# # EPage 453 # # Written by: # -- # John L. Weatherwax 2009-04-21 # # email: wax@alum.mit.edu # # Please send comments and especially bug reports to the # above email address. # #----- # Ex 12.44: # t_value = qt( 1-0.025, 25 ) 7.592 + c(-1,+1) * t_value * 0.179 s = sqrt( 18.736/25 ) 7.592 + c(-1,+1) * t_value * sqrt( 0.179^2 + s^2 ) # Ex 12.45: # beta_0 = 72.96 beta_1 = 0.041 y_hat = beta_0 + beta_1 * 125 n = 20 alpha = .1 t_value = qt( 1 - 0.5 * alpha, n-2 ) sum_x = 2817.9 sum_xx = 415949.85 S_xx = sum_xx - sum_x^2 / n x_bar = sum_x / n s = 0.665 s_y_hat = s * sqrt( 1 / n + (125 - x_bar)^2 / sum_xx ) ci = y_hat + c(-1,+1) * t_value * s_y_hat pi = y_hat + c(-1,+1) * t_value * sqrt( s_y_hat^2 + s^2 ) alpha = 0.01 t_value = qt( 1 - 0.5 * alpha, n-2 ) ci = y_hat + c(-1,+1) * t_value * s_y_hat print(ci) # Ex 12.46: # n = 13 S_xx = 3022050 - 6130^2 / n y_hat = -0.311 + 0.00143 * 500 x_bar = 6130 / n s_y_hat = 0.131 * sqrt( 1/n + ( 500 - x_bar )^2 / S_xx ) alpha = 0.05 t_value = qt( 1 - 0.5 * alpha, n-2 ) ci = y_hat + c(-1,+1) * t_value * s_y_hat s_beta_hat = 0.131 / sqrt( S_xx ) ci = 0.00143 + c(-1,+1) * t_value * s_beta_hat print(ci) # when x = 400 the s_y_hat changes: y_hat = -0.311 + 0.00143 * 400 s_y_hat = 0.131 * sqrt( 1/n + ( 400 - x_bar )^2 / S_xx ) y_hat + c(-1,+1) * t_value * s_y_hat # Ex 12.47: # DF = read.csv( "../../Data/CH12/ex12-16.txt", header=TRUE, quote="'" ) n = 16 x_bar = mean(DF$x) S_xx = sum( DF$x^2 ) - sum( DF$x )^2 / n y_hat = -1.128 + 0.82697 * 40 alpha = 0.05 t_value = qt( 1 - 0.5 * alpha, n-2 ) ci = y_hat + c(-1,+1) * t_value * sqrt( 1.44^2 + 5.24^2 ) print(ci) y_hat = -1.128 + 0.82697 * 50 s_beta_hat = 5.24 * sqrt( 1 / n + ( 50 - x_bar )^2 / S_xx ) ci = y_hat + c(-1,+1) * t_value * sqrt( s_beta_hat^2 + 5.24^2 ) print(ci) # Ex 12.48: # DF = read.csv( "../../Data/CH12/ex12-48.txt", header=TRUE, quote="'" ) n = length(DF$x) m = lm( y ~ x, data=DF ) sm = summary(m) print(sm) # The CI for 10 beta_1: # s_beta_1 = 0.05088 # standard error of beta_1 alpha = 0.05 t_value = qt( 1 - 0.5 * alpha, n-2 ) ci = 10 * m$coefficients[2] + c(-1,+1) * t_value * ( 10 * s_beta_1 ) # The CI/PI for Y|x=100 and Y|x=200: # x_bar = mean(DF$x) S_xx = sum( DF$x^2 ) - sum( DF$x )^2 / n s_y_hat = sm$sigma * sqrt( 1 / n + ( 100 - x_bar )^2 / S_xx ) ci_100 = m$coefficients[1] + m$coefficients[2] * 100 + c(-1,+1) * t_value * s_y_hat pi_100 = m$coefficients[1] + m$coefficients[2] * 100 + c(-1,+1) * t_value * sqrt( s_y_hat^2 + sm$sigma^2 ) s_y_hat = sm$sigma * sqrt( 1 / n + ( 200 - x_bar )^2 / S_xx ) ci_200 = m$coefficients[1] + m$coefficients[2] * 200 + c(-1,+1) * t_value * s_y_hat pi_200 = m$coefficients[1] + m$coefficients[2] * 200 + c(-1,+1) * t_value * sqrt( s_y_hat^2 + sm$sigma^2 ) # Ex 12.49: # y_hat = 0.5 * ( 462.1 + 597.7 ) n = 10 t_value = qt( 1 - 0.5*0.05, n-2 ) s_hat_y = ( 597.7 - 462.1 )/(2*t_value) t_value = qt( 1 - 0.5*0.01, n-2 ) y_hat + c(-1,+1) * t_value * s_hat_y # Ex 12.50: # DF = read.csv( "../../Data/CH12/ex12-50.txt", header=TRUE, quote="'" ) beta_1_hat = -0.4327 SSE = 0.059842 n = 13 s = sqrt( SSE / (n-2) ) S_xx = sum( DF$x^2 ) - sum( DF$x )^2 / n s_beta_1_hat = s / sqrt( S_xx ) alpha = 0.05 t_value = qt( 1 - 0.5 * alpha, n-2 ) ci = 0.1 * beta_1_hat + c(-1,+1) * t_value * ( 0.1 * s_beta_1_hat ) m = lm( y ~ x, data=DF ) x_bar = mean( DF$x ) y_hat = predict( m, newdata=data.frame(x=0.5) ) s_y_hat = s * sqrt( 1/n + (0.5 - x_bar)^2 / S_xx ) ci = y_hat + c(-1,+1) * t_value * s_y_hat print(ci) pi = y_hat + c(-1,+1) * t_value * sqrt( s_y_hat^2 + s^2 ) print(pi) # Ex 12.51: # DF = data.frame( x=c(0.1,0.16,0.31,0.37,0.37,0.46,0.50,0.50,0.60,0.70,0.75,0.80,0.90,1.00,1.07,1.08,1.11,1.30,1.37,1.54) ) n = length( DF$x ) x_bar = sum( DF$x ) / n abs( c(0.4,1.2) - x_bar ) beta_0 = 1.06930 beta_1 = -0.64884 x = 0.4 y_hat = beta_0 + beta_1 * x SSE = 0.1981 s = sqrt( SSE / (n-2) ) # also given in the output S_xx = sum( DF$x^2 ) - sum( DF$x )^2 / n s_y_hat = s * sqrt( 1 / n + (x-x_bar)^2 / S_xx ) t_value = qt( 1 - 0.5 * 0.05, n-2 ) ci = y_hat + c(-1,+1) * t_value * s_y_hat print(ci) x = 1.2 y_hat = beta_0 + beta_1 * x s_y_hat = s * sqrt( 1 / n + (x-x_bar)^2 / S_xx ) pi = y_hat + c(-1,+1) * t_value * sqrt( s_y_hat^2 + s^2 ) print(pi) # Ex 12.52: # DF = read.csv( "../../Data/CH12/ex12-52.txt", header=TRUE, quote="'" ) n = length(DF$x) x_bar = mean(DF$x) S_xx = sum(DF$x^2) - sum(DF$x)^2 / n plot( DF$x, DF$y ) m = lm( y ~ x, data=DF ) sm = summary(m) print(sm) t_value = qt( 1 - 0.5 * 0.05, n-2 ) ci = m$coefficient[2] + c(-1,+1) * t_value * sm$coefficients[2,2] print(ci) x_value = 3.0 y_hat = predict( m, newdata=data.frame(x=x_value) ) s_y_hat = sm$sigma * sqrt( 1/n + (x_value - x_bar)^2 / S_xx ) ci = y_hat + c(-1,+1) * t_value * s_y_hat print(ci) pi = y_hat + c(-1,+1) * t_value * sqrt( s_y_hat^2 + sm$sigma^2 ) print(pi) # Ex 12.53: # x_bar = 1640.1 / 15 # Ex 12.54: # DF = read.csv( "../../Data/CH12/ex12-54.txt", header=TRUE, quote="'" ) plot( DF$x, DF$y ) m = lm( y ~ x, data=DF ) sm = summary(m) print(sm) n = length(DF$x) x_bar = mean(DF$x) S_xx = sum(DF$x^2) - sum(DF$x)^2 / n y_hat = predict( m, newdata=data.frame(x=75) ) s_y_hat = sm$sigma * sqrt( 1/n + (75-x_bar)^2 / S_xx ) t_value = qt( 1 - 0.5 * , n-2 ) ci = y_hat + c(-1,+1) * t_value * s_y_hat print(ci) # Ex 12.56: # DF = read.csv( "../../Data/CH12/ex12-56.txt", header=TRUE, quote="'" ) plot( DF$Torque, DF$Load ) n = length( DF$Torque ) x_bar = mean( DF$Torque ) S_xx = sum( DF$Torque^2 ) - sum( DF$Torque )^2 / n summary( lm( Load ~ Torque, data=DF ) ) alpha = 0.05 t_value = qt( 1 - 0.5 * alpha, n-2 ) mean( DF$Load ) + c(-1,+1) * t_value * sd( DF$Load ) / sqrt( n ) y_hat = 152.44 + 178.23 * 2.0 s_y_hat = 73.2141^2 * ( 1/15 + (2.0 - x_bar)^2 / S_xx ) pi = y_hat + c(-1,+1) * t_value * sqrt( s_y_hat^2 + 73.2141^2 )