Section 2.3 Solutions

 

2.42

  1. The slope of the regression is 2.59. Hence, as the revenue of an NBA team increases by 1 million, the predicted value of the team increases by 2.59 million.

 

  1. Predicted value for Los Angeles Lakers:  million dollars. Then the residual (i.e., the error) is 447 – 407.31 = 39.69 million dollars.

 

  1. Note that . Hence, the least-squares regression line explains 85.84% of the variation in NBA team value. Because this value is so high, we can conclude the regression line is pretty successful in its predictions.

 

 

 

2.43

  1. For each additional year, the predicted discharge of the Mississippi River increases by 4.2255 cubic kilometers.

 

  1. Predicted value for 1780:  cubic kilometers. It’s impossible to discharge a negative amount of water, indicating that this extrapolation is nonsense. The regression line should be used for predictions only within the range of the data collected.

 

  1. Predicted value for 1990:  cubic kilometers. Then the residual (i.e., the error) is 680 – 617 = 63 cubic kilometers.

 

  1. There are high spikes in the timeplot in the years 1973 and 1993, indicating an unusually high discharge (which would go along with a flood).

 

 

 

2.55

We are interested in predicting a husband’s height from his wife’s height. Hence, husband height is the response (y) variable and wife height is the explanatory (x) variable. Then the slope of the regression line is  and the y-intercept is  Thus, the equation of the regression line (shown in the graph below) is  For a woman 67 inches tall, the predicted height of her husband is 33.67 + 0.54(67) = 69.85 inches.

 

 

 

2.56

The equation for the least-squares regression line is

 (plugging in the formula for the y-intercept). Note what happens when we plug in  as out x-value:

Hence, when  is our x-value,  is our y-value. That is, the point  is always on the least-square regression line.

 

 

2.58

Professor Friedman wants to predict final exam scores from pre-exam totals. Hence, final exam score is the response (y) variable and pre-exam total is the explanatory (x) variable.

 

  1. The slope of the regression line is  and the y-intercept is

 

  1. For Julie’s pre-exam total of 300, the predicted final exam score is 30.2 + (0.16)(300) = 78.2.

 

  1. The least-squares regression line only explains 36% of the variation in final exam scores. This is not very high, so Julie’s final exam score may be quite different from the predicted value. She has a legitimate argument.

 

 

2.59

Since is 0.16, r is either 0.4 or –0.4. Since there is a positive association between class attendance and grades, r must be 0.4.