420Pr1 - Practice Problems 1. A marketing firm wishes to...

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Practice Problems 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. Alex obtains a random sample of seven students from STAT 100 class he taught in Fall Semester of 2007 and wants to use it to see if there is a relationship between the number of absences and students’ final grade. The data and the scatterplot are given below. Number of Absences, x Final Grade Percentage, y 6 70 2 92 15 47 9 72 11 49 5 96 8 78 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 regression line to the scatter plot. b) Use the equation of the least-squares regression line obtained in part (a) to predict the final grade percentage for a student who missed 20 classes. c) Use the equation of the least-squares regression line obtained in part (a) to predict the final grade percentage for a student who missed 7 classes. d) In which prediction ( part (b) or part (c) ) would we have more confidence? Why?
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e) Use the equation of the least-squares regression line obtained in part (a) to find the residuals. f)*
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This note was uploaded on 02/10/2009 for the course STAT 420 taught by Professor Stepanov during the Spring '08 term at University of Illinois at Urbana–Champaign.

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420Pr1 - Practice Problems 1. A marketing firm wishes to...

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