Question

(a) The production manager of ABC Co. Ltd. has found that the relationship between production cost (y in $) and lot size (x in units) of a certain product is linear. A random sample of eight lots is taken and the results are summarized as below: Xx=941, Xx? = 325751, y=9570, y =32849700 and X xy = 3271030 (i) Find the least squares linear regression equation for predicting production cost on lot size. (4 marks) (ii) Calculate the coefficient of determination and interpret your result. (4 marks) (b) The ISB Manufacturing Company is developing a regression equation to relate the monthly maintenance costs (y, $) of their machines to monthly production time (x, machine-hours), number of inspections per month (x,) and age of machine (xz, years). A random sample of eighteen machines is selected and a regression equation is fitted with partial results provided below: Intercept Coefficients 353.5612 5.3650 3.4331 -5.4293 p-value (Sig.) 0.0149 0.1333 0.0546 0.0425 (i) Write down the fitted multiple linear regression equation. (2 marks) (ii) Test whether the overall model is significant at 5% level of significance given that the coefficient of determination is 0.7741. (5 marks) (iii) Test whether the production time is a significant explanatory variable at 10% level of significance. (3 marks) (iv) It is known that the number of inspections and age of machine are highly correlated. Discuss the reliability of the regression coefficients. (2 marks)

Answer 1

6 a)

i)

n = 8

linear regression equation

Y = b0 + b1 * X

Slope b1 = SSxy / SSxx

SSxx = SUM of X^2 - (SUM of x)^2/n = 325751 - 941^2/8 = 215065.9

SSxy = SUM of xy - (SUM of x * SUM of y)/n = 3271030 - (941*9570)/8 = 2145359

Slope b1 = 2145359/215065.9 = 9.9754

Intercept b0 = Mean Y - Mean X * Slope

Mean X = SUM of X / n = 941/8 = 117.625

Mean Y = SUM of Y / n = 9570/8 = 1196.25

Intercept b0 = 1196.25-117.625*9.9754 = 22.8936

Y = 22.8936 + 9.9754 * X

ii)

Coefficient of correlation r = SSxy/SQRT(SSxx*SSyy)

SSyy = SUM of y^2 - (SUM of y)^2/n = 32849700 - (9570^2)/8 = 21401588

r = 2145359/SQRT(215065.9*21401588) = 0.99998 (neraly 1)

Coefficient of determination r^2 = 0.999959 (nearly 1)

100% of variation in Y variable is explained by regression analysis

or

100% of variation in Y variable is explained by variation in X variable

Given n38, [x =947, IYE 9570 Iq" = 325751, [y: 32849700 2 and 2xy = 3271030 i) linear regression equation - y = bot box . Slope (61) = ssay / Ssxx and Entercep (bo)= Mecn y-(mean x) a slope we have to find Shar, ssyy and stay. Step z [x_61% = 325751-(941778 = 215065.9 Styy = [Y? – (54/n = 39549700–14570 21401588 ssay = Ixy - ([xx[Y/n = 3271030- (941* 9570) 6 = 2145359

bi = 2145359/ 215065.9 bi = 9.9754 bo = Mean y - (Mean x) slope FY -(EX) bi 59570 - (244) 9.9754 bo = 1196.25 - 17.625 * 9.9754 = 22.8936 y: 22.8936 + 9.9754 X " coefficient of correlation & ssay 2145359 / 215065.98 21401558 = 0.99978= 1 (neaily) Coefficiant of determinations or 0.999969 of variation in Y vatable is exp by Yegretion Analysis