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Chapter 10: Q3BSC (page 468)

Coefficient of Determination Using the heights and weights described in Exercise 1, the linear correlation coefficient r is 0.394. Find the value of the coefficient of determination. What practical information does the coefficient of determination provide?

Short Answer

Expert verified

The value of the coefficient of determination is 0.155.

It means that 15.5% variation in the weight of males is explained by the linear association between height and weight. Other factors, as well as random variation, account for the remaining 84.5%.

Step by step solution

01

Given information

The linear correlation coefficient between height and weight is 0.394.

02

Coefficient of determination

The coefficient of determination is the square of the linear correlation coefficient between the response variable and the predictor variable.

Here, the linear correlation coefficient (r) between height and weight is 0.394.

Thus,

\(\begin{array}{c}{\rm{Coefficient}}\;{\rm{of}}\;{\rm{Determination}} = {r^2}\\ = {0.394^2}\\ = 0.155\end{array}\)

Therefore, the coefficient of determination is 0.155.

03

Practical interpretation of the coefficient of determination

The actual interpretation of this value is that approximately 15.5% variation in the response variable 鈥渨eights of males鈥� is explained by the linear relationship between weight and height.

The remaining variation in y is \(100\% - 15.5\% = 84.5\% \). It can be explained by other variables/factors or random variation.

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Most popular questions from this chapter

Interpreting r. In Exercises 5鈥�8, use a significance level of A = 0.05 and refer to the accompanying displays.

5. Bear Weight and Chest Size Fifty-four wild bears were anesthetized, and then their weights and chest sizes were measured and listed in Data Set 9 鈥淏ear Measurements鈥� in Appendix B; results are shown in the accompanying Statdisk display. Is there sufficient evidence to support the claim that there is a linear correlation between the weights of bears and their chest sizes? When measuring an anesthetized bear, is it easier to measure chest size than weight? If so, does it appear that a measured chest size can be used to predict the weight?

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)

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.)

Pizza and the Subway The 鈥減izza connection鈥� is the principle that the price of a slice of pizza in New York City is always about the same as the subway fare. Use the data listed below to determine whether there is a significant linear correlation between the cost of a slice of pizza and the subway fare.

Year

1960

1973

1986

1995

2002

2003

2009

2013

2015

Pizza Cost

0.15

0.35

1

1.25

1.75

2

2.25

2.3

2.75

Subway Fare

0.15

0.35

1

1.35

1.5

2

2.25

2.5

2.75

CPI

30.2

48.3

112.3

162.2

191.9

197.8

214.5

233

237.2

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.)

Old Faithful Listed below are duration times (seconds) and time intervals (min) to the next eruption for randomly selected eruptions of the Old Faithful geyser in Yellowstone National Park. Is there sufficient evidence to conclude that there is a linear correlation between duration times and interval after times?

Duration

242

255

227

251

262

207

140

Interval After

91

81

91

92

102

94

91

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.)

Manatees Listed below are numbers of registered pleasure boats in Florida (tens of thousands) and the numbers of manatee fatalities from encounters with boats in Florida for each of several recent years. The values are from Data Set 10 鈥淢anatee Deaths鈥� in Appendix B. Is there sufficient evidence to conclude that there is a linear correlation between numbers of registered pleasure boats and numbers of manatee boat fatalities?

Pleasure Boats

99

99

97

95

90

90

87

90

90

Manatee Fatalities

92

73

90

97

83

88

81

73

68
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