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Chapter 13: Problem 1

Let \(x\) be the size of a house (sq \(\mathrm{ft}\) ) and \(y\) be the amount of natural gas used (therms) during a specified period. Suppose that for a particular community, \(x\) and \(y\) are related according to the simple linear regression model with \(\beta=\) slope of population regression line \(=.017\) \(\alpha=y\) intercept of population regression line \(=-5.0\) a. What is the equation of the population regression line? b. Graph the population regression line by first finding the point on the line corresponding to \(x=1000\) and then the point corresponding to \(x=2000\), and drawing a line through these points. c. What is the mean value of gas usage for houses with 2100 sq \(\mathrm{ft}\) of space? d. What is the average change in usage associated with a 1-sq-ft increase in size? e. What is the average change in usage associated with a 100-sq-ft increase in size? f. Would you use the model to predict mean usage for a 500-sq-ft house? Why or why not? (Note: There are no small houses in the community in which this model is valid.)

Short Answer

Expert verified
a. The equation of the population regression line is \(y = -5 +0.017x\). \n b. The points (1000, 12) and (2000, 29) are plotted on the regression line. \n c. The mean value of gas usage for houses with 2100 sq. ft of space is \(30.7\) therms. \n d. The average change in usage associated with a 1-sq-ft increase is \(0.017\) therms. \n e. The average change in usage associated with a 100-sq-ft is \(1.7\) therms. \n f. The model should not be used to predict mean usage for a 500-sq-ft house because it's outside the range of observed data.

Step by step solution

01

Derive the regression equation

Given that \(y\) = \(\alpha\) + \(\beta\)x is the formula for simple linear regression, the equation of the population regression line is \(y = -5 + 0.017x\).
02

Determine the points to plot on the graph

To graph the population regression line, two points on the line need to be determined. \nFor \(x = 1000\) sq. ft, substitute \(x=1000\) into \(y = -5 + 0.017x\) to get \(y = -5 + 0.017*\ 1000 = 12\) therms. \nFor \(x = 2000\) sq. ft, substitute \(x=2000\) into \(y = -5 + 0.017x\) to get \(y = -5 + 0.017*\ 2000 = 29\) therms.
03

Plot the points and draw the line

Plot the points (1000,12) and (2000,29) on a graph and draw a line through them to represent the population regression line.
04

Predict the mean value of gas usage for a house of 2100 sq ft

Plug \(x=2100\) into the regression equation \(y = -5 + 0.017x\) to get \(y = -5 + 0.017 * 2100 = 30.7\) therms.
05

Find the average change in usage for a 1-sq-ft increase

The average change in usage associated with a 1-sq-ft increase in size is the slope of the regression line, \(\beta = 0.017\) therms/sq. ft
06

Find the average change in usage for a 100-sq-ft increase

The average change in usage associated with a 100-sq-ft increase in size is \(\beta * 100 = 0.017* 100 = 1.7\) therms.
07

Evaluate the model’s appropriateness for predicting mean usage for a 500-sq-ft house

Given that there are no small houses in the community, the model may not be applicable or may yield inaccurate predictions when applied to a 500-sq-ft house. This is due to extrapolation -- using the model to predict for values outside the range of observed data.

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