Chapter 11

Hypothesis Testing IV (Chi Square)

Basic Logic

Chi Square is a test of significance based on bivariate tables.

We are looking for significant differences between the actual cell

frequencies in a table (fo) and those

that would be expected by random

chance (fe).

The relationship of homicide rate and gun sales

Low homicide

High homicide

Totals

Low gun sales

8 5 13

High gun sales

4 8 12

Totals 12 13 25

Tables

Notice the following about these tables 1. Table must have a title 2. Independent vrble must go into columns and if percentaged, must percentage within columns3. Subtotals are called marginals.4. N is reported at the intersection of row and

column marginals.

TablesTitle

Rows Column 1 Column 2

Row 1 cell a cell b Row Marginal 1

Row 2 cell c cell d Row Marginal 2

Column Marginal 1

Column Marginal 2

N

Example of Computation

Problem 11.2 Are the homicide rate and volume of gun

sales related for a sample of 25 cities?

Example of Computation The bivariate table showing the relationship

between homicide rate (columns) and gun sales (rows). This 2x2 table has 4 cells.

Low High

High 8 5 13

Low 4 8 12

12 13 25

Example of Computation

Use Formula 11.2 to find fe.

Multiply column and row marginals for each cell and divide by N. For Problem 11.2

(13*12)/25 = 156/25 = 6.24 (13*13)/25 = 169/25 = 6.76 (12*12)/25 = 144/25 = 5.76 (12*13)/25 = 156/25 = 6.24

Example of Computation Expected frequencies:

Low High

High 6.24 6.76 13

Low 5.76 6.24 12

12 13 25

Example of Computation A computational table helps organize the

computations.

fo fe fo - fe (fo - fe)2 (fo - fe)2 /fe

8 6.24

5 6.76

4 5.76

8 6.24

25 25

Example of Computation

Subtract each fe from each fo. The total of this column must be zero.

fo fe fo - fe (fo - fe)2 (fo - fe)2 /fe

8 6.24 1.76

5 6.76 -1.76

4 5.76 -1.76

8 6.24 1.76

25 25 0

Example of Computation Square each of these values

fo fe fo - fe (fo - fe)2 (fo - fe)2 /fe

8 6.24 1.76 3.10

5 6.76 -1.76 3.10

4 5.76 -1.76 3.10

8 6.24 1.76 3.10

25 25 0

Example of Computation Divide each of the squared values by the fe for that

cell. The sum of this column is chi square

fo fe fo - fe (fo - fe)2 (fo - fe)2 /fe

8 6.24 1.76 3.10 .50

5 6.76 -1.76 3.10 .46

4 5.76 -1.76 3.10 .54

8 6.24 1.76 3.10 .50

25 25 0 χ2 = 2.00

Step 1 Make Assumptions and Meet Test Requirements

Independent random samples LOM is nominal

Note the minimal assumptions. In particular, note that no assumption is made about the shape of the distribution of the parameters. The chi square test is non-parametric.

Step 2 State the Null Hypothesis

H0: The variables are independent Another way to state the H0, more

consistent with previous tests: H0: fo = fe

Step 2 State the Null Hypothesis

H1: The variables are dependent Another way to state the H1:

H1: fo ≠ fe

Step 3 Select the S. D. and Establish the C. R.

Sampling Distribution = χ2

Alpha = .05 df = (r-1)(c-1) = 1 χ2 (critical) = 3.841

Calculate the Test Statistic

χ2 (obtained) = 2.00

Step 5 Make a Decision and Interpret the Results of the Test

χ2 (critical) = 3.841 χ2 (obtained) = 2.00 The test statistic is not in the Critical

Region. Fail to reject the H0. There is no significant relationship

between homicide rate and gun sales.

Interpreting Chi Square

The chi square test tells us only if the variables are independent or not.

It does not tell us the pattern or nature of the relationship.

To investigate the pattern, compute %s within each column and compare across the columns.

Interpreting Chi Square Cities low on homicide rate were low in gun sales

and cities high in homicide rate were high in gun sales.

As homicide rates increase, gun sales increase. This relationship is not significant . The apparent pattern may be sampling error.

Low High

Low 8 (66.7%) 5 (38.5%) 13

High 4 (33.3%) 8 (61.5%) 12

12 (100%) 13 (100%) 25

The Limits of Chi Square

Like all tests of hypothesis, chi square is sensitive to sample size. As N increases, obtained chi square

increases. With large samples, trivial relationships

may be significant.

Remember: significance is not the same thing as importance.

Additional limits

If there are more than four categories in either variable, the use of chi square is questionable.

If one of the cells has a frequency less than 5 (as in our example), the use of chi square is questionable