17	13	7

Test the null hypothesis that there is no association between the year in school and the gender using a 1% significance level. Be sure to specify the test statistic with degrees of freedom, the P-value or critical value, and your conclusion. Please no computer software answers! Thank you!

Answer 1

Solution:

    We are given that: The 90 students in a statistics class are categorized by gender and by the year in school. The numbers are listed in the following table:

Year in School	Freshman	Sophmore	Junior	Senior
Gender
Male	1	4	8	17
Female	23	17	13	7

We have to test the null hypothesis that there is no association between the year in school and the gender using a 1% significance level.

Step 1) State H0 and H1:

H0: there is no association between the year in school and the gender

Vs

H1: there is association between the year in school and the gender.

Step 2) Test statistic:

Formula:

where Oij = Observed frequencies and Eij = Expected Frequencies

	Year in School	Freshman	Sophmore	Junior	Senior	Row Totals
Gender	Male	1	4	8	17	R1=30
	Female	23	17	13	7	R2=60
Column Totals		C1=24	C2=21	C3=21	C4=24	N= 90

Thus we need to make following table:

Oij	Eij	Oij^2 / Eij
1	8	0.125
4	7	2.285714
8	7	9.142857
17	8	36.125
23	16	33.0625
17	14	20.64286
13	14	12.07143
7	16	3.0625

Thus Chi-square test statistic is:

Step 3) Chi-square critical value:

df = (R-1)x (C-1) = ( 2-1) x ( 4-1) = 1 x 3 = 3

df = 3

Level of significance = 1% = 0.01

Chi-square critical value = 11.345

Step 4) Decision rule: Reject H0, if Chi-square test statistic value > Chi-square critical value, otherwise we fail to reject H0.

Since Chi-square test statistic value = > Chi-square critical value = 11.345, we reject H0.

Step 5) Conclusion: Since we have rejected H0, we conclude that there is association between the year in school and the gender using a 1% significance level.