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Simple Linear Regression Analysis - Correlation Coefficient

Q: A very broad consensus has emerged around the proposition that global warming is a reality with likely serious global consequences. Moreover, while there is still not unanimity that is desirable to cut emissions from coal and petroleum. Finally, many energy economist and political leaders are advocating a multipronged approach to providing alternative energy including nuclear, natural gas, clean coal, and renewable sources from solar and wind. Municipalities and states have been asked by the Department of Energy to assess their energy requirements for the each of the alternative fuels. In particular, they have decided to focus initially on natural gas, given the enormity of U.S. reserves and its relative cleanliness. The regression output for selected municipalities in Illinois for 10 reporting periods (weeks) see below. The dependent variable is consumption of natural gas in millions of cubic feet (Fuelcons) and the independent variables is the temperature (Temp), measure in degrees Fahrenheit.
Regression Analysis

r² 0.836 n 10
r -0.915 k 1
Std. Error 0.766 Dep. Var. FuelCons

ANOVA table
Source SS df MS F p-value
Regression 24.0074 1 24.0074 40.89 .0002
Residual 4.6966 8 0.5871
Total 28.7040 9

Regression output confidence interval
variables coefficients std. error t (df=8) p-value 95% lower 95% upper
Intercept 15.2569 0.7913 19.282 5.43E-08 13.4323 17.0816
Temp -0.1173 0.0183 -6.395 .0002 -0.1595 -0.0750

Predicted values for: FuelCons
95% Confidence Interval 95% Prediction Interval
Temp Predicted lower upper lower upper Leverage
44 10.0976 9.5254 10.6698 8.2404 11.9548 0.105

Coefficient of determination = ______ (round to three decimal places)
Interpretation of the coefficient of determination:
A. This is the proportion of the total variation in temperature (in degrees Fahrenheit) that is explained by simple linear regression model.
B. This is the point estimate of the change in fuel consumption (in millions of cubic feet) associated with each degree (Fahrenheit) increase in temperature.
C. This is a measure of the variability of the observed values of fuel consumption from their predicted values at particular temperatures.
D. This is the proportion of the total variation in fuel consumption that is explained by the simple linear regression model.
E. This value has no practical interpretation.
Select your choice: _______ (A, B, C, D, E)
Correlation coefficient = _______ (round to three decimal places)
Interpretation of the correlation coefficient:
A. This tells us that there is a strong negative relationship between fuel consumption and temperature.
B. This is the point estimate of the change in fuel consumption (in millions of cubic feet) associated with each degree (Fahrenheit) increase in temperature.
C. Because of the negative value, this tells us to drop the temperature variable and look for other ways to explain what drives fuel consumption.
D. This tells us the proportion of the total variation in the 10 fuel consumption values that is explained by the simple linear regression model.
E. This value has no practical interpretation.
Select your choice: _______ (A, B, C, D, E)

Regression Analysis

r² 0.836 n 10
r -0.915 k1
Std. Error 0.766 Dep. Var. FuelCons

Solution Preview

Coefficient of determination = ___0.836___ (round to three decimal places)
Interpretation of the coefficient of determination:
A. This is the proportion of the total variation in temperature (in degrees Fahrenheit) that is explained by simple linear regression model.
B. This is the point estimate of the change in fuel consumption (in millions of cubic feet) associated with each degree (Fahrenheit) increase in temperature.
C. This is a measure of the variability of the observed values of fuel consumption from their predicted values at particular temperatures.
D. This is the proportion of the total ...

Solution Summary

The solution identifies correlation coefficient and coefficient of determination from a regression analysis output and provides interpretation of correlation coefficient and coefficient of determination.

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