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Math3080-001 Spring 2009 Exam #2 1.) Name:_______________________________ In simple linear regression analysis, _________________, measures the amount of variation in observed y values that is not explained by the model - that is, the amount of y-data variation that cannot be attributed to a linear relationship, and ____________________ measures the total amount of variation in observed y values. 2.) An experiment to measure the macroscopic magnetic relaxation time in crystals ( µ sec) as a function of the strength of the external biasing magnetic field (KG) yielded the following data x y 11.0 187 12.5 225 15.2 305 17.2 318 19.0 367 x y 22.0 400 24.2 435 25.3 450 27.0 506 29.0 558 The summary statistics are ∑x i = 223.2, ∑ yi = 4116, ∑ xi2 = 4877.50, ∑ xi yi = 90, 096.1, ∑y a.) Compute the equation of the estimated regression line. 2 i 20.8 365 = 1, 666, 782. b.) Suppose it is possible to make a single observation at each of the n=20 values x1 = 11.0, x2 = 11.5 , . . . , x20 = 20.5 . If a major objective is to estimate β1 as accurately as possible, would the experiment with n=20 be preferable to the one with n=11? Explain. 3.) Suppose that in a certain chemical process the expected value of y = reaction time (hours) is related to x = temperature (o F) in the chamber in which the reaction takes place according to a simple linear regression model with equation y = 5.00 - .01x and σ = .075. a.) What is the expected change in reaction time for a 10 o F increase in temperature? b.)Let Y1 and Y2 denote observations of reaction time when x = x1 and x = x2 respectively. If x2 − x1 = 1 o F, then what is P(Y2 > Y1 ) ? 4.) An investigation of the relationship between x = traffic flow (1000’s of cars per 24 hours) and y = lead content in bark on trees near the highway ( µ g / g dry wt) yielded the data in the accompanying table. 8.3 227 x y 8.3 312 12.1 12.1 17.0 362 521 640 17.0 539 17.0 728 24.3 945 24.3 738 = 5,390,382, ∑x y 24.3 33.6 759 1263 The summary statistics are: n = 11, ∑x i = 198.3, ∑y i = 7034 , ∑x 2 i = 4198.03, ∑y 2 i i i = 149,354.4 SSE = 76,492.54, and SST = 892,458.73 In addition, the least squares estimates are given by: βˆ0 = −12.84159, and βˆ1 = 36.18385 a.) What proportion of observed variation in lead content can be explained by the approximate linear relationship between the variables? b.) Does it appear that there is a useful linear relationship between the two variables? Conduct a test at significance level α = .01. 5.) The simple linear regression model provides a very good fit to a data set on rainfall volume and runoff volume. The equation of the least squares line is 2 yˆ = −1.128 + .82697 x , r = .975 , s = 5.24 , and n=15. a.) Use the fact that sYˆ = 1.44 when rainfall volume is 50 m 3 to calculate a 95% CI for true average runoff volume b.) Calculate a 95% PI for runoff when rainfall volume is 50 m 3 . How does this interval compare to the one you found in (a)? Why?