2.10 The Chi-Squared Test of Independence

The Chi-Squared Test is used to find if two factors from the same sample are independent.

Some examples:

• Income and voting intentions
• Gender and money earning capacity

The test examines the difference between the observed and expected values

, where is the observed frequency, is the expected frequency

Ex 1 The following table shows the results from a survey of 400 people. (these are the f₀ values)

	Regular Exercise	No Exercise	Sum
Male	112	104	216
Female	96	88	184
Sum	208	192	400

We are going to examine whether or not regular exercise and gender are independent

We have the f₀ values, we need to calculate the f_e values.

Consider a general 2 x 2 table

	A1	A2	Sum
B1	p	q	w
B2	r	s	x
Sum	y	z	n

So, for independence,

To change a probability to an expected value, multiply by n.

This means the expected value of is

So our general table is going to look like

	A1	A2	Sum
B1			w
B2			x
Sum	y	z	n

This table tells us the general f_e values

	A1	A2	Sum
B1			216
B2			184
Sum	208	192	400

On the GDC,

f0	fe	f0 - fe	(f0 - fe)^2	(f0 - fe)^2/ fe
112	112.3
104	103.7
96	95.7
88	88.3
			Total =	0.00363

Since is very small (more on this later), then the two factors have been determined to be independent. This calculation is dependent on the degrees of freedom for our table.

, where r is the number of rows and c is the number of columns.

More on this next class.

p. 593# 1-4

Review of Probs: p. 490# 1-9, p. 492# 1-9, (p. 493# 1-9)