Question

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1. We want to predict to number of days till an Ebola outbreak spreads to a gorilla habitat based on the distance (in miles) the habitat is from the source of the outbreak. The data are: Distance: Days: 3 21 4 33 4 41 4 43 5 46 4 It may be helpful to know that SSxx = 9.5, SSyy = 1301.33, SSxy = 107, Exi = 21, and Ey= 188 a) (10 pts.) Give the least squares line and interpret the coefficients in terms of the original problem. b) (7 pts.) Calculate the coefficient of determination and interpret its value in terms of the original problem. c) (2 pts.) Predict the number of days until an Ebola outbreak spreads to a gorilla habitat that is 4.5 miles from the source of the outbreak.

Answer 1

х Y X*Y X2 y2 4 4 1 16 1 3 21 63 9 441 4 33 132 16 1089 4 41 164 16 1681 4 43 172 16 1849 5 46 230 25 2116 Sum = 21 188 765 83 7192 Based on the above table, the following is calculated: X = X; 21 6 = 3.5 n n2 1 Y = Y = 188 6 = 31.333333333333 n =1 1 SSxx *-- (2x) = 83 – 212/6 = 9.5 2 n 72 1 SSyy = y; ΣΥ, = 7192 – 1882 /6 = 1301.3333333333 n =1 i=1 1 SSxy = 3xx -- (Ex) () 765 – 21 x 188/6 = 107 Therefore, based on the above calculations, the regression coefficients (the slope m, and the y-intercept n) are obtained as follows: m = SSXY SSxx 107 9.5 = 11.2632 n=Y - X.m= 31.333333333333 - 3.5 x 11.2632 = -8.0877 Therefore, we find that the regression equation is: Y = -8.0877 + 11.2632X

b.

Obs. X Y X? Y? XY 1 1 4 1 16 4 2 3 21 9 441 63 3 4 33 16 1089 132 4 4 41 16 1681 164 5 4 43 16 1849 172 6 5 46 25 2116 230 Sum= 21 188 83 7192 765 Based on the table above, we compute the following sum of squares that will be used in the calculation of the correlation coefficient SSxx Σx? (2x) 83 x 441 9.5 2 1 SSYY Y? Y1 n2 i=1 11 1 7192 x 35344 6 1301.3333 1 SSXY (*) (2) n2 i=1 1 765 x 3948 6 107 Now, the correlation coefficient is computed using the following expression SSXY SSXX SSyy 107 19.5 x 1301.3333 0.9623 Then, the coefficient of determination, or R-Squared coefficient (Rº), is computed by simply squaring the correlation coefficient that was found above. So we get: R2 0.96232 0.9261 Therefore, based on the sample data provided, it is found that the coefficient of determination is R² = 0.9261. This implies that approximately 92.61% of variation in the dependent variable is explained by the independent variable

c.

Yhat = -8.0877 + 11.2632 X

Yhat = -8.0877 + 11.2632 * 4.5

Yhat = 42.5967

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