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 Solutions
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?”
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