Chapter 7: Exercises 1, 2, 6, 9, 10, 11, 13, 15 Part 2: Analyzing and Interpreting Bivariate

Chapter 7: Exercises 1, 2, 6, 9, 10, 11, 13, 15 Part 2: Analyzing and Interpreting Bivariate Statistics: Regression and Prediction Answer the following questions from the Coladarci (2011) text: Chapter 8: Exercises 2, 6, 12, 16, 17

1. Give examples, other than those mentioned in this chapter, of pairs of variables you would expect to show:

(a) a positive association

(b) a negative association

(c) no association at all

1. Why is it important to inspect scatterplots?

X Y

11 12

9 8

8 10

6 7

4 4

3 6

12

6. (a) Using the x and y data in Problem 3 (above), divide each value of X by 2 and construct a scatterplot showing the relationship between X and Y.

(a) How do your impressions of the new scatterplot compare with your impressions of the original plot?

(b) What is the covariance between X and Y?

(c) How is the covariance affected by this transformation?

(d) What is the Pearson r between X and Y? How does this compare with the initial r from Problem 4?

(*4. (a) Using the data in Problem 3, determine r from both the defining formula and the calculating formula. (a) Interpret r within the context of the coefficient of determination. Do not have to do this one.)

(e) What generalizations do these results permit regarding the effect of linear transformations (e.g., halving each score) on the degree of linear association between two variables?

9. The covariance between X and Y is 72, SX ¼ 8 and SY ¼ 11. What is the value of r?

10. r ¼ :47, SX ¼ 6, and SY ¼ 4. What is the covariance between X and Y?

11. For a particular set of scores, SX ¼ 3 and SY ¼ 5. What is the largest possible value ofthe covariance? (Remember that r can be positive or negative.)

13. Does a low r necessarily mean that there is little \association" between two variables?

(Explain.)

15. Some studies have found a strong negative correlation between how much parents helptheir children with homework (X) and student achievement (Y). That is, children whoreceive more parental help on their homework tend to have lower achievement thankids who receive little or no parental help. Discuss the possible explanations for whythese two variables would correlate negatively. Although one cannot infer causality froma correlation, which explanation do you find most persuasive?

2. The relationship between student performance on a state-mandated test administered

in the fourth grade and again in the eighth grade has been analyzed for a large group

of students in the state. Ellen obtains a score of 540 on the fourth-grade test. From

this, her performance on the eighth-grade test is predicted (using the regression line)

to be 550.

(a) In what sense can the value 550 be considered an estimated mean?

(b) Why is it an estimated rather than an actual mean?

6. Following are the scores on a teacher certification test administered prior to hiring (X)and the principal’s ratings of teacher effectiveness after three months on the job (Y) fora group of six first-year teachers (A–F):

A B C D E F

Test score (X): 14 24 21 38 34 49

Principal rating (Y): 7 4 10 8 13 11

(a) Compute the summary statistics required for determining the regression equationfor predicting principal ratings from teacher certification test scores.

(b) Using values from Problem 6a calculate the intercept and slope; state the regressionequation.

(c) Suppose that three teachers apply for positions in this school, obtaining scores of 18,32, and 42, respectively, on the teacher certification test. Compute their predictedratings of teacher effectiveness.

(d) If in fact these data were real, what objections would you have to using the equationfrom Problem 6b for prediction in a real-life situation?

12. (No calculations are necessary for this problem.) Suppose the following summary statisticsare obtained from a large group of individuals: X ¼ 52:0, SX ¼ 8:7, Y ¼ 147:3,

SY ¼ 16:9. Dorothy receives an X score of 52. What is her predicted Y score if:

(a) r = 0?

(b) r = .55?

(c) r = +.38?

(d) r = 1.00?

(e) State the principle that emerges from your answers to Problems 12a to 12d.

(f) Show how Formula (8.5) illustrates this principle.

16. At the end of Section 8.3, we asked you to consider how the location of Student 26 would affect the placement of the regression line in Figure 8.4.

(a) Imagine you deleted this case, recalculated intercept and slope, and drew in the new regression line. Where do you think the new line would lie relative to the original regression line? Why? (Refer to the least-squares criterion.)

(b) How should the removal of Student 26 affect the magnitude of the intercept? The slope?

(c) With Student 26 removed, the relevant summary statistics are X ¼ 69:45,

SX ¼ 9:68, Y ¼ 100:83, SY ¼ 14:38, r ¼ :79. Calculate the new intercept and slope.

(d) As accurately as possible, draw in the new regression line using the figure below

(from which Student 26 has been deleted). How does the result compare with your response to Problems 16a and 16b?

17. At the end of the section on “setting up the margin of error," we asked if you can see from Table A in Appendix C how we got “1.00" and “2.58" for 68% and 99% confidence, respectively. Can you?