Question

26. Short Answer Question We have a dataset with n= 10 pairs of observations (Li, Yi), and n2 n Ti = 683, yi = 813, i=1 i=1 12 n r* = 47,405, tiyi = 56,089, y = 66, 731. Σ- Σ - i=1 What is an approximate 99% confidence interval for the mean response at Io = 90? 27. Short Answer Question We have a dataset with n = 10 pairs of observations (L'i, yi), and n2 Xi = 683, Yi = 813, i=1 n n 12 * = 47,405, Riyi = 56,089, y = 66, 731. i=1 i=1 i=1 What is an approximate 95% prediction interval for the response yo at To = 90? 28. Short Answer Question We have a dataset with n = 10 pairs of observations (L'i, yi), and n2 n Y; = 683, yi = 813, i=1 i=1 n n2 ** = 47, 405, Iiyi = 56,089, y = 66, 731. What is an approximate 95% confidence interval for the intercept of the line of best fit?

Answer 1

Ʃx =	683
Ʃy =	813
Ʃxy =	56089
Ʃx² =	47405
Ʃy² =	66731
Sample size, n =	10
x̅ = Ʃx/n = 683/10 =	68.3
y̅ = Ʃy/n = 813/10 =	81.3
SSxx = Ʃx² - (Ʃx)²/n = 47405 - (683)²/10 =	756.1
SSyy = Ʃy² - (Ʃy)²/n = 66731 - (813)²/10 =	634.1
SSxy = Ʃxy - (Ʃx)(Ʃy)/n = 56089 - (683)(813)/10 =	561.1

Sum of Square error, SSE = SSyy -SSxy²/SSxx = 634.1 - (561.1)²/756.1 = 217.70903

Standard error, se = √(SSE/(n-2)) = √(217.70903/(10-2)) = 5.2167

Slope, b = SSxy/SSxx = 561.1/756.1 = 0.7420976

y-intercept, a = y̅ -b* x̅ = 81.3 - (0.7421)*68.3 = 30.614734

Regression equation :

ŷ = 30.6147 + (0.7421) x

Q26:

Predicted value of y at x = 90

ŷ = 30.6147 + (0.7421) * 90 = 97.4035

Critical value, t_c = T.INV.2T(0.01, 8) = 3.3554

99% Confidence interval :

Lower limit = ŷ - tc*se*√((1/n) + ((x-x̅)²/(SSxx)))

= 97.4035 - 3.3554*5.2167*√((1/10) + ((90 - 68.3)²/(756.1))) = 82.522

Upper limit = ŷ + tc*se*√((1/n) + ((x-x̅)²/(SSxx)))

= 97.4035 + 3.3554*5.2167*√((1/10) + ((90 - 68.3)²/(756.1))) = 112.285

Q27:

Critical value, t_c = T.INV.2T(0.05, 8) = 2.306

95% Prediction interval :

Lower limit = ŷ - tc*se*√(1 + (1/n) + ((x-x̅)²/(SSxx)))

= 97.4035 - 2.306*5.2167*√(1 + (1/10) + ((90 - 68.3)²/(756.1))) = 81.614

Upper limit = ŷ + tc*se*√(1 + (1/n) + ((x-x̅)²/(SSxx)))

= 97.4035 + 2.306*5.2167*√(1 + (1/10) + ((90 - 68.3)²/(756.1))) = 113.193

Q28:

95% Confidence interval for Intercept:

Lower limit = βₒ - tc*se*√((1/n) + (x̅²/SSxx))

= 30.6147 - 2.306*5.2167*√((1/10) + (68.3²/756.1)) = 0.493

Upper limit = βₒ + tc*se*√((1/n) + (x̅²/SSxx))

= 30.6147 + 2.306*5.2167*√((1/10) + (68.3²/756.1)) = 60.736