# The degrees of freedom for a contingency table with

January 30, 2017

Question
____ 1. The degrees of freedom for a contingency table with 12 rows and 12 columns is

a.

144

b.

121

c.

12

d.

120

___ 2. The following information was obtained from matched samples.

The daily production rates for a sample of workers before and after a training program are shown below. Test the hypothesis that the program is effective.

Worker

Before

After

1

20

22

2

25

23

3

27

27

4

23

20

5

22

25

6

20

19

7

17

18

The

a.

null hypothesis should be rejected

b.

null hypothesis should not be rejected

c.

alternative hypothesis should be accepted

d.

None of these alternatives is correct.

____ 3. In order to determine whether or not there is a significant difference between the hourly wages of two companies, the following data have been accumulated.

Company A

Company B

Sample size

80

60

Sample mean

\$16.75

\$16.25

Population standard deviation

\$1.00

\$0.95

The p-value is

a.

0.0013

b.

0.0026

c.

0.0042

d.

0.0084

____ 4. A statistics teacher wants to see if there is any difference in the abilities of students enrolled in statistics today and those enrolled five years ago. A sample of final examination scores from students enrolled today and from students enrolled five years ago was taken. You are given the following information.

Today

Five Years Ago

.0/msohtmlclip1/01/clip_image001.png”>

82

88

s2

112.5

54

n

45

36

What is the conclusion that can be reached about the difference in the average final examination scores between the two classes? (Use a .05 level of significance.)

a.

There is a statistically significant difference in the average final examination scores between the two classes.

b.

There is no statistically significant difference in the average final examination scores between the two classes.

c.

It is impossible to make a decision on the basis of the information given.

d.

There is a difference, but it is not significant.

____ 5.In order to determine whether or not a particular medication was effective in curing the common cold, one group of patients was given the medication, while another group received sugar pills. The results of the study are shown below.

Patients Cured

Patients Not Cured

70

10

20

50

We are interested in determining whether or not the medication was effective in curing the common cold.

The p-value is

a.

less than .005

b.

between .005 and .01

c.

between .01 and .025

d.

between .025 and .05

____ 6.

Source

of Variation

Sum

of Squares

Degrees

of Freedom

Mean

Square

F

Between Treatments

2,073.6

4

Between Blocks

6,000

5

1,200

Error

20

288

Total

29

The null hypothesis is to be tested at the 5% level of significance. The p-value is

a.

greater than 0.10

b.

between 0.10 to 0.05

c.

between 0.05 to 0.025

d.

between 0.025 to 0.01

____ 7. In order to estimate the difference between the average hourly wages of employees of two branches of a department store, the following data have been gathered.

Downtown Store

North Mall Store

Sample size

25

20

Sample mean

\$9

\$8

Sample standard deviation

\$2

\$1

A 95% interval estimate for the difference between the two population means is

a.

0.078 to 1.922

b.

1.922 to 2.078

c.

1.09 to 4.078

d.

1.078 to 2.922

____ 8. The results of a recent poll on the preference of teenagers regarding the types of music they listen to are shown below.

Music Type

Teenagers Surveyed

Teenagers Favoring

This Type

Pop

800

384

Rap

900

450

The 95% confidence interval for the difference between the two proportions is

a.

384 to 450

b.

0.48 to 0.5

c.

0.028 to 0.068

d.

-0.068 to 0.028

____ 9. The F ratio in a completely randomized ANOVA is the ratio of

a.

MSTR/MSE

b.

MST/MSE

c.

MSE/MSTR

d.

MSE/MST

____ 10. In an analysis of variance problem involving 3 treatments and 10 observations per treatment, SSE = 399.6. The MSE for this situation is

a.

133.2

b.

13.32

c.

14.8

d.

30.0

____ 11. A goodness of fit test is always conducted as a

a.

lower-tail test

b.

upper-tail test

c.

middle test

d.

None of these alternatives is correct.

____ 12. The management of a department store is interested in estimating the difference between the mean credit purchases of customers using the store’s credit card versus those customers using a national major credit card. You are given the following information.

