Question

Arnold Sorenson, production manager of New Frontiers Instruments Inc., was interested in estimating the mathematical relationship between the number of electronic assemblies produced during an 8-hour shift and the average cost per assembly. This function would then be used to estimate cost for various production order bids and to determine the production level that would minimize average cost. Arnold collected data from nine shifts during which the number of assemblies ranged from 100 to 900. In addition, he obtained the average cost per unit for the days from the accounting department. The data appear in the Production Cost worksheet in Multiple Regression Modeling data below

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a) Draw a scatter plot of Mean Cost per Unit (Y) versus Number of Units (X). Does there appear to be a linear relationship?

b) Fit a simple regression model using Mean Cost per Unit as the Y or dependent variable and Number of Units as the X or independent variable. Is the model significant at ALPHA = 0.05? What is the R-Squared for this model? Interpret the R-Squared value in the context of the problem.

c) For the simple regression model in part b), draw a scatter plot of the residuals (Y axis) versus Number of Units (X axis). What does this scatter plot suggest regarding the relationship between Mean Cost per Unit and Number of Units?

d) Calculate the Durbin Watson statistic. Do the residuals show autocorrelation?

e) Fit a curvilinear model using Mean Cost per Unit as the Y or dependent variable with independent variables of Number of Units, and the (Number of Units)^2 quadratic term. Is the model significant at  = 0.05? What is the R-Squared for this model? Interpret the R-Squared value in the context of the problem.

f) For the multiple regression model in part e), draw scatter plots of the residuals (Y axis) versus Number of Units (X axis) and the residuals (Y axis) versus (Number of Units)^2 (X axis). What do these scatter plot suggest regarding the relationship between Mean Cost per Unit and Number of Units?

g) Using the model in part e), estimate the change in Mean Cost per Unit when the Number of Units increases from 200 to 250.

h) Using the model in part e), estimate the change in Mean Cost per Unit when the Number of Units increases from 800 to 850.

Number of Units	Mean Cost per Unit
100	5.11
210	4.42
290	4.07
415	3.52
509	3.33
613	3.44
697	3.77
806	4.07
908	4.28

Answer 1

SOLUTION

Arnold Sorenson, production manager of New Frontiers Instruments Inc., was interested in estimating the mathematical relationship between the number of electronic assemblies produced during an 8-hour shift and the average cost per assembly. This function would then be used to estimate cost for various production order bids and to determine the production level that would minimize average cost. Arnold collected data from nine shifts during which the number of assemblies ranged from 100 to 900. In addition, he obtained the average cost per unit for the days from the accounting department. The data appear in the Production Cost worksheet in HW8 data workbook on Moodle.

a) Fit a simple regression model using Mean Cost per Unit as the Y or dependent variable and Number of Units as the X or independent variable. Is the model significant at a = 0.05? What is the R-Squared for this model? Interpret the R-Squared value in the context of the problem.

Regression Analysis
	r²	0.174	n	9
	r	-0.418	k	1
	Std. Error	0.548	Dep. Var.	y
ANOVA table
Source	SS	df	MS	F	p-value
Regression	0.4433	1	0.4433	1.48	.2634
Residual	2.0992	7	0.2999
Total	2.5425	8
Regression output					confidence interval
variables	coefficients	std. error	t (df=7)	p-value	95% lower	95% upper
Intercept	4.4330	0.3994	11.100	1.07E-05	3.4886	5.3773
x	-0.0009	0.0007	-1.216	.2634	-0.0025	0.0008

Y=4.4330-0.0009x

R square =0.174

17.4% of variance in y is explained by the model.

b) For the simple regression model in part a), draw a scatter plot of the residuals (Y axis) versus Number of Units (X axis). What does this scatter plot suggest regarding the relationship between Mean Cost per Unit and Number of Units?

The plot suggests there is curvilinear relation between x and y.

c) Fit a multiple regression model using Mean Cost per Unit as the Y or dependent variable with independent variables of Number of Units, and the (Number of Units)^2 quadratic term. Is the model significant at a = 0.05? What is the R-Squared for this model? Interpret the R-Squared value in the context of the problem.

Regression Analysis
	R²	0.962
	Adjusted R²	0.949	n	9
	R	0.981	k	2
	Std. Error	0.127	Dep. Var.	y
ANOVA table
Source	SS	df	MS	F	p-value
Regression	2.4459	2	1.2230	75.97	.0001
Residual	0.0966	6	0.0161
Total	2.5425	8
Regression output					confidence interval
variables	coefficients	std. error	t (df=6)	p-value	95% lower	95% upper
Intercept	5.9084	0.1614	36.601	2.78E-08	5.5134	6.3034
x	-0.0088	0.0007	-12.040	1.99E-05	-0.0106	-0.0070
x*x	0.00000793	0.00000071	11.154	3.10E-05	0.00000619	0.00000967

Y=5.9084-0.0088x-0.00000793x*x

R square =0.962

96.2% of variance in y is explained by the model.

d) For the multiple regression model in part c), draw scatter plots of the residuals (Y axis) versus Number of Units (X axis) and the residuals (Y axis) versus (Number of Units)^2 (X axis). What do these scatter plot suggest regarding the relationship between Mean Cost per Unit and Number of Units?

The plot shows that quadratic model fits data well.

e) Using the model in part c), estimate the change in Mean Cost per Unit when the Number of Units increases from 200 to 250.

For 200, Y =5.9084-0.0088*200-0.00000793*200^2 =3.8312

For 250, Y =5.9084-0.0088*250-0.00000793*250^2 = 3.212775

Change = 3.8312-3.212775 = 0.618425

The change is decrease of 0.618425.

f) Using the model in part c), estimate the change in Mean Cost per Unit when the Number of Units increases from 800 to 850.

For 800, Y =5.9084-0.0088*800-0.00000793*800^2 =-6.2068

For 850, Y =5.9084-0.0088*850-0.00000793*850^2 = -7.30103

Change = -6.2068-(-7.30103) = 1.09423

The change is decrease of 1.09423.