Regression and Correlation
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Two problems, one may require the working of two problems for correlation. Data and all questions included (if needed). The only questions required are 15.7 and 15.22.
15.7
As Communications and the media change and become more important in the economy, the radio industry has become an area of concern. One of the major communications companies wanted to look at the relationship between the number of radio stations a company owned and the revenues generated by radio. It collected the following data:
American Radio Companies
Company Number of Revenues
Stations (Billions)
Westinghouse/Infinity 83 1.05
Jacor 57 0.31
Clear Channel 104 0.31
Evergreen 35 0.30
Disney/ABC 21 0.29
American Radio Systems 63 0.23
SFX 67 0.22
Chancellor 41 0.21
Cox 38 0.21
Bonneville 20 0.12
a) Create a scatter plot of the data. Do you think that there is a relationship between the number of stations that a company owns and the revenues generated by radio?
b) Find the linear regression line for the data.
c) Use the regression line to predict the radio revenues for Chancellor and Westinghouse/Infinity. Which prediction is better?
d) Leave out the data point for Westinghouse/Infinity and recalculate the regression line. Do you think that this line will do a better job of predicting radio revenues? Why or why not?
15.22 (Extra Question)
Important (15.22 Requires 15.3/15.18)
Questions for 15.3/15.18 only provided if needed to solve 15.22.
15.3
As part of an international study on energy consumption, data were collected on the number of cars in a country and the total travel in kilometers. The data for 12 of the countries are shown below:
(millions) (billion km)
Country Total Cars Travel
US 142.35 3140.29
Finland 1.82 34.66
Denmark 1.66 30.76
Britain 21.32 352.76
Australia 8.53 138.22
Sweden 3.32 53.21
Netherlands 5.53 83.69
France 23.27 348.20
Norway 1.59 23.54
Italy 26.12 367.85
Germany 43.75 608.52
Japan 40.25 439.30
a) Create a scatter plot of the data. Do you think that there is a linear relationship between number of kilometers traveled and the number of cars?
b) Find the least-squares regression line for the data. Interpret the value of the slope.
c) Does the intercept make sense for these data? Why or why not?
d) Plot the regression line on the same plot with the data? Does the line make you feel confident about predicting travel as a function of the number of cars?
e) Use the regression line to predict the number of kilometers traveled for Sweden and Japan. How well do the predictions agree with the original data?
15.18
For the data on the number of automobiles in a country and the total number of kilometers traveled, the relevant data are
Ex= 319.51 Ex(squared) = 25,598.8975 Sy|x = 156.0981 n=12
a) Find 99% confidence intervals for the number of kilometers traveled for X = 1.5 and 10 (million) cars.
b) Find 99% prediction intervals for the two values of X from part (a).
15.22
Look at the model you found for the data on the number of automobiles in a country and the total number of kilometers traveled.
a) Make a residual plot of the data versus the independent variables.
b) What does the residual plot tell you about the assumption of a linear relationship? The assumption of equal variances?
c) Using the residual plot, the computer output, or a plot of the values of the independent variable, do you think that any observation had a lot of influence on the model? If so, which point(s) are they?
d) Do you think that the assumption of normality is reasonable?
f) Considering your answers to parts (a) - (d), do you think that the linear model is appropriate for these data? Why or why not?
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Solution Summary
Two problems, one may require the working of two problems for correlation. Data and all questions included (if needed). The only questions required are 15.7 and 15.22.
15.7
As Communications and the media change and become more important in the economy, the radio industry has become an area of concern. One of the major communications companies wanted to look at the relationship between the number of radio stations a company owned and the revenues generated by radio. It collected the following data:
American Radio Companies
Company Number of Revenues
Stations (Billions)
Westinghouse/Infinity 83 1.05
Jacor 57 0.31
Clear Channel 104 0.31
Evergreen 35 0.30
Disney/ABC 21 0.29
American Radio Systems 63 0.23
SFX 67 0.22
Chancellor 41 0.21
Cox 38 0.21
Bonneville 20 0.12
a) Create a scatter plot of the data. Do you think that there is a relationship between the number of stations that a company owns and the revenues generated by radio?
b) Find the linear regression line for the data.
c) Use the regression line to predict the radio revenues for Chancellor and Westinghouse/Infinity. Which prediction is better?
d) Leave out the data point for Westinghouse/Infinity and recalculate the regression line. Do you think that this line will do a better job of predicting radio revenues? Why or why not?
15.22 (Extra Question)
Important (15.22 Requires 15.3/15.18)
Questions for 15.3/15.18 only provided if needed to solve 15.22.
15.3
As part of an international study on energy consumption, data were collected on the number of cars in a country and the total travel in kilometers. The data for 12 of the countries are shown below:
(millions) (billion km)
Country Total Cars Travel
US 142.35 3140.29
Finland 1.82 34.66
Denmark 1.66 30.76
Britain 21.32 352.76
Australia 8.53 138.22
Sweden 3.32 53.21
Netherlands 5.53 83.69
France 23.27 348.20
Norway 1.59 23.54
Italy 26.12 367.85
Germany 43.75 608.52
Japan 40.25 439.30
a) Create a scatter plot of the data. Do you think that there is a linear relationship between number of kilometers traveled and the number of cars?
b) Find the least-squares regression line for the data. Interpret the value of the slope.
c) Does the intercept make sense for these data? Why or why not?
d) Plot the regression line on the same plot with the data? Does the line make you feel confident about predicting travel as a function of the number of cars?
e) Use the regression line to predict the number of kilometers traveled for Sweden and Japan. How well do the predictions agree with the original data?
15.18
For the data on the number of automobiles in a country and the total number of kilometers traveled, the relevant data are
Ex= 319.51 Ex(squared) = 25,598.8975 Sy|x = 156.0981 n=12
a) Find 99% confidence intervals for the number of kilometers traveled for X = 1.5 and 10 (million) cars.
b) Find 99% prediction intervals for the two values of X from part (a).
15.22
Look at the model you found for the data on the number of automobiles in a country and the total number of kilometers traveled.
a) Make a residual plot of the data versus the independent variables.
b) What does the residual plot tell you about the assumption of a linear relationship? The assumption of equal variances?
c) Using the residual plot, the computer output, or a plot of the values of the independent variable, do you think that any observation had a lot of influence on the model? If so, which point(s) are they?
d) Do you think that the assumption of normality is reasonable?
f) Considering your answers to parts (a) - (d), do you think that the linear model is appropriate for these data? Why or why not?
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