TOPIC 8 - LEAST-SQUARES REGRESSION
8-1: Feeding Fido
(a) FAT = 100 - .3 (200) = 40 g
(b) FAT = 100 - 3 (120) = 64 g
(c)
(d) decrease, by 3 g.
8-2 Air Fares (cont.)
(a) $140
(b) $240
(c) slope b = (140-240)/(300-1500) = 1/12 = .083
(d) intercept a = y1 - b x1 = 140 - .083 (300) = 115
(e) FARE = .083 (DISTANCE) + 115
| Mean | Std. dev. | Correlation | |
|---|---|---|---|
| Airfare (y) | 166.9 | 59.5 |
|
| Distance (x) | 713 | 413 |
(a) b = 0.117; a = 83.3
(b) airfare = 83.3 + .117 distance
(c) 118.50
(d) 259.30
(e) 
(f) (Asks for prediction)
(g) 188.90
(a) 416.80
(b)
| Distance | 900 | 901 | 902 | 903 |
|---|---|---|---|---|
| Predicted Airfare | 188.90 | 189.02 | 189.13 | 189.25 |
(c) yes; 0.117; this number is the slope coefficient of the least squares line.
(d) 11.70
(a) 150.88
(b) 27.12
(c)
| Destination | Distance | Airfare | Fitted | Residual |
|---|---|---|---|---|
| Atlanta | 576 | 178 | 150.88 | 27.12 |
| Boston | 370 | 138 | 126.70 | 11.30 |
| Chicago | 612 | 94 | 155.10 | -61.10 |
| Dallas/Fort Worth | 1216 | 278 | 226.00 | 52.00 |
| Detroit | 409 | 158 | 131.27 | 26.73 |
| Denver | 1502 | 258 | 259.57 | -1.56 |
| Miami | 946 | 198 | 194.30 | 3.70 |
| New Orleans | 998 | 188 | 200.41 | -12.41 |
| New York | 189 | 98 | 105.45 | -7.45 |
| Orlando | 787 | 179 | 175.64 | 3.36 |
| Pittsburgh | 210 | 138 | 107.92 | 30.08 |
| St. Louis | 737 | 98 | 169.77 | -71.77 |
(d) St. Louis; distance: 737 airfare: 98; residual: 71.77; overestimate
(e) greater
(f) below
(g) mean: 0; standard deviation: 36.1
(h) 0.368
(i) 0.632
(j) The sum equals one.
(k) .632
8-8 Car's Fuel Efficiency (cont.)
(a) b = -.959 (6.70)/590 = -.0109, a = 27.56 - (-.0109)(3208) = 62.53
(b) The least-squares line predicts a car's fuel efficiency to drop by 100(-.0109) = 1.09 miles per gallon for each 100 additional pounds of weight.
(c) r2 = (-.959)2 = .920, so 92% of the variability in cars' miles per gallon is explained by the weight variable.
8-9 Governor Salaries (cont.)
(a) Gov salary = 28569.7+2.70875 (Avg pay)
(b) r2 =.212, so 21.2% of the variability in Gov salary values is explained by Avg pay.
(c) North Carolina has the largest positive residual. This means that NC has a unusually large governor salary given its average pay.
(d) Connecticut has the largest (in absolute value) negative residual. This state has an usually small governor salary given its average pay.
(e) Alaska has the largest fitted value, since it has the largest Avg pay.
8-10 Basketball Rookie Salaries
(a) salary = 2657443 - 98957 (pick number)
(b) r2 = .787, so 78.7% of the variation in salaries is explained by pick number.
(c) player drafted number 12 has fitted salary
2657443 - 98957 (12) = 1469959
His residual would be
residual = observed salary - fitted salary = 1370000 - 1469959 = -99956
(d) fitted yearly salary for player drafted 15 would be 2657443 - 98957 (15) = 1173088
fitted yearly salary for player drafted 25 would be 2657443 - 98957 (25) = 183518
(e) the regression line predicts the salary to drop by $98957 for each additional pick number.
8-14 Beatles Hit Songs
(a) There is a general negative relationship between WEEKS on the chart and PEAK position.
(b) If a song peaks at #20, we would predict (using the regression line) that it would stay
WEEKS = 11.54 - .124 (20) = 9.06 approx 9
(c) "Hey Jude" stayed 19 weeks on the chart and (since it peaked at 1) we would predict to stay
WEEKS = 11.54 - .124 (1) = 11.42
So the residual is
RESIDUAL = 19 - 11.42 = 7.58
(d) "If I Feel" has a positive residual around +2, "Don't Let Me Down" has a negative residual around 2.
(e) The negative residuals are located in the middle of the plot for moderate PEAK values; the positive residuals are located for very small and very large PEAK values. This pattern suggests a line fit is not appropriate.