Use technology to calculate the equation of the leastsquares regression line relating y price to x screen size Lesson 27 Assessing a Regression Model Objectives Use a residual plot to determine whether a regression model is appropriate ID: 1026953
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2. Do nowHere are prices and screen sizes (in inches, measured diagonally) for 7 different sizes of one brand of LED HD television. Use technology to calculate the equation of the least-squares regression line relating y = price to x = screen size.
3. Lesson 2.7: Assessing a Regression Model
4. ObjectivesUse a residual plot to determine whether a regression model is appropriate.Interpret the standard deviation of the residuals. Interpret .
5. We can also use residuals to assess whether a regression model is appropriate by making a residual plot.A residual plot is a scatterplot that plots the residuals on the vertical axis and the explanatory variable on the horizontal axis.
6. Interpreting a Residual PlotTo determine whether the regression model is appropriate, look at the residual plot.If there is no leftover pattern in the residual plot, the regression model is appropriate. If there is a leftover pattern in the residual plot, the regression model is not appropriate.
7. Below is a scatterplot showing the relationship between Super Bowl number and the cost of a 30-second commercial for the years 1967–2013, along with the least-squares regression line. The resulting residual plot is shown in b.
8. The least-squares regression line clearly doesn’t fit this association very well! In the early years, the actual cost of an ad is always greater than the line predicts, resulting in positive residuals. From Super Bowl 11 to Super Bowl 33, the actual cost is always less than the line predicts, resulting in negative residuals. After Super Bowl 33, the actual cost is almost always greater than the line predicts, again resulting in positive residuals. This positive-negative-positive pattern in the residual plot indicates that the linear form of our model doesn’t match the form of the association. A curved model might be better in this case.
9. The scatterplot (a) showing the Ford F-150 data from Lesson 2.5, along with the corresponding residual plot (b). Looking at the scatterplot, the line seems to be a good fit for the association. You can “see” that the line is appropriate by the lack of a leftover pattern in the residual plot. In fact, the residuals look randomly scattered around the residual = 0 line.
10. You try!In Lesson 2.5, we used a least-squares regression line to model the relationship between the amount of soda remaining and the tapping time for cans of vigorously shaken soda. Here is the residual plot for that model. Use the residual plot to determine whether the regression model is appropriate.
11. Because there is no leftover pattern in the residual plot, the least-squares regression line is an appropriate model for relating the amount of soda remaining to the tapping time.
12. The standard deviation of the residuals s measures the size of a typical residual. That is, s measures the typical distance between the actual y values and the predicted y values.
13. You Try! In Lesson 2.5, we used a least-squares regression line to model the relationship between the price of a Ford F-150 and the number of miles it had been driven. The standard deviation of the residuals for this model is s = $5740. Interpret this value.
14. The actual price of a Ford F-150 is typically about $5740 away from its predicted price using the least-squares regression line with x = miles driven.The typical distance between the actual y values and the predicted y values is 5740.
15. Besides the standard deviation of the residuals s, we can also use the coefficient of determination to measure how well the regression line makes predictions.The coefficient of determination measures the percent reduction in the sum of squared residuals when using the least-squares regression line to make predictions, rather than the mean value of y. In other words, measures the percent of the variability in the response variable that is accounted for by the least-squares regression line.
16. You try!In Lesson 2.5, we used a least-squares regression line to model the relationship between the amount of soda remaining (in milliliters) and the tapping time (in seconds) for cans of vigorously shaken soda. Interpret the value =0.85 for this model.
17. 85% of the variability in the amount of soda remaining (milliliters) is accounted for by the least-squares regression line with x = tapping time (seconds).