Chi-square Tests

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1.  Jury Selection
Checking Conditions: We have a random sample; the sample size is less than 10% of the county population; all expected cells are larger than 5, so a Chi-squared test is appropriate.
Stating Hypotheses:
Ho:  For each age group, the proportion of jurors is consistent with the county proportion.
Ha:  The proportion of jurors for at least one age group is inconsistent with the county proportions.


Calculating the test statistic:
Age
Observed
Expected
(O-E)
(O-E)2/E
21 to 40
5
0.42(66) = 27.72
-22.72
18.62
41 to 50
9
15.18
-6.18
5.55
51 to 60
19
10.56
8.44
6.75
61 or over
33
12.52
20.46
33.38
totals
66
66.00
0
61.2556

Chi-squared statistic = 2.516 and df = 3, p-value is almost 0.


Writing a conclusion in context:
It is almost impossible for a jury to differ this much from the county age distribution by chance. Our sample provides evidence that grand juries in Alameda County are not selected at random.
(In a footnote, the authors tell us that grand juries are nominated by judges, who prefer older jurors.)

2.  Pre-school Attendance and Pre-algebra Achievement
We use a Chi-squared test since we have one random sample,  the sample size is less than 10% of the population of all 7th graders in the district, and each expected cell count is large enough (greater than 5).


Ho:  Pre-algebra achievement is independent of pre-school attendance.
Ha:  There is a relationship between Pre-algebra achievement and pre-school attendance
Expected counts Below grade level At grade level Advanced
Pre-school 5.6 8.4 6
No Pre-school 8.4 12.6 9

Chi-squared =  2.85 with p-value = .239, much larger than alpha = .05.


Since our p-value is so high, our sample does not provide significant evidence that pre-algebra achievement is related to pre-school attendance. This study alone would not support funding pre-school education for all students in the district.


3.  Evaluating Textbooks
We use a Chi-squared test since we have two random samples which were independently chosen; the sample size is less than 10% of the population of all algebra students in the district; and each expected cell count is large enough (greater than 5).


Ho:  At each proficiency level, the proportions are the same for students who used the new text and those who used the traditional text.
Ha: There is a difference in the proportions for students who used the new text and those who used the traditional text.
Expected counts Below grade level At grade level Advanced
Pre-school 5.6 8.4 6
No Pre-school 8.4 12.6 9

Chi-squared =  2.85 with p-value = .239, much larger than alpha = .05.
Since our p-value is so large, our sample provides no significant evidence that the proportions at each level are different for students who used the new or traditional algebra text. On the basis of this study, we would probably not recommend new textbooks.

4.  Summary
a)  In Jury Selection, we examine how well the sample fit our model or its goodness-of-fit.
b)  For the Pre-school and Pre-algebra, each student in our only sample is classified by two attributes. If the variables are independent, the observed counts will be consistent with the expected counts. We want to know if those attributes are related or independent.
c)  In the Textbooks problem, we select students in two samples and we examine a single attribute to see if the proportions are equal or homogeneous for each group.
Although the mechanics for a test of independence and for a test of homogeneity are the same, the methods for selecting the samples differ. We ask different questions. “Are the attributes independent?” rather than “Are the groups homogeneous (or alike) with respect to this attribute?”