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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 “weights of males” 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

Effects of an Outlier Refer to the Minitab-generated scatterplot given in Exercise 11 of

Section 10-1 on page 485.

a. Using the pairs of values for all 10 points, find the equation of the regression line.

b. After removing the point with coordinates (10, 10), use the pairs of values for the remaining 9 points and find the equation of the regression line.

c. Compare the results from parts (a) and (b).

Interpreting the Coefficient of Determination. In Exercises 5–8, 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.

Weight , Waist r = 0.885 (x = weight of male, y = waist size of male)

Time and Motion In a physics experiment at Doane College, a soccer ball was thrown upward from the bed of a moving truck. The table below lists the time (sec) that has lapsed from the throw and the height (m) of the soccer ball. What do you conclude about the relationship between time and height? What horrible mistake would be easy to make if the analysis is conducted without a scatterplot?

Time (sec)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

Height (m)

0.0

1.7

3.1

3.9

4.5

4.7

4.6

4.1

3.3

2.1

Testing for a Linear Correlation. In Exercises 13–28, 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

Best Multiple Regression Equation For the regression equation given in Exercise 1, the P-value is 0.000 and the adjusted \({R^2}\)value is 0.925. If we were to include an additional predictor variable of neck size (in.), the P-value becomes 0.000 and the adjusted\({R^2}\)becomes 0.933. Given that the adjusted \({R^2}\)value of 0.933 is larger than 0.925, is it better to use the regression equation with the three predictor variables of length, chest size, and neck size? Explain.
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