We are interested in using the pH of the lake water (which is easy to measure) to predict the average mercury level in fish from the lake, which is hard to measure. Let x be the pH of the lake water and Y be the average mercury level in fish from the lake. A sample of n = 10 lakes yielded the following data: Observation (i) pH (x;) Average mercury level (y¡) 0.15 1 3 4 6 7 8 10 8.2 8.4 7.0 7.2 7.3 6.4 9.1 5.8 7.6 8.1 0.04 0.40 0.50 0.27 0.81 0.04 0.83 0.05 0.19 Suppose we fit the data with the following regression model: Y-α+ βα;+ εi, i-1,. . . , 10 , where ɛ; ~ N(0, o²) are independent. We have the following quantities: = J= Σ% = 0.328, ΣΗ= 572.71 , ΣΗ 1.8922 , Σ" T;J; 22.218. Η ΣΗ = 7.51, n n i=1 i=1 Some R output that may help. > p1 <- c(0.01, 0.025, 0.05, 0.1, o.9, 0.95, 0.975, 0.99) > qt (p1, 8) [1] -2.896 -2.306 -1.860 -1.397 1.397 1.860 2.306 2.896 > qt (p1, 9) [1] -2.821 -2.262 -1.833 -1.383 1.383 1.833 2.262 2.821 (a) Find the ordinary least squares (OLS) estimates (denoted as â and B) of the regression coefficients (a and 3).

We are interested in using the pH of the lake water (which is easy to measure) to predict the average mercury level in fish from the lake, which is hard to measure. Let x be the pH of the lake water and Y be the average mercury level in fish from the lake. A sample of n = 10 lakes yielded the following data: Observation (i) pH (x;) Average mercury level (y¡) 0.15 1 3 4 6 7 8 10 8.2 8.4 7.0 7.2 7.3 6.4 9.1 5.8 7.6 8.1 0.04 0.40 0.50 0.27 0.81 0.04 0.83 0.05 0.19 Suppose we fit the data with the following regression model: Y-α+ βα;+ εi, i-1,. . . , 10 , where ɛ; ~ N(0, o²) are independent. We have the following quantities: = J= Σ% = 0.328, ΣΗ= 572.71 , ΣΗ 1.8922 , Σ" T;J; 22.218. Η ΣΗ = 7.51, n n i=1 i=1 Some R output that may help. > p1 <- c(0.01, 0.025, 0.05, 0.1, o.9, 0.95, 0.975, 0.99) > qt (p1, 8) [1] -2.896 -2.306 -1.860 -1.397 1.397 1.860 2.306 2.896 > qt (p1, 9) [1] -2.821 -2.262 -1.833 -1.383 1.383 1.833 2.262 2.821 (a) Find the ordinary least squares (OLS) estimates (denoted as â and B) of the regression coefficients (a and 3).

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter7: Analytic Trigonometry

Section7.6: The Inverse Trigonometric Functions

Problem 92E

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