### Chapter 9: Categorical Association: Chi-Square Test for Independence

### Chi-Square Test for Independence: Purpose, Hypotheses, and Assumptions

Chi-Square Test for Independence

The **Chi-Square Test for Independence **is used to determine whether there is a dependency (relationship) between two *categorical variables *in the population.

Two variables are said to be **independent **when the value obtained for one variable is not related to the value for the other variable.

The hypotheses of a *Chi-Square Test for **Independence *are:

- #H_0:# The variables are independent.
- #H_a:# The variables are dependent.

The data for two categorical variables (e.g. eye color and gender) is typically displayed in a *two-way frequency table**:*

Blue | Brown | Green | Other | Total | |

Men | 24 | 11 | 4 | 9 | 48 |

Women | 20 | 12 | 7 | 8 | 47 |

Total | 44 | 23 | 11 | 17 | 95 |

The following assumptions are required to hold in order for a *Chi-Square Test for Independence* to produce valid results:

- Both variables are
**categorical**(qualitative) in nature. - The measurement categories are
**mutually exclusive**, which means that each observation can be classified into one and only one category. **Random sampling**is used to draw the sample.- All cells should have an expected frequency of at least #1#.
- The majority of cells #(\geq 80\%)# should have an expected frequency of at least #5#.

If either assumption #4# or #5# is not satisfied, you must combine some categories.

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