# Consider the results of a simple linear regression

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Consider the results of a simple linear regression model of y = revenues on x = sales: a. Test whether the number of cars sold is an important predictor variable (use signi fi cance level 0.05). b. Calculate a 95% con fi dence interval for the regression coef fi cient of number of cars sold. students, GPA, and their GMAT scores taken before entering the MBA program are given below. Use the GMAT scores as a predictor of GPA, and conduct a regression of GPA on GMAT scores. x = GMAT y = GPA 560 3.20 540 3.44 520 3.70 580 3.10 520 3.00 620 4.00 660 3.38 630 3.83 550 2.67 550 2.75 600 2.33 537 3.75 a. Obtain and interpret the coef fi cient of determination R 2 . b. Calculate the fi tted value for the second person. c. Test whether GMAT is an important predictor variable (use signi fi cance level 0.05). 2.10. The following are the results of a regression of fuel ef fi ciency (gallons per 100 miles traveled) on the weight (in pounds) of the car. A total of 45 cars were considered. c. Calculate a 90% con fi dence interval for 2.9. Grade point averages of 12 graduating MBA students, GPA, and their GMAT scores taken before entering the MBA program are given below. Use the GMAT scores as a predictor of GPA, and conduct a regression of GPA on GMAT scores. x = GMAT y = GPA 560 3.20 540 3.44 520 3.70 580 3.10 520 3.00 620 4.00 660 3.38 630 3.83 550 2.67 550 2.75 600 2.33 537 3.75 a. Obtain and interpret the coef fi cient of determination R 2 . b. Calculate the fi tted value for the second person. c. Test whether GMAT is an important predictor variable (use signi fi cance level 0.05). 2.10. The following are the results of a regression of fuel ef fi ciency (gallons per 100 miles traveled) on the weight (in pounds) of the car. A total of 45 cars were considered.
the regression coef fi cient of number of cars sold.
d. Obtain the coef fi cient of determination.
e. Determine the standard deviation among revenues ( y ), after factoring in the explanatory variable sales ( x ). Compare this standard deviation to the standard deviation of y without considering the explanatory variable.
f. Estimate the revenues for BMW.
Abraham Abraham ˙ C02 November 8, 2004 0:36 Exercises 59 a. Determine an approximate 95% prediction interval for the fuel ef fi ciency of an automobile weighing 2000 pounds. The computer output does not give you the information to construct exact prediction intervals. Approximate the prediction intervals, assuming that the sample size n is large enough to allow you to ignore the parameter estimation uncertainty. b. Determine an approximate 95% prediction interval for the fuel ef fi ciency of an automobile weighing 1500 pounds. 2.11. Discuss the functional relationship between the coef fi cient of determination R 2 and the F ratio. 2.12. Occasionally, a model is considered in which the intercept is known to be zero a priori. Such a model is given by y i = β 1 x i + i , i = 1 , 2 ,..., n where the errors i follow the usual assumptions. a. Obtain the LSEs ( ˆ β 1 , s 2 ) of ( β 1 2 ) . b. De fi ne e i = y i ˆ β 1 x i . Is it still true that n i = 1 e i = 0? Why or why not? c. Show that V ( ˆ β 1 ) = σ 2 / n i = 1 x 2 i . 2.13. The data listed in the fi le sriver include the water content of snow on April 1 ( x ) and the water yield from April to July ( y ) in the Snake River watershed in Wyoming. Information on n = 17 years (from 1919 to 1935) is listed (see Weisberg, 1980). a. Fit a regression through the origin ( y = β 1 x + ), and fi nd ˆ β 1 and s 2 . Obtain a 95% con fi dence interval for β 1 .