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# The pearson r and spearman rho correlation coefficients are related.

10. The Pearson r and Spearman rho correlation coefficients are related. Is this statement correct? Explain.

15. In a large introductory sociology course, a professor gives two exams. The professor wants to determine whether the scores students receive on the second exam are correlated with their scores on the first exam. To make the calculations easier, a sample of eight students is selected. Their scores are shown in the accompanying table.

Student          Exam 1         Exam 2

1                      60                60

2                      75                100

3                      70                80

4                      72                68

5                      54                73

6                      83                97

7                      80                85

8                      65                90

a. Construct a scatter plot of the data, using exam 1 score as the X variable. Does the relationship look linear?

b. Assuming a linear relationship exists between scores on the two exams, compute the value for Pearson r.

c. How well does the relationship account for the scores on exam 2?

18. An educator has constructed a test for mechanical aptitude. He wants to determine how reliable the test is over two administrations spaced by 1 month. A study is conducted in which 10 students are given two administrations of the test, with the second administration being 1 month after the first. The data are given in the following table.

1                    10                              10

2                    12                              15

3                    20                              17

4                    25                              25

5                    27                              32

6                    35                              37

7                    43                              40

8                    40                              38

9                    32                              30

10                  47                              49

a. Construct a scatter plot of the paired scores.

b. Determine the value of r.

c. Would it be fair to say that this is a reliable test? Explain using r2.

22. A social psychologist conducts a study to determine the relationship between religion and selfesteem. Ten eighth graders are randomly selected for the study. Each individual undergoes two tests, one measuring self-esteem and the other religious involvement. For the self-esteem test, the lower the score is, the higher self-esteem is; for the test measuring religious involvement, the higher the score is, the higher religious involvement is. The selfesteem test has a range from 1 to 10 and the religious involvement test ranges from 0 to 50. For the purposes of this question, assume both tests are well standardized and of interval scaling. The following data are collected.

Subject        Religious Involvement        Self-Esteem

1                   5                                          8

2                   25                                        3

3                   45                                        2

4                   20                                        7

5                   30                                        5

6                   40                                        5

7                   1                                          4

8                   15                                       4

9                   10                                       7

10                 35                                       3

a. If a relationship exists such that the more religiously involved one is, the higher actual self-esteem is, would you expect r computed on the provided values to be negative or positive? Explain.

b. Compute r. Were you correct in your answer to part a?

Chapter 7:

9. Given the set of paired X and Y scores,

X  7  10  9  13  7  11  13

Y  1   2   4   3   3   4    5

a. Construct a scatter plot of the paired scores. Does the relationship appear linear?

b. Determine the least-squares regression line for predicting Y given X.

c. Draw the regression line on the scatter plot.

d. Using the relationship between X and Y, what value would you predict for Y if X  12? (Round to two decimal places.)

The predicted Y is,

12. A statistics professor conducts a study to investigate the relationship between the performance of his studentson exams and their anxiety. Ten students fromhis class are selected for the experiment. Just before taking the final exam, the 10 students are given ananxiety questionnaire. Here are final exam and anxietyscores for the 10 students:

Student No.1 2 3 4 5 6 7 89 10

Anxiety 28 4135 39 31 42 50 46 45 37

Final Exam 82 5863 89 92 64 55 70 51 72

a. On a piece of graph paper, construct a scatter plot of the paired scores. Use anxiety as the X variable.

b. Describe the relationship shown in the graph.

c. Assuming the relationship is linear, compute the value of Pearson r.

d. Determine the least-squares regression line for predicting the final exam score, given the anxiety level. Should be Y be positive or negative? Why?

SUMMARY OUTPUT

Regression Statistics

Multiple R

0.6908

R Square

0.4772

0.4118

Standard Error

10.8653

Observations

10

ANOVA

df

SS

MS

F

Significance F

Regression

1

861.9592

861.9592

7.3013

0.0270

Residual

8

944.4408

118.0551

Total

9

1806.4000

Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Intercept

125.8830

21.1109

5.9629

0.0003

77.2013

174.5648

Anxiety

-1.4285

0.5287

-2.7021

0.0270

-2.6476

-0.2094

e. Draw the least-squares regression line of part d on the scatter plot of part a.

f. Based on the data of the 10 students, if a student has an anxiety score of 38, what value would you predict for her final exam score? (Round to two decimal places.)

g. Calculate the standard error of estimate for predicting final exam scores from anxiety scores.

13. The sales manager of a large sporting goods store hasrecently started a national advertising campaign. Hehas kept a record of the monthly costs of the advertisingand the monthly profits. These are shown here The entries are in thousands of dollars.

Month Jan.Feb.Mar.Apr.May Jun. Jul.

MonthlyAdvertising Cost10.0 14.011.4 15.6 16.8 11.2 13.2

Monthly Profit125 200160 150 210 110 125

a. Assuming a linear relationship exists, derive the least-squares regression line for predicting monthly profits from monthly advertising costs.

SUMMARY OUTPUT

Regression Statistics

Multiple R

0.6958

R Square

0.4841

0.3809

Standard Error

30.3395

Observations

7

ANOVA

df

SS

MS

F

Significance F

Regression

1

4318.9930

4318.9930

4.6921

0.0825

Residual

5

4602.4356

920.4871

Total

6

8921.4286

Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Intercept

11.6599

66.8350

0.1745

0.8683

-160.1449

183.4648

10.8284

4.9990

2.1661

0.0825

-2.0219

23.6787

The output tells us that the regression equation is,

Monthly Profit = 11.6599+10.8284*Monthly Advertising Cost

b. In August, the manager plans to spend \$17,000 on advertising. Based on the data, how much profit should he expect that month? (Round to the nearest \$1000.)

c. Given the relationship shown by the paired scores, can you think of a reason why the manager doesn’t spend a lot more money on advertising?

18. In Chapter 6, Problem 22 (p. 153), data were presentedfrom a study conducted to determine the relationshipbetween religious involvement and self-esteem. Thedata are again presented below.

Subject   ReligiousInvolvement Self-Esteem

1 5 8

2 25 3

3 45 2

4 20 7

5 30 5

6 40 5

7 1 4

8 15 4

9 10 7

10 35 3

a. Assuming a linear relationship, derive the leastsquaresregression line for predicting self-esteemfrom religious involvement.

SUMMARY OUTPUT

Regression Statistics

Multiple R

0.5626

R Square

0.3165

0.2310

Standard Error

1.7440

Observations

10

ANOVA

df

SS

MS

F

Significance F

Regression

1

11.2667

11.2667

3.7041

0.0905

Residual

8

24.3333

3.0417

Total

9

35.6000

Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Intercept

6.4885

1.0363

6.2614

0.0002

4.0989

8.8782

Religious Involvement

-0.0747

0.0388

-1.9246

0.0905

-0.1642

0.0148

b. Using this regression line, what value of selfesteemwould you predict for an eighth graderwhose value of religious involvement is 43?

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