Chi-square Tests

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 Practice Problems

1. Jury Selection (adapted from the Freedman, Pisani, Purves classic text)
One study of grand juries in Alameda County, California, compared the demographic characteristics of jurors with the general population, to see if jury panels were representative. The results for age are shown below. The investigators wanted to know if the 66 jurors were selected at random from the population of Alameda County. (Only persons over 21 and over are considered; the county age distribution is known from Public Health Department data.) The study was published in the UCLA Law Review.

Age Count-wide % # of jurors observed # of jurors expected (O-E) (O-E)2/E
21-40 42% 5      
41-50 23% 9      
51-60 16% 19      
over 60 19% 33      
Total 100% 66      

Do we have evidence that grand juries are selected at random for the population of Alameda County?

2.  Pre-school Attendance and Pre-algebra Achievement
(these are contrived data, based on a real study)
In these times of educational reform, attention has been focused on pre-school for all children. Since many districts are facing budget cuts, funding pre-school programs may impact other offerings. Before making their recommendations, administrators in a large urban district take a random sample of 50 seventh graders and compare the pre-algebra achievement levels of those who attended pre-school and those who did not. If achievement is independent of attending pre-school then the proportions at each level should be equal. Use the counts in the frequency table to determine if there is an association between attending pre-school and pre-algebra achievement.
  Below grade level At grade Level Advanced
Pre-school 8 6 6
No Pre-school 6 15 9

3. Evaluating Textbooks
Does the new math program improve student performance?
Suppose you take a random sample of 20 students who are using a new algebra text which features group work and unit summaries and a second sample of 30 students who are using a more traditional text. You compare student achievement on the state test given to all students at the end of the course. Use the frequency table to determine if the proportions from each group are equal at each performance level.
  Below grade level At grade level Advanced
New text 8 6


Old text 6 15 9

4.  Summary

The practice problems illustrate the three different Chi-squared tests. Identify each and determine how you can distinguish among the tests.

5. Remember those depressed people forced to look at sad pictures in Unit 4? No? Well, a group of subjects were classified by whether or not they were depressed. All subjects were shown "sad" pictures and their responses were measured. The researchers were interested in knowing whether there were a greater proportion of strong emotional responses among the depressed people than among the non-depressed people. Should you do a test of homogeneity or a test of independence? Why?

Check your solutions here.