Question

The following sample observations were randomly selected: (Round the final answers to 4 decimal places.)

X: 16 14 9 17 12 11 20 18

Y: 14 21 1 13 11 16 14 15

a. Determine the 95% confidence interval for the mean predicted when X = 6.

b. Determine the 95% prediction interval for an individual predicted when X = 6.

Answer 1

Least Squares Regression Equation: y^^= b_0 + b_1x b_1 = n sigma xy - (sigma x)(sigma y)/n sigma x^2 - (sigma x)^2 = 8(1606) - (117)(105)/8(1811) - (117)^2 = 12848 - 12285/14488 - 13689 = 563/799 almostequalto 0.704631 almostequalto 0.7046 Slope term, b_1 = 0.7046 b_0 = y^- - b_1x^- = sigma y/n - b_1 sigma x/n 105/8 -(0.704631) 117/8 almostequalto 2.819775 almostequalto 2.8198 Intercept term, b_0 = 2.8198 Least Squares Regression Eq^n: y^^= 2.8198 + (0.7046)x SS_xx = sigma x^2 - (sigma x)^2/n = 1811 - (117)^2/8 = 99.875 SS_yy = sigma y^2 -(sigma y)^2/n = 1605 - (105)^2/8 = 226.875 SS_xy = sigma xy - (sigma x)(sigma y)/n = 1606 - (117)(105)/8 = 70.375 SST = SS_yy = 226.875 SSR = SS^2_xy/SS_xx = 70.375^2/99.875 = 49.588392 SSE = SST - SSR = 226.875 - 49.588392 = 177.286608 95% for mu_y at x_0 = 6 95% CI of mean value of y at given x_0 is: y^^plusminus E, where E is margin of error. y^^= b_0 + b_1x_0 y^^= (2.8198) + (0.7046)(6) almostequalto 7.0476 E = t_a/2, df = n - 2 times s_e squareroot (1/n + (x_0 - x^-)^2/S_xx) = 2.447