Chapter 9: Inference for Two–Way Tables

Section 9.2: The Chi–Square Test

This test uses the observed counts and expected counts. The chi–square statistic is: .

The fraction part of the statistic is calculated for each cell of a table. Then these fraction parts are added together to form the chi–square statistic. This is a measure of how far the observed counts are from the expected counts. As the distance between theses counts increases, the chi–square statistic increases.

Large values of χ² are evidence against H₀.

The χ² statistic is easy to find on the TI83 calculator.

The chi–square distributions

The chi–square distributions are a family of right–skewed distributions that take on only positive values. A specific chi–square distribution is described by its degrees of freedom: d.f. = (r – 1)(c – 1).

The mean of the chi–square distribution is equal to d.f.!

The P–value is the area to the right of χ² under the chi–square density function.

More uses of the chi–square test

The chi–square test can be used to compare:

1. two or more populations
2. two or more independent SRSs from same population
3. individuals in a single SRS whom are classified according to both of the categorical variables

The chi–square test:

1. State the hypotheses for the "many–tailed test":
   • H₀: p₁ = p₂ = p₃ = ··· = p_n
     H_a: not all of p₁, p₂, p₃, ..., p_n are equal
2. Calculate the P–value
3. Make your conclusion

Cell counts required for the chi–square test

The following are required in order to use the chi–square test:

1. no more than 20% of expected counts are less than 5
2. all individual expected counts are at least 1
3. all four counts in a 2 × 2 table are at least 5

The chi–square test and the z test

We can make any r × c table into a r × 2 table by making two columns labeled, successes and failures. Then, if r = 2 (i.e., a 2 × 2 table), there are two options to do this comparison:

1. chi–square test with d.f. = 1
2. two–sample proportion z test

• the z test permits a one–tailed test
• the z test is related to the CI for p₁ – p₂