420Pr3 - 1 A marketing firm wishes to determine whether or...

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1. A marketing firm wishes to determine whether or not there is a relationship between the number of television commercials broadcast and the sales of its product. The data, obtained from 5 different cities, are shown in the following table. Number of TV Commercials x Sales Units y 3 7 7 19 5 13 9 15 6 11 Σ x = 30, Σ y = 65, Σ x 2 = 200, Σ y 2 = 925, Σ x y = 420, Σ ( x x ) 2 = 20, Σ ( y y ) 2 = 80, Σ ( x x ) ( y y ) = Σ ( x x ) y = 30. Consider the model Y i = β 0 + β 1 x i + ε i ., where ε i ’s are i.i.d. N ( 0, σ 2 ). a) Find the equation of the least-squares regression line. Add the least-squares regression line to the scatter plot. b) In Anytown, 20 commercials aired. What is your prediction of the sales? Why is it dangerous to predict sales for this particular value of x . c) Find an estimate for σ , the standard deviation of the observations about the true regression line? d) What proportion of the observed variation in the sales is explained by a straight-line relationship with the number of television commercials for the product? e) Construct a 90% confidence interval for β 1 . f) Test for the significance of the regression at a 5% level of significance. That is, test H 0 : β 1 = 0 vs. H 1 : β 1 0 at a 5% level of significance.
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g) Construct a 95% prediction interval for the sales corresponding to x = 8 TV commercials. h) Test H 0 : μ ( x = 8 ) = 20 vs. H 1 : μ ( x = 8 ) < 20 at a 10% level of significance. i) Test H 0 : β 0 = 0 vs. H 1 : β 0 0 at a 10% level of significance. 2. An agronomist experimented with different amounts of liquid fertilizer on a sample of equal-size plots. The amount of fertilizer and the yields are: Plot Amount of Fertilizer (tons) Yield (hundreds of bushels) A 0 4 B 1 4.5 C 1 5.5 D 1.5 5.5 E 2 4.5 F 2.5 6.5 G 3 5.5 H 3 7
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This note was uploaded on 04/29/2010 for the course STAT stat 420 taught by Professor Stepanov during the Spring '07 term at University of Illinois at Urbana–Champaign.

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420Pr3 - 1 A marketing firm wishes to determine whether or...

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