Problem 4 Given the data in Table 6.9, the response variable Y is the number of supervisors. and the predictor variable X is the number of supervised workers. Based on empirical observation, it is hypothesized that the standard deviation of the error term 6; is proportional to ry: of = ker;, k >0 . Use the weighted least squares (WLS) method to fit the model. Provide the regression equation. . Use data transformation method to transform Y to Y' = Y/X, and transform X to X' = 1/X (see equations 6.11 and 6.12), and then use the ordinary least squares (OLS) method to regress Y"' on X'. Provide the regression equation. . Compare the results from the above two methods and conclude if the two methods are equivalent. You can compare the residual vs fitted value plot side by side and conclude if they have the same effect in terms of removing heteroscedasticity.\f6.6 REMOVAL OF HETEROSCEDASTICITY In many industrial, economic, and biological applications, when unequal error variances are encountered, it is often found that the standard deviation of residuals tends to increase as the predictor variable increases. Based on this empirical obser vation, we will hypothesize in the present example that the standard deviation of the residuals is proportional to X (some indication of this is available from the plot of the residuals in Figure 6.14): Var(E; ) = k x;, k >0. (6.10) Dividing both sides of (6.9) by I;, we obtain yi Bo + B1+ Ei (6.11) Ti Now, define a new set of variables and coefficients, Y'= Y 86 = BI, Bi = Bo, E X ' X' = X ' X In terms of the new variables (6.11) reduces to 1 = Bo+ Biz,te. (6.12) Note that for the transformed model, Var(e) is constant and equals 2. If our assumption about the error term as given in (6.10) holds, to fit the model properly we must work with the transformed variables: Y/X and 1/ X as response and predictor variables, respectively. If the fitted model for the transformed data is 36 + 81 /X, the fitted model in terms of the original variables is Y = Bi + BOX. (6.13)