1. Test of independence: Contingency Table
Hypotheses test for independence

2. Data Collection
– Individuals are selected randomly from a population, and responses are cross-classified according to two characteristics of the individual.
Example: Individuals are classified as either left or right handed, and as either blue or brown eyes.
– The purpose is to see if there is a relationship between the two characteristics.
For instance, is there a relationship between dominant hand and eye color?

3. Test of Independence: Contingency Tables
1. Set up the null and alternative hypotheses.
H0: Two variables are independent
Ha: Two variables are dependent
2. Select a random sample and record the observed
frequency, fij , for each cell of the contingency table.
3. Compute the expected frequency, eij , for each cell.

4. Test of Independence: Contingency Tables
4. Compute the test statistic.
5. Determine the rejection rule.
Reject H0 if p -value < a or
where  is the significance level and, with n rows and m columns, there are (n - 1)(m - 1) degrees of freedom.
Note: The test statistic follows Chi-squared distribution only if all expected cell
counts are greater than or equal to 5.

5. Contingency Table (Independence) Test
Example: Finger Lakes Homes (B)
Each home sold by Finger Lakes
Homes can be classified according to
price and to style. Finger Lakes’
manager would like to determine if
the price of the home and the style of
the home are independent variables.

6. Contingency Table (Independence) Test
Example: Finger Lakes Homes (B)
The number of homes sold for
each model and price for the past two
years is shown below. For convenience,
the price of the home is listed as either
$99,000 or less or more than $99,000.
Price Colonial Log Split-Level A-Frame
< $99,
> $99,

7. Contingency Table (Independence) Test
Hypotheses
H0: Price of the home is independent of the
style of the home that is purchased
Ha: Price of the home is not independent of the
style of the home that is purchased

8. Contingency Table (Independence) Test
Expected Frequencies
Price Colonial Log Split-Level A-Frame Total
< $99K
> $99K
Total 55 45

9. Contingency Table (Independence) Test
Rejection Rule
With  = .05 and (2 - 1)(4 - 1) = 3 d.f.,
Reject H0 if p-value < .05 or 2 > 7.815
Test Statistic
= =

10. Contingency Table (Independence) Test
Conclusion Using the p-Value Approach
Area in Upper Tail
c2 Value (df = 3)
Because c2 = is between and 9.348, the
area in the upper tail of the distribution is between
.05 and .025.
The p-value < a . We can reject the null hypothesis.

11. Contingency Table (Independence) Test
Conclusion Using the Critical Value Approach
c2 = > 7.815
We reject, at the .05 level of significance,
the assumption that the price of the home is
independent of the style of home that is
purchased.