Question
In: Statistics and Probability
15.30 Note: This exercise requires a computer and statistical software. The data mentioned below can be...
15.30 Note: This exercise requires a computer and
statistical software. The data mentioned below can be found in
several formats on your Premium Website under "Chapter
datasets".
For the top law firms in the world in terms of profit per equity
partner, data file XR15012 lists the number of equity partners and
the gross revenue ($ millions) for the most recent fiscal year.
Given these data, determine the least-squares equation for
predicting gross revenue on the basis of the number of equity
partners, then interpret its slope. (SOURCE: law.com,
August 8, 2009.)
Please round your answers to 1 decimal place.
a. The point estimate for gross revenue when there are 200 equity partners is $ million
b. For an individual law firm having 200 equity partners, the lower bound of the 90% prediction interval for gross revenue is $ million
c. For an individual law firm having 200 equity partners, the upper bound of the 90% prediction interval for gross revenue is $ million
c. For all law firms having 200 equity partners, the lower bound of the 90% confidence interval for their mean gross revenue is $ million.
d. For all law firms having 200 equity partners, the upper bound of the 90% confidence interval for their mean gross revenue is $ million.
| partners | gross_revenue |
| 76 | 578.5 |
| 90 | 610.5 |
| 167 | 985.0 |
| 126 | 839.0 |
| 79 | 384.5 |
| 401 | 2358.5 |
| 109 | 966.0 |
| 76 | 587.0 |
| 111 | 651.0 |
| 440 | 2588.5 |
| 112 | 642.5 |
| 220 | 1310.0 |
| 79 | 419.5 |
| 163 | 836.5 |
| 395 | 2660.5 |
| 159 | 478.0 |
| 137 | 709.5 |
| 421 | 2170.0 |
| 445 | 2005.5 |
| 363 | 2034.5 |
| 133 | 603.0 |
| 189 | 894.0 |
| 192 | 1175.0 |
| 131 | 646.5 |
| 142 | 844.5 |
| 194 | 975.0 |
| 265 | 907.5 |
| 183 | 921.0 |
| 287 | 1373.0 |
| 144 | 772.0 |
| 224 | 934.0 |
| 92 | 430.5 |
| 142 | 537.5 |
| 125 | 577.5 |
| 85 | 431.0 |
| 156 | 628.0 |
| 174 | 611.0 |
| 281 | 978.0 |
| 136 | 464.0 |
| 187 | 1074.0 |
| 119 | 531.0 |
| 251 | 1033.0 |
| 129 | 485.0 |
| 134 | 615.5 |
| 332 | 1386.0 |
| 139 | 743.5 |
| 254 | 733.0 |
| 231 | 959.0 |
| 124 | 470.5 |
| 288 | 1200.0 |
| 200 | 577.5 |
| 144 | 579.0 |
| 185 | 697.5 |
| 242 | 894.5 |
| 158 | 594.5 |
| 83 | 372.5 |
| 318 | 1183.0 |
| 276 | 1134.5 |
| 199 | 752.5 |
| 711 | 2188.0 |
| 99 | 367.5 |
| 235 | 596.0 |
| 296 | 880.0 |
| 107 | 391.0 |
| 178 | 467.0 |
| 187 | 457.0 |
| 144 | 461.0 |
| 145 | 781.0 |
| 320 | 944.0 |
| 85 | 354.5 |
| 148 | 518.0 |
| 138 | 491.0 |
| 137 | 475.0 |
| 268 | 892.0 |
| 109 | 462.5 |
| 174 | 590.0 |
| 92 | 329.0 |
| 238 | 720.5 |
| 130 | 478.0 |
| 211 | 508.0 |
| 248 | 653.5 |
| 155 | 530.5 |
| 123 | 375.0 |
| 243 | 755.0 |
| 510 | 1441.0 |
| 356 | 649.5 |
| 159 | 394.5 |
| 147 | 263.5 |
| 169 | 412.0 |
| 196 | 612.5 |
| 215 | 442.5 |
| 176 | 469.0 |
| 192 | 367.0 |
| 174 | 349.0 |
| 211 | 456.5 |
| 84 | 228.0 |
| 203 | 357.0 |
| 146 | 315.0 |
| 64 | 193.5 |
| 248 | 305.0 |
Solutions
Expert Solution
the leaset square regression line is given as
gross_revenue(y)=a+b*partners(x)=27.99+3.81*x
(a) for x=200, y=27.99+3.81*200=789.99
(b)
(1-alpha)100% prediction interval of y^(for given x)=y^
t(alpha/2,error df)*Se*sqrt(1+1/n+(x-x-
)2/Sxx)
90% prediction interval=
=789.991.66*301.83*sqrt(1+1/100+(200-196.08)*(200-196.08)/1097245)=789.99503.54
=(286.45, 1293.53)
(c) (1-alpha)100%confidence interval of y^(for given x)=y^
t(alpha/2,error df)*Se*sqrt(1/n+(x-
)2/Sxx)
90%confidence interval=
=789.991.66*301.83*sqrt(1/100+(200-196.08)*(200-196.08)/1097245)=789.9950.14=(739.49,740.13
following regression analysis information has been generated using ms-excel
| SUMMARY OUTPUT | ||||||
| Regression Statistics | ||||||
| Multiple R | 0.80030665 | |||||
| R Square | 0.640490734 | |||||
| Adjusted R Square | 0.636822272 | |||||
| Standard Error | 301.8331351 | |||||
| Observations | 100 | |||||
| ANOVA | ||||||
| df | SS | MS | F | Significance F | ||
| Regression | 1 | 15906061.89 | 15906062 | 174.5938 | 1.7E-23 | |
| Residual | 98 | 8928117.661 | 91103.24 | |||
| Total | 99 | 24834179.55 | ||||
| Coefficients | Standard Error | t Stat | P-value | Lower 95% | Upper 95% | |
| Intercept | 27.98837174 | 64.05685582 | 0.43693 | 0.663123 | -99.1304 | 155.1071 |
| X Variable 1 | 3.807408345 | 0.288147621 | 13.2134 | 1.7E-23 | 3.235589 | 4.379228 |
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