11-1. Exercises

1. BASIC

Ex 1. Find $χ_{0.01}^2$ for each of the following number of degrees of freedom.

1. $df=5$

2. $df=11$

3. $df=25$

Ex 2. Find $χ_{0.05}^2$ for each of the following number of degrees of freedom.

1. $df=6$

2. $df=12$

3. $df=30$

Ex 3. Find $χ_{0.10}^2$ for each of the following number of degrees of freedom.

1. $df=6$

2. $df=12$

3. $df=30$

Ex 4. Find $χ_{0.01}^2$ for each of the following number of degrees of freedom.

1. $df=7$

2. $df=10$

3. $df=20$

Ex 5. For $df=7$ and $α=0.05$ , find

1. $χ^2_α$

2. $χ^2_{α/2}$

Ex 6. For $df=17$ and $α=0.01$ , find

1. $χ^2_α$

2. $χ^2_{α/2}$

Ex 7. A data sample is sorted into a 2 × 2 contingency table based on two factors, each of which has two levels.

Factor 1
Factor 2	Level 1	Level 2	Row Total
Level 1	20	10	R
Level 2	15	55	R
Column Total	C	C	n

1. Find the column totals, the row totals, and the grand total, n, of the table.

2. Find the expected number E of observations for each cell based on the assumption that the two factors are independent (that is, just use the formula $E=(R×C)∕n$ .

3. Find the value of the chi-square test statistic $χ^2$ .

4. Find the number of degrees of freedom of the chi-square test statistic.

Ex 8. A data sample is sorted into a 3 × 2 contingency table based on two factors, one of which has three levels and the other of which has two levels.

	Factor 1
Factor 2	Level 1	Level 2	Row Total
Level 1	20	10	R
Level 2	15	55	R
Level 3	10	20	R
Column Total	C	C	n

1. Find the column totals, the row totals, and the grand total, n, of the table.

2. Find the expected number E of observations for each cell based on the assumption that the two factors are independent (that is, just use the formula $E=(R×C)∕n$ .

3. Find the value of the chi-square test statistic $χ^2$ .

4. Find the number of degrees of freedom of the chi-square test statistic.

2. APPLICATIONS

Ex 9. A child psychologist believes that children perform better on tests when they are given perceived freedom of choice. To test this belief, the psychologist carried out an experiment in which 200 third graders were randomly assigned to two groups, A and B. Each child was given the same simple logic test. However in group B, each child was given the freedom to choose a text booklet from many with various drawings on the covers. The performance of each child was rated as Very Good, Good, and Fair. The results are summarized in the table provided. Test, at the 5% level of significance, whether there is sufficient evidence in the data to support the psychologist’s belief.

	Group
Performance	A	B
Very Good	32	29
Good	55	61
Fair	10	13

Ex 10. In regard to wine tasting competitions, many experts claim that the first glass of wine served sets a reference taste and that a different reference wine may alter the relative ranking of the other wines in competition. To test this claim, three wines, A, B and C, were served at a wine tasting event. Each person was served a single glass of each wine, but in different orders for different guests. At the close, each person was asked to name the best of the three. One hundred seventy-two people were at the event and their top picks are given in the table provided. Test, at the 1% level of significance, whether there is sufficient evidence in the data to support the claim that wine experts’ preference is dependent on the first served wine.

	Top Pick
First Glass	A	B	C
A	12	31	27
B	15	40	21
C	10	9	7

Ex 11. Is being left-handed hereditary? To answer this question, 250 adults are randomly selected and their handedness and their parents’ handedness are noted. The results are summarized in the table provided. Test, at the 1% level of significance, whether there is sufficient evidence in the data to conclude that there is a hereditary element in handedness.

	Number of Parents Left-Handed
Handedness	0	1	2
Left	8	10	12
Right	178	21	21

Ex 12. Some geneticists claim that the genes that determine left-handedness also govern development of the language centers of the brain. If this claim is true, then it would be reasonable to expect that left-handed people tend to have stronger language abilities. A study designed to text this claim randomly selected 807 students who took the Graduate Record Examination (GRE). Their scores on the language portion of the examination were classified into three categories: low, average, and high, and their handedness was also noted. The results are given in the table provided. Test, at the 5% level of significance, whether there is sufficient evidence in the data to conclude that left-handed people tend to have stronger language abilities.

	GRE English Scores
Handedness	Low	Average	High
Left	18	40	22
Right	201	360	166

Ex 13. It is generally believed that children brought up in stable families tend to do well in school. To verify such a belief, a social scientist examined 290 randomly selected students’ records in a public high school and noted each student’s family structure and academic status four years after entering high school. The data were then sorted into a 2 × 3 contingency table with two factors. Factor 1 has two levels: graduated and did not graduate. Factor 2 has three levels: no parent, one parent, and two parents. The results are given in the table provided. Test, at the 1% level of significance, whether there is sufficient evidence in the data to conclude that family structure matters in school performance of the students.

	Academic Status
Family	Graduated	Did Not Graduate
No parent	18	31
One parent	101	44
Two parents	70	26

Ex 14. A large middle school administrator wishes to use celebrity influence to encourage students to make healthier choices in the school cafeteria. The cafeteria is situated at the center of an open space. Everyday at lunch time students get their lunch and a drink in three separate lines leading to three separate serving stations. As an experiment, the school administrator displayed a poster of a popular teen pop star drinking milk at each of the three areas where drinks are provided, except the milk in the poster is different at each location: one shows white milk, one shows strawberry-flavored pink milk, and one shows chocolate milk. After the first day of the experiment the administrator noted the students’ milk choices separately for the three lines. The data are given in the table provided. Test, at the 1% level of significance, whether there is sufficient evidence in the data to conclude that the posters had some impact on the students’ drink choices.

	Student Choice
Poster Choice	Regular	Strawberry	Chocolate
Regular	38	28	40
Strawberry	18	51	24
Chocolate	32	32	53

3. LARGE DATA SET EXERCISE

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data8.xls

Ex 15. Large Data Set 8 records the result of a survey of 300 randomly selected adults who go to movie theaters regularly. For each person the gender and preferred type of movie were recorded. Test, at the 5% level of significance, whether there is sufficient evidence in the data to conclude that the factors “gender” and “preferred type of movie” are dependent.