Question

"In Exercises 17 and 18, use the data to (a) find the coefficient of determination r^2 and interpret
the result, and (b) find the standard error of estimate s_e and interpret the result.

18. [APPLET] The table shows the cooking areas (in square inches) of 18 gas grills and their prices (in dollars). The regression equation is y = 1.501x - 341.501. (Source: Lowe's)

Accepted Answer

Step 1: Calculate the coefficient of determination (r²). This involves first finding the correlation coefficient (r) between the cooking area (x) and the price (y). You can use the formula for r: r = \frac{n\sum xy - \sum x \sum y}{\sqrt{(n\sum $$x^2$$ - (\sum $$x)^2)(n$$\sum $$y^2$$ - (\sum $$y)^2$$)}}, where n is the number of data points. Then square the correlation coefficient to get r². Step 2: Interpret the coefficient of determination (r²). Explain that r² represents the proportion of the variance in the dependent variable (price) that is predictable from the independent variable (cooking area). For example, an r² of 0.8 means 80% of the variation in price is explained by the cooking area. Step 3: Calculate the standard error of estimate (s_e). Use the regression equation \hat{y} = 1.501x - 341.501 to find predicted prices for each cooking area. Then compute the residuals (differences between observed and predicted prices). The formula for s_e is s_e = \sqrt{\frac{\sum (y - \hat{y}$$)^2$$}{n - 2}}, where n is the number of data points. Step 4: Interpret the standard error of estimate (s_e). Explain that s_e measures the typical distance that the observed prices fall from the regression line. A smaller s_e indicates that the regression line fits the data points more closely. Step 5: Summarize the findings by relating both r² and s_e to the quality of the regression model in predicting grill prices based on cooking area.

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Concetti chiave

Coefficient of Determination (r²)

Standard Error of Estimate (sₑ)

Linear Regression Equation