1. A company's workforce was cross-classified according to job type and gender. That classification appears in the following contingency (two-way) table.
|
|
Men
|
Women
|
Total
|
|
Management/Sales
|
39
|
47
|
86
|
|
Research/Development
|
28
|
22
|
50
|
|
Labor
|
188
|
116
|
304
|
|
Total
|
255
|
185
|
440
|
Using a Chi-Square test and the 0.05 level of significance, determine if there is a significant association between gender and the job type. Your test should include appropriate hypotheses, the computation of a test statistic, mention of the number of degrees of freedom involved, the p-value, a decision (reject H0 or not) with justification, and a conclusion.
2. A research study about student retention is interested in assessing whether there is an association between the students' initial major and the program they transferred to. The data collected by the study is as follows:
|
Initial Major
|
Program transferred to
|
Total
|
|
Engineering
|
Management
|
Liberal Arts
|
Other
|
|
Biology
|
13
|
25
|
158
|
202
|
398
|
|
Chemistry
|
16
|
15
|
19
|
64
|
114
|
|
Mathematics
|
3
|
11
|
20
|
38
|
72
|
|
Physics
|
9
|
5
|
14
|
33
|
61
|
|
Total
|
41
|
56
|
211
|
337
|
645
|
a. Complete the following table of expected counts:
|
Initial Major
|
Program transferred to
|
|
Engineering
|
Management
|
Liberal Arts
|
Other
|
|
Biology
|
25.30
|
|
130.20
|
207.95
|
|
Chemistry
|
|
9.90
|
37.29
|
59.56
|
|
Mathematics
|
4.58
|
6.25
|
|
37.62
|
|
Physics
|
3.88
|
5.30
|
19.96
|
|
b. Use Chi-Square test with the significance level 1% to check whether there is a significant relationship between the students' initial major and the program they transferred to. Provide the hypotheses, the test statistic, its degrees of freedom, the p-value, and the conclusion in context.
3. If both parents have hybrid genotypes Aa, a simplified genetic theory predicts that the child will have exactly one of the genotypes AA, Aa, and aa with respective probabilities 0.25, 0.50, and 0.25. In a random sample of 24 offsprings each having both parents with hybrid genotype Aa, it was found that 10 had genotype AA, 10 had genotype Aa, and 4 had genotype aa. By carrying out a Chi-Square goodness of fit test on this sample data, at level of significance α= 0.10, investigate whether the simplified genetic theory is incorrect. Clearly indicate the hypotheses, the test statistic, its degrees of freedom, the p-value, and the conclusion in context.
4. A forestry researcher is interested in assessing whether the longleaf pine trees are distributed randomly across a certain tract. The tract was divided into four equal quadrants. A random sample of 100 longleaf pine trees was taken and the number of trees in each quadrant was counted. Below are the data:
|
Quadrant
|
Number of Trees
|
|
Q1
|
18
|
|
Q2
|
22
|
|
Q3
|
39
|
|
Q4
|
21
|
a. If the longleaf pine trees are randomly distributed, we expect to find equal number of trees in each quadrant. Find the expected count for each quadrant.
b. Perform a goodness of fit test to determine if the longleaf pine trees are randomly distributed. Clearly indicate the null hypothesis, the test statistic, its degrees of freedom, and the p-value.
c. Based on the answer to the previous part, is the evidence significant at level 10%? At level 5%? At level 1%?