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Chapter 11
Hypothesis Testing IV (Chi Square)
Chapter Outline
Introduction Bivariate Tables The Logic of Chi Square The Computation of Chi Square The Chi Square Test for Independence The Chi Square Test: An Example
Chapter Outline
An Additional Application of the Chi Square Test: The Goodness-of-Fit Test
The Limitations of the Chi Square Test Interpreting Statistics: Family Values
and Social Class
In This Presentation
The basic logic of Chi Square. The terminology used with bivariate
tables. The computation of Chi Square with
an example problem. The Five Step Model
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).
Tables
Must have a title. Cells are intersections of columns and
rows. Subtotals are called marginals. N is reported at the intersection of row
and column marginals.
Tables
Columns are scores of the independent variable. There will be as many columns as there
are scores on the independent variable. Rows are scores of the dependent
variable. There will be as many rows as there are
scores on the dependent variable.
Tables
There will be as many cells as there are scores on the two variables combined.
Each cell reports the number of times each combination of scores occurred.
TablesTitle
Rows Columns
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 sampling distribution. 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 high in gun sales
and cities high in homicide rate were low in gun sales.
As homicide rates increase, gun sales decrease. This relationship is not significant but does have a clear pattern.
Low High
High 8 (66.7%) 5 (38.5%) 13
Low 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.