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Unformatted text preview: Midterm 2 Practice Problems With Solutions Problem 1: Television Ads: You own a chain of stores that sells television sets and you want to know whether your advertising is increasing your sales. Let Y be the number of TVs you sell in a given month, and let X be the amount of money you spend on advertising in a given month in thousands of dollars. You have data on advertising expenditures and sales for n=42 months and have fit a simple linear regression of Y on X. The printout for this regression is given below along with a few useful summary statistics. Use it to answer the following questions: The regression equation is TV Sales = 48.4 + 10.2 Ad-Spending Parameter Estimates Predictor Coef Stdev t-ratio p Constant 48.40 17.61 2.75 0.009 Ad Spending 10.2457 0.5224 19.61 0.000 RMSE = 38.54 R-sq = 90.6% R-sq(adj) = 90.3% Analysis of Variance SOURCE DF SS MS F p Regression 1 571411 571411 384.70 0.000 Error 40 59413 1485 Total 41 630824 1 Summary Statistics For Number of Televisions Sold Per Month N MEAN MEDIAN STDEV MIN MAX TV Sales 42 373.5 363.0 124.0 146.0 609.0 (a) What percentage of variability in television sales is explained by advertising expenditures? Does the model do a good job in this respect? Explain. Solution: The percentage of variability explained is given by R 2 = 90 . 6% or R 2 adj = 90 . 3% from the printout above. The second number is more accurate since it takes the degrees of freedom into account. However, we accepted either one. They are actually very similar in this case–the regression has explained just over 90% of the variability in television sales. Since the maximum is 100% this seems like a pretty high amount. I would say the regression is doing a good job of explaining the variability in sales. (b) Do advertising expenditures give good predictions for the number of television sales? Briefly justify your answer using appropriate numbers from the printouts. Solution: The key to answering this question is to compare the average error we are making in our predictions with the values we are trying to predict. The average error is given by RMSE = 38.54. Since Y is television sales, this means that when we try to predict the number of TVs sold each month we are off by about 38 sets. On average (see the “Summary Statistics Table” on the printout) we sell around 373.5 TVs per month with a low one month of 146 and a high another month of 609. So it looks like typically we make an error of around 10% (38.54/373.5) in our predictions. Personally I think this is a fairly big error–I am off by several dozen TVs–but we gave credit for saying it wasn’t a big error as long as you made it clear you understood that you needed to compare RMSE to the Y values. We gave a small amount of partial credit for certain other answers (such as p-values). However, remember that R 2 , SSR, etc. do NOT tell you whether the regression gives good predictions–they simply say whether you have explained a lot. Even if you explain a lot the remaining error may be significant.explain a lot the remaining error may be significant....
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This note was uploaded on 03/05/2008 for the course BIOSTAT 100B taught by Professor Sugar during the Winter '07 term at UCLA.
- Winter '07