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Chapter 11 (10 in 2nd Can. Ed.)

Hypothesis Testing Using the Chi Square (χ2) Distribution

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Outline:

• The basic logic of Chi Square. • The terminology used with bivariate tables. • The computation of Chi Square with an

example problem using the Five Step Model

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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 data are often presented in a table format. If starting with raw data on two variables, a bivariate table must be created first.

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Bivariate 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.

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Tables (cont.)

• 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.

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Tables (cont.)

• 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.

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What your table should look like: Title

Rows Columns Total

Row 1 cell a cell b Row Marginal 1

Row 2 cell c cell d Row Marginal 2

Total Column Marginal 1

Column Marginal 2

N

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The Chi Square Distribution • The chi square distribution is asymmetric and its values

are always positive (Appendix C, Healey, p. 498). Degrees of freedom are based on the table and are calculated as (rows-1)X(columns-1).

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Example: Healey #11.2 (#10.2 in 2nd) • Question: Are the homicide rate and volume of gun sales

related for a sample of 25 cities?

• The bivariate table showing the relationship between homicide rate (columns) and gun sales (rows). This 2x2 table has 4 cells.

GUN SALES Low High Totals

High 8 5 13

Low 4 8 12

Totals 12 13 N = 25

HOMICIDE RATE

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Solution Using 5-Step Method
**Step 1** Make Assumptions and Meet Test Requirements

• Independent random samples • Level of measurement is nominal • Note that no assumption is made about the

shape of the sampling distribution. When the
distribution is normal, a **parametric test** (Z- or
t-test, ANOVA) can be used.

• The chi square test is **non-parametric**. It can
be used when normality is not assumed.

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**Step 2** State the Null and Alternate
Hypothesis

• H0: The variables are independent – You can also say: H0: fo = fe

• H1: The variables are dependent – Or: H1: fo ≠ fe

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**Step 3** Select the Sampling Distribution and
Establish the Critical Region

• Because normality is not assumed and our data are in tabular form, our Sampling Distribution = χ2

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

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**Step 4** Calculate the Test Statistic
• Formula:

χ2 (obtained) =

Method: 1. Find expected frequencies for each cell.

2. Complete computational table to find χ2 (obtained)

∑ −
*e
*

*eo
*

*f
ff *2)(

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1. Find expected frequencies for each cell.

• To find fe =

• Multiply column and row marginals for each cell and divide by N.

• (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

*N
*marginalcolumn marginal row ×

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Observed and Expected Frequencies for each cell (Note that totals are

unchanged): HOMICIDE RATE

GUN SALES Low High Total

High fo = 8 fe = 6.24 fo = 5

fe = 6.76 13

Low fo = 4 fe = 5.76 fo = 8

fe = 6.24 12

Total 12 13 N = 25

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2. Complete Computational Table
• A table like this will help organize the computations:
• (a) Add values for **fo **and** fe **for each cell to table.

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

8 6.24

5 6.76

4 5.76

8 6.24

**Total** 25 25

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Computational Table (cont.)

• (b) 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

**Total **25 25 0

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Computational Table (cont.) • (c) 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

**Total** 25 25 0

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Computational Table (cont.) • (d) Divide each of the squared values by the fe for that cell. • (e) The sum of this column is chi square.

• χ2 (obtained) = 2.00

**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

**Total** 25 25 0 χ2 = 2.00

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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.

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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 % within each column and compare across the columns.

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Interpreting Chi Square (cont.) • 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.

GUN SALES Low High Total

High 8 **(66.7%) **5 **(38.5%) **13

Low 4 **(33.3%) **8 **(61.5%) **12

Total 12 **(100%) **13 **(100%) **N = 25

HOMICIDE RATE

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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. To correct for this, when N>1000, set your alpha = .01.

• Remember: *significance* is not the same thing
as *importance*.

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Yates’ Correction for Continuity • The chi square statistic is sensitive to small cell sizes.

Whenever any of your cell sizes are <5, a slightly modified formula (Yates’ correction for continuity) to calculate chi square should be used (see Healey Formula 11.4)

• Modified Formula (Note:.5 is deducted from the absolute value of fo – fe before squaring)

(obtained) =

∑ −−

*e
*

*eo
*

*f
ff *2)5.0(2

cχ

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Using SPSS to Calculate Chi Square
1. Open data set in SPSS.
2. Go to **Analyze>Descriptive Statistics>Crosstabs**.
3. Move your dependent variable into the **Rows** box, and

your independent variable into the **Columns** box.
4. Click **Statistics** and check box for **Chi square**.
5. Click **Cells** and select **Column** in percentages box.
6. Click **Continue** and **OK**.

Note: The “rule of thumb” for analyzing your % data is “Percentage Down, Compare Across” When analyzing your %, always compare the categories of your dependent (Row) variable across the columns of your independent (Column) variable.

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Practice Question:

• Try Healey #11.3 (#10.3 in 2nd) parts a and b.

• Use the five step model to calculate a full solution to this question.

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