Today we talked about the Chi-squared Test for Association. This is a test to see if an association between two categorical variables is statistically significant. It uses the Chi-Squared Distribution which is a new probability distribution for us. To work with this distribution, I recommend either using the Probability Distributions app on your phone or using a chi-squared distribution table.
Here are the steps for the test:
These are always:
There is a formula for finding the χ² test statistic, but in practice people usuaully just use software (just like with standard deviations and correlation coefficients). I recommend using the χ²-Test App on my website.
Chi-squared distributions have degrees of freedom df = (#rows -1)(#columns-1). Once you know that, you can use either the Probabilities Distribution app or the χ²-Distribution Table to find the p-value.
This works just like any other hypothesis test. The lower the p-value, the more significant the results, and the more convinced you should be that the null hypothesis can be rejected. In this case, that would mean that there is an association between the two variables (in the population and not just in the sample).
We did the following examples in class.
The General Social Survey asked people ``In general how do you feel about your time? Do you always feel rushed, sometimes feel rushed, or almost never feel rushed?" The results are shown below.
| Male | Female | |
|---|---|---|
| Always | 116 | 188 |
| Sometimes | 229 | 284 |
| Never | 82 | 78 |
Find the \(\chi^2\) statistic, the degrees of freedom, and the p-value, then explain what it means about whether there is a difference between how rushed men and women feel in the population.
There is evidence that the opinions of Americans on evolution have changed in the last few years. Here is data from Gallup comparing the results of two polls, one from 2001 and one from 2017. The counts are the number of people who agree with each of the statements.
| 2001 | 2017 | Total | |
|---|---|---|---|
| Humans evolved with God’s guidance | 380 | 384 | 764 |
| Humans evolved God, played no part | 123 | 192 | 315 |
| God created humans in their present form | 462 | 384 | 846 |
| Other/no opinion | 51 | 51 | 102 |
| Total | 1016 | 1011 | 2027 |
The chi-squared test statistic for this table is χ² = 22.3. What does this mean? Which entries in the table contributed the most to the chi-squared statistic?