Question

A subsidiary of Elektra Electronics has developed new software that allows Windows-based personal computers to run all Apple (i.e., Mac, iPhone, iPad, etc.) and Droid applications. Elektra has collected preliminary data on the weekly total cost of producing the new product at a number of different levels of production. Cost data are available in the worksheet entitled Software Cost.

a) Generate a scatterplot in order to understand the nature of the relationship between weekly quantity produced and weekly total cost. Use this information to complete the statements below.

According to the scatterplot, weekly total cost  --- remains the same decreases sharply increases sharply at first, then it  --- decreases sharply increases sharply levels off for a while, and then it begins to  --- increase again levels off decrease again , as the quantity produced increases. There appears to be  bend(s) or curve(s) in the data which suggests that a  --- 2nd order polynomial 3rd order polynomial reciprocal transformation logarithmic transformation 4th order polynomial regression model is appropriate.

b) Use the data to fit three separate regression models. For the first model, fit the 2nd order polynomial regression model to predict weekly total cost. For the second model, fit the 3rd order polynomial regression model to predict weekly total cost. For the third model, fit the 4th order polynomial regression model to predict weekly total cost.

Provide summary measures for each model separately in the table below. (Enter your R² values as percents to two decimal places and enter your standard errors to three decimal places.)

Model	R2	R2adj	se
Second-Order Model	%	%
Third-Order Model	%	%
Fourth-Order Model	%	%

According to your analysis so far, summarize your results.

According to R2adj, the second-order model is clearly worse than either the third or fourth-order models, however, results are not so clear concerning whether the third or fourth-order model is best.According to R2adj, the second-order model is clearly superior to either the third or fourth-order models.    According toR2adj, the third-order model is clearly superior to either the second or fourth-order models.According toR2adj, the fourth-order model is clearly worse than either the second or third-order models, however, results are not so clear concerning whether the second or third-order model is best.According to R2adj, the third-order model is clearly worse than either the second or fourth-order models, however, results are not so clear concerning whether the second or fourth-order model is best.According to R2adj, the fourth-order model is clearly superior to either the second or third-order models.

c) Perform the appropriate statistical test to test whether the fourth-order model explains a statistically significant amount of variation in total weekly cost above and beyond of that explained by the third-order model. Use a 5% significance level.

State the appropriate test statistic name, degrees of freedom, test statistic value, and the associated p-value (Enter your degrees of freedom as a whole number, the test statistic value to three decimal places, and the p-value to four decimal places).

---Select--- G t F z p (  ) = , p  ---Select--- ≤ > = ≥ <  

State your decision.

The fourth-order model explains a significant amount of variation in total weekly cost compared to the third-order model. Therefore, the fourth-order term in the model is needed and the fourth-order model is best.The fourth-order model explains an insignificant amount of variation in total weekly cost compared to the third-order model. Therefore, the fourth-order term in the model is needed and the fourth-order model is best.    The fourth-order model explains a significant amount of variation in total weekly cost compared to the third-order model. Therefore, the fourth-order term in the model is not needed and a simpler model is preferred.The fourth-order model explains an insignificant amount of variation in total weekly cost compared to the third-order model. Therefore, the fourth-order term in the model is not needed and a simpler model is preferred.

d) Regardless of your results above, assume that the third-order model is best. Based on this estimated total cost function, provide the estimated marginal total cost function (Enter all function coefficients to four decimal places).

C'(x) =

e) Compute the following quantities WITHOUT any intermediate rounding. In other words, do NOT use the rounded version of the function you reported above in part d. Instead, use the one stored in you EXCEL worksheet. Enter your answers to two decimal places.

How quickly is the weekly total cost increasing when the level of production is 125 units per week?

dollars per unit

How quickly is the weekly total cost increasing when the level of production is 400 units per week?

dollars per unit

How quickly is the weekly total cost increasing when the level of production is 700 units per week?

dollars per unit

Quantity	Cost
110	15670.76
500	23405.66
120	18380.88
510	23145.39
340	23191.53
620	24464.14
350	22262.63
130	17012.08
510	23712.91
60	13728.09
80	14777.4
360	21645.45
110	15983.57
120	16254.58
230	20811.9
500	22145.26
650	25873.01
510	23094.15
90	14473.49
30	10810.92
250	20647.02
160	17342.48
710	27978.15
580	23724.68
560	23031.68
510	24193.06
340	22032.14
430	21703.87
660	25802.86
160	19589.52
360	23020.11
510	22350.73
520	22894.49
670	26330.24
510	23090.06
80	14072.29
740	30801.06
660	26305.13
250	20907.3
610	24660.3
380	21717.19
10	7132.79
760	30773.76
460	22831.66
380	22242.87
380	21411.74
250	18686.31
540	22751.03
430	21957.92
360	23092.64
550	23171.41
30	9890.3
300	20918.65
30	13103.14
120	17733.49
700	28261.58
690	27784.98
620	24911.21
800	36389.01
800	32989.9
360	21781.93
230	20480.66
520	21205.32
520	22512.59
510	24730.43
460	19778.44
370	22718.31
370	21864.31
390	22794.56
130	18682.11
80	14434.91
250	18966.93
130	17563.93
640	28181.57
400	22574.92
800	34484.12
430	22946.85
560	23943.21
620	24346.69
660	26324.26
30	10622.52
430	22247.62
550	23766.4
460	22794.75
30	11170.02
800	35079.01
110	15928.94
130	17397.95
300	23065.57
440	23248.85
400	21781.95
170	18099.79
120	16266.64
120	17441.66
540	22637.82
480	22443.35
160	17973.58
250	20820.46
510	22105.08
110	16005.45
520	22771.04
300	21372
380	21949.94
510	21284.16
620	23692.35
340	22329.97
550	23186.41
110	16214.73
380	21830.46
60	13411.67
360	21810.07
110	14902.48
430	22888.18
190	21034.96
430	22459.69
500	22541.96
620	24826.46
760	31807.33
250	21103.83
120	16178.64
370	22762.99
400	19607.21
70	12830.7
380	21656.21
510	22429.83
400	21939.05
20	9201.14
640	25381.11
740	29910.93
500	20920.47

Answer 1

Steps in Excel:
Insert option on menu bar. Select Scatterplot

From the Scatterplot:
the nature of the relationship between weekly quantity produced and weekly total cost is not linear.

According to the scatterplot, weekly total cost increases sharply at first, then it remains nearly the same for a while and then and after sometime again increases sharply.
There appears three curves in the data which suggests that a higher order polynomial-2nd order polynomial/ 3rd order polynomial/ reciprocal transformation/ logarithmic transformation/ 4th order polynomial regression model will be appropriate.

b)

Method,Steps and outputs in Excel:

The Analysis ToolPak is an Excel add-in program that provides data analysis tools for financial, statistical and engineering data analysis.

To load the Analysis ToolPