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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

Exercises 13鈥�28 use the same data sets as Exercises 13鈥�28 in Section 10-1. In each case, find the regression equation, letting the first variable be the predictor (x) variable. Find the indicated predicted value by following the prediction procedure summarized in Figure 10-5 on page 493.

Using the listed old/new mpg ratings, find the best predicted new

mpg rating for a car with an old rating of 30 mpg. Is there anything to suggest that the prediction is likely to be quite good?

\({s_e}\)Notation Using Data Set 1 鈥淏ody Data鈥� in Appendix B, if we let the predictor variable x represent heights of males and let the response variable y represent weights of males, the sample of 153 heights and weights results in\({s_e}\)= 16.27555 cm. In your own words, describe what that value of \({s_e}\)represents.

In Exercises 9 and 10, use the given data to find the equation of the regression line. Examine the scatterplot and identify a characteristic of the data that is ignored by the regression line.

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.)

Internet and Nobel Laureates Listed below are numbers of Internet users per 100 people and numbers of Nobel Laureates per 10 million people (from Data Set 16 鈥淣obel Laureates and Chocolate鈥� in Appendix B) for different countries. Is there sufficient evidence to conclude that there is a linear correlation between Internet users and Nobel Laureates?

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0.1

Confidence Intervals for a Regression Coefficients A confidence interval for the regression coefficient b1 is expressed

\(\begin{array}{l}{b_1} - E < {\beta _1} < {b_1} + E\\\end{array}\)

Where

\(E = {t_{\frac{\alpha }{2}}}{s_{{b_1}}}\)

The critical t score is found using n 鈥�(k+1) degrees of freedom, where k, n, and sb1 are described in Exercise 17. Using the sample data from Example 1, n = 153 and k = 2, so df = 150 and the critical t scores are \( \pm \)1.976 for a 95% confidence level. Use the sample data for Example 1, the Stat diskdisplay in Example 1 on page 513, and the Stat Crunchdisplay in Exercise 17 to construct 95% confidence interval estimates of \({\beta _1}\) (the coefficient for the variable representing height) and\({\beta _2}\) (the coefficient for the variable representing waist circumference). Does either confidence interval include 0, suggesting that the variable be eliminated from the regression equation?
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