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.
17. The table shows the times (in seconds) to accelerate from 0 to 60 miles per hour and the top speeds (in miles per hour) for eight electric cars. The regression equation is y =- 14.399x + 196.996. (Source: Car and Driver)

Accepted Answer

Step 1: Calculate the coefficient of determination, r². This is done by squaring the correlation coefficient r, which measures the strength and direction of the linear relationship between the acceleration time (x) and the top speed (y). If r is not given, you can calculate it using the formula r = (S_xy) / (sqrt(S_xx * S_yy)), where S_xy is the sum of the products of deviations, S_xx is the sum of squared deviations of x, and S_yy is the sum of squared deviations of y. Step 2: Interpret the coefficient of determination r². This value represents the proportion of the variance in the dependent variable (top speed, y) that is predictable from the independent variable (acceleration time, x). For example, an r² of 0.85 means 85% of the variation in top speed can be explained by the acceleration time. Step 3: Calculate the standard error of estimate, s_e. Use the formula se = \sqrt{\frac{\sum (y_i - \hat{y}_$$i)^2$$}{n - 2}}, where y_i are the observed values, \hat{y}_i are the predicted values from the regression equation y = -14.399x + 196.996, and n is the number of data points (8 in this case). Step 4: Interpret the standard error of estimate s_e. This value measures the average distance that the observed values fall from the regression line. A smaller s_e indicates that the data points are closer to the regression line, meaning better predictive accuracy. Step 5: Summarize your findings by explaining how well the regression model fits the data based on r² and s_e, and what this implies about the relationship between acceleration time and top speed for these electric cars.

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Coefficient of Determination (r²)

Standard Error of Estimate (sₑ)

Linear Regression Equation