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Chapter 10: Q5BSC (page 468)

Interpreting the Coefficient of Determination. In Exercises 5鈥�8, use the value of the linear correlation coefficient r to find the coefficient of determination and the percentage of the total variation that can be explained by the linear relationship between the two variables.

Crickets and Temperature r = 0.874 (x = number of cricket chirps in 1 minute, y = temperature in 掳F)

Short Answer

Expert verified

The coefficient of determination is 0.764.

The percentage of variation that can be explained by the linear association between the number of cricket chirps in 1 minute and the temperature is 76.4%.

23.6% of the variation in the response variable (temperature) is explained by other factors and random variation.

Step by step solution

01

Given information

The linear correlation coefficient between the number of cricket chirps in 1 min and the temperature is 0.874.

02

Coefficient of determination

The coefficient of determination is obtained by taking the square of linear correlation coefficient between the two variables.

Here, the linear correlation coefficient (r) between the number of cricket chirps in 1 minute and the temperature is 0.874.

Thus,

\(\begin{array}{c}{\rm{Coefficient}}\;{\rm{of}}\;{\rm{determination}} = {r^2}\\ = {0.874^2}\\ = 0.764\end{array}\)

Therefore, the value of the coefficient of determination is 0.764.

03

Percentage of variation

Here,

\(\begin{array}{c}{r^2} = 0.764\\ = \frac{{76.4}}{{100}} \times 100\% \\ = 76.4\% \end{array}\)

Therefore, the percentage of the variation explained by the linear association of the number of cricket chirps in 1 min and the temperature is 76.4%.

The remaining \(100\% - 76.4\% = 23.6\% \) variation is explained by other factors and random variation.

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Most popular questions from this chapter

Critical Thinking: Is the pain medicine Duragesic effective in reducing pain? Listed below are measures of pain intensity before and after using the drug Duragesic (fentanyl) (based on data from Janssen Pharmaceutical Products, L.P.). The data are listed in order by row, and corresponding measures are from the same subject before and after treatment. For example, the first subject had a measure of 1.2 before treatment and a measure of 0.4 after treatment. Each pair of measurements is from one subject, and the intensity of pain was measured using the standard visual analog score. A higher score corresponds to higher pain intensity.

Pain Intensity Before Duragesic Treatment

1.2

1.3

1.5

1.6

8

3.4

3.5

2.8

2.6

2.2

3

7.1

2.3

2.1

3.4

6.4

5

4.2

2.8

3.9

5.2

6.9

6.9

5

5.5

6

5.5

8.6

9.4

10

7.6










Pain Intensity After Duragesic Treatment

0.4

1.4

1.8

2.9

6

1.4

0.7

3.9

0.9

1.8

0.9

9.3

8

6.8

2.3

0.4

0.7

1.2

4.5

2

1.6

2

2

6.8

6.6

4.1

4.6

2.9

5.4

4.8

4.1










Matched Pairs The methods of Section 9-3 can be used to test a claim about matched data. Identify the specific claim that the treatment is effective, then use the methods of Section 9-3 to test that claim.

Testing for a Linear Correlation. In Exercises 13鈥�28, construct a scatterplot, and find the value of the linear correlation coefficient r. Also find the P-value or the critical values of r from Table A-6. Use a significance level of A = 0.05. Determine whether there is sufficient evidence to support a claim of a linear correlation between the two variables. (Save your work because the same data sets will be used in Section 10-2 exercises.)

Oscars Listed below are ages of Oscar winners matched by the years in which the awards were won (from Data Set 14 鈥淥scar Winner Age鈥� in Appendix B). Is there sufficient evidence to conclude that there is a linear correlation between the ages of Best Actresses and Best Actors? Should we expect that there would be a correlation?

Actress

28

30

29

61

32

33

45

29

62

22

44

54

Actor

43

37

38

45

50

48

60

50

39

55

44

33

Testing for a Linear Correlation. In Exercises 13鈥�28, construct a scatterplot, and find the value of the linear correlation coefficient r. Also find the P-value or the critical values of r from Table A-6. Use a significance level of A = 0.05. Determine whether there is sufficient evidence to support a claim of a linear correlation between the two variables. (Save your work because the same data sets will be used in Section 10-2 exercises.)

POTUS Media periodically discuss the issue of heights of winning presidential candidates and heights of their main opponents. Listed below are those heights (cm) from severalrecent presidential elections (from Data Set 15 鈥淧residents鈥� in Appendix B). Is there sufficient evidence to conclude that there is a linear correlation between heights of winning presidential candidates and heights of their main opponents? Should there be such a correlation?

President

178

182

188

175

179

183

192

182

177

185

188

188

183

188

Opponent

180

180

182

173

178

182

180

180

183

177

173

188

185

175

Interpreting\({R^2}\)In Exercise 2, the quadratic model results in = 0.255. Identify the percentage of the variation in Super Bowl points that can be explained by the quadratic model relating the variable of year and the variable of points scored. (Hint: See Example 2.) What does the result suggest about the usefulness of the quadratic model?

Interpreting\({R^2}\)For the multiple regression equation given in Exercise 1, we get \({R^2}\)= 0.928. What does that value tell us?
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