11.39 To run a manufacturing operation efficiently, it is necessary to know how long it takes employees to manufacture the product. Without such information, the cost of making the product cannot be determined. Furthermore, management would not be able to establish an effective incentive plan for its employees, because it would not know how to set work standards

(Chase and Aquilano, Production and Operations Man-agement, 1979). Estimates of production time are frequently obtained by using time studies. The data in the following table were obtained from a recent time study of a sample of 15 employees on an automobile assembly line. The recorded values are y time to complete task (minutes) and x months of experience.

a The computer printout for fitting the model is shown below.

Find the least-squares prediction equation.

y = b0 + b1x + b2x2 + e Y = b0 + b1x + e

b Plot the fitted equation on a scatterplot of the data. Is there sufficient evidence to support the inclusion of the quadratic term in the model? Explain.
c Test the null hypothesis that against the alternative hypothesis that . Use . Does the quadratic term make an important contribution to the model?
d Your conclusion in part

(c) should have been to drop the quadratic term from the model.
Do so, and fit the “reduced model”
to the data.
e Define ‚ in the context of this exercise. Find a 90% confidence interval for ‚ in the reduced model of part (d).

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Time to Complete Months of Time to Complete Months of Task y Experience x Task y Experience x 10 24 17 3 24 20 1 18 1 15 10 16 7 11 15 16 9 11 17 17 7 19 3 18 5 11 20 10 20 13 9 Predictor Coef stdev t-ratio P Constant 20.0911 0.7247 27.72 0.000 x -0.6705 0.1547 -4.33 0.001 x*x 0.006326 1.51 0.158 0.009535 s = 1.091 r-sq = 91.6% R-sq (adj) = 90.2% Analysis of Variance SOURCE DF SS MS F P Regression 2 156.119 78.060 65.59 0.000 Error Total 12 14.281 14 170.000 1.190