Store’s Card

Major Credit Card

Sample size

64

49

Sample mean

\$140

\$125

Population standard deviation

\$10

\$8

A 95% confidence interval estimate for the difference between the average purchases of the customers using the two different credit cards is

a.

49 to 64

b.

11.68 to 18.32

c.

125 to 140

d.

8 to 10

____ 13. The critical F value with 6 numerator and 60 denominator degrees of freedom ata = .05 is

a.

3.74

b.

2.25

c.

2.37

d.

1.96

____14. Part of an ANOVA table is shown below.

Source of

Variation

Sum of

Squares

Degrees of

Freedom

Mean

Square

F

Between

Treatments

180

3

Within

Treatments

(Error)

TOTAL

480

18

If at 95% confidence, we want to determine whether or not the means of the populations are equal, the p-value is

a.

between 0.01 to 0.025

b.

between 0.025 to 0.05

c.

between 0.05 to 0.1

d.

greater than 0.1

____ 15.Salary information regarding male and female employees of a large company is shown below. Conduct a test of significance to test the difference in the salaries of males and females.

Male

Female

Sample Size

64

36

Sample Mean Salary (in \$1,000)

44

41

Population Variance (.0/msohtmlclip1/01/clip_image002.png”>)

128

72

The p-value is

a.

0.0668

b.

0.0334

c.

1.336

d.

1.96

16. The results of a recent study regarding smoking and three types of illness are shown in the following table.

Illness

Non-Smokers

Smokers

Totals

Emphysema

20

60

80

Heart problem

70

80

150

Cancer

30

40

70

Totals

120

180

300

We are interested in determining whether or not illness is independent of smoking.

a.

State the null and alternative hypotheses to be tested.

b.

Show the contingency table of the expected frequencies.

c.

Compute the test statistic.

d.

The null hypothesis is to be tested at 95% confidence. Determine the critical value for this test. What do you conclude?

e.

Determine the p-value and perform the test.

17. Recently, a local newspaper reported that part time students are older than full time students. In order to test the validity of its statement, two independent samples of students were selected.

Full Time

Part Time

.0/msohtmlclip1/01/clip_image001.png”>

26

24

s

2

3

n

42

31

a.

Give the hypotheses for the above.

b.

Determine the degrees of freedom.

c.

Compute the test statistic.

d.

At 95% confidence, test to determine whether or not the average age of part time students is significantly more than full time students.

18. From a poll of 800 television viewers, the following data have been accumulated as to their levels of education and their preference of television stations. We are interested in determining if the selection of a TV station is independent of the level of education.

Educational Level

High School

Bachelor

TOTAL

50

150

80

280

Commercial Stations

150

250

120

520

TOTAL

200

400

200

800

a.

State the null and the alternative hypotheses.

b.

Show the contingency table of the expected frequencies.

c.

Compute the test statistic.

d.

The null hypothesis is to be tested at 95% confidence. Determine the critical value for this test.

e.

Determine the p-value and perform the test.

19. In a completely randomized experimental design, 7 experimental units were used for the first treatment, 9 experimental units for the second treatment, and 14 experimental units for the third treatment. Part of the ANOVA table for this experiment is shown below.

Source of

Variation

Sum of

Squares

Degrees of

Freedom

Mean

Square

F

Between

Treatments

_____?

_____?

_____?

4.5

Error (Within

Treatments)

_____?

_____?

4

Total

_____?

_____?

a.

Fill in allthe blanks in the above ANOVA table.

b.

At 95% confidence using both the critical value and p-value approaches, test to see if there is a significant difference among the means.

20. The following table shows the results of recent study regarding gender of individuals and their selected field of study.

Field of study

Male

Female

TOTAL

Medicine

80

40

120

60

20

80

Engineering

160

40

200

TOTAL

300

100

400

We want to determine if the selected field of study is independent of gender.

a.

Compute the test statistic.

b.

Using the p-value approach at 90% confidence, test to see if the field of study is independent of gender.

c.

Using the critical method approach at 90% confidence, test for the independence of major and gender.

21. The data below represents the fields of specialization for a randomly selected sample of undergraduate students. We want to determine whether there is a significant difference in the fields of specialization between regions of the country.

Northeast

Midwest

South

West

Total

54

65

28

93

240

Engineering

15

24

8

33

80

Liberal Arts

65

84

33

98

280

Fine Arts

13

15

7

25

60

Health Sciences

3

12

4

21

40

150

200

80

270

700

a.

Determine the critical value of the chi-squarec2 for this test of independence.

b.

Calculate the value of the test statistic.

c.

What is the conclusion for this test? Leta = .05.

22. During the first few weeks of the new television season, the evening news audience proportions were recorded as ABC- 31%, CBS- 34%, and NBC- 35%. A sample of 600 homes yielded the following viewing audience data.

Number of Homes

ABC

150

CBS

200

NBC

250

We want to determine whether or not there has been a significant change in the number of viewing audience of the three networks.

a.

State the null and alternative hypotheses to be tested.

b.

Compute the expected frequencies.

c.

Compute the test statistic.

d.

The null hypothesis is to be tested at 95% confidence. Determine the critical value for this test. What do you conclude?

e.

Determine the p-value and perform the test.

23. The management of Regional Hospital has made substantial improvements in their hospital and would like to test and determine whether there has been a significant decrease in the average length of stay of their patients in their hospital. The following data has been accumulated from before and after the improvements. At 95% confidence, test to determine if there has been a significant reduction in the average length of stay.

After

Before

Sample size

45

58

Mean (in days)

4.6

4.9

Standard Deviation (s)

0.5

0.6

a.

Formulate the hypotheses.

b.

Compute the test statistic.

c.

Using the p-value approach, test to see if the average length of stay in RFH is significantly less than the average length of stay in GH. Leta = 0.05.

24. During the primary elections of 1996, candidate A showed the following pre-election voter support in Tennessee and Mississippi.

Voters Surveyed

Voters Favoring

Candidate A

Tennessee

500

295

Mississippi

700

357

a.

Develop a 95% confidence interval estimate for the difference between the proportions of voters favoring candidate A in the two states.

b.

Is there conclusive evidence that one of the two states had a larger proportion of voters’ support? If yes, which state? Explain.

25. In order to determine whether or not a driver’s education course improves the scores on a driving exam, a sample of 6 students were given the exam before and after taking the course. The results are shown below.

Let d = Score After – Score Before.

Student

Score

Before the Course

Score

After the Course

1

83

87

2

89

88

3

93

91

4

77

77

5

86

93

6

79

83

a.

Compute the test statistic.

b.

At 95% confidence using the p-value approach, test to see if taking the course actually increased scores on the driving exam.

26. Three universities in your state decided to administer the same comprehensive examination to the recipients of MBA degrees from the three institutions. From each institution, MBA recipients were randomly selected and were given the test. The following table shows the scores of the students from each university.

Northern

Central

Southern

University

University

University

75

85

80

80

89

81

84

86

84

85

88

79

81

83

85

Ata = 0.01, test to see if there is any significant difference in the average scores of the students from the three universities. (Note that the sample sizes are not equal.) Use both the critical and p-value approaches.

27. The results of a recent poll on the preference of voters regarding the presidential candidates are shown below.

Voters Surveyed

Voters Favoring

This Candidate

Candidate A

200

150

Candidate B

300

195

a.

Develop a 90% confidence interval estimate for the difference between the proportion of voters favoring each candidate.

b.

Does your confidence interval provide conclusive evidence that one of the candidates is favored more? Explain.

28. A dietician wants to see if there is any difference in the effectiveness of three diets. Eighteen people were randomly chosen for the test. Then each individual was randomly assigned to one of the three diets. Below you are given the total amount of weight lost in six months by each person.

Diet A

Diet B

Diet C

14

12

25

18

10

32

20

22

18

12

12

14

20

16

17

18

12

14

a.

State the null and alternative hypotheses.

b.

Calculate the test statistic.

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