Question

2. A random sample of 395 people were surveyed and each person was asked to report the highest education level they obtained. The data that resulted from the survey is summarized in the following table:
    High School   Bachelors   Masters   Ph.d.   Total
Female   60   54   46   41   201
Male   40   44   53   57   194
Total   100   98   99   98   395
a. Are gender and education level dependent at 5% level of significance? (6mks)
b.State and explain two methods of studying correlation                    (4mks

Answer 1

Solution

Part (a)

Solution is based on the theory of Chi-square Test for Independence.

Final answers are given below. Back-up Theory and Details of calculations follow at the end.

DF	3	α	0.05	χ2crit	7.8147
χ2cal	8.0061	p-value	0.0459	Reject H0

Since null hypothesis of independence is rejected, we conclude that

gender and education level are NOT independent. Answer 1

Back-up Theory and Details of calculations

Suppose we have a contingency table with r rows representing r levels/grades of one attribute and c columns representing c levels/grades of another attribute.

The Chi-square Test of Independence is designed to test if the two attributes are associated.

Hypotheses

Null H₀: The two attributes, namely gender and education level are independent .

Vs

Alternative H₁: The two attributes are not independent

Test Statistic

χ² = ∑(i = 1 to r, j = 1 to c){(Oij - Eij)²/Eij}, where Oij and Eij are respectively, the observed and the expected frequencies of the ijth cell of the contingency table.

Under H₀, Eij = (Oi. x O.j)/O.., where Oi.,O.j, and O.. are respectively the ith row total, jth column total and grand total.

Calculations

Oij
	j1	j2	j3	j4	Oi.
i1	60	54	46	41	201
i2	40	44	53	57	194
O.j	100	98	99	98	395
Eij = (Oi. X O.j)/N
	j1	j2	j3	j4	Ei.
i1	50.8861	49.8684	50.3772	49.8684	201
i2	49.1139	48.1316	48.6228	48.1316	194
E.j	100	98	99	98	395
χ2ij
	j1	j2	j3	j4	Total
i1	1.6323	0.3423	0.3803	1.5771	3.9321
i2	1.6912	0.3547	0.3941	1.6340	4.0740
Total	3.3236	0.6970	0.7744	3.2111	8.0061

Distribution

Under H₀, χ² ~ χ²_n, where n = {(r - 1)(s - 1)}

Critical Value

Given level of significance as α, critical value is the upper α% point of χ²_n.

p-value

P(χ²_n > χ²_cal)

Critical value and p-value obtained using Excel Function: Statistical CHIINV and CHIDIST are given in the above table.

Decision

Since χ²_cal > χ²_crit, or equivalently, p-value < α, H₀is rejected.

Part (b)

Two methods of studying correlation

1. Chi-square Test for Independence which will ascertain if there is association

2. Theory of correlation and regression which will first ascertain if there is association and if so establish the actual correlation relationship. Answer 2

DONE