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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":
  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 P–values from both tests will be equal! In this case, it is recommended to use the z test because:


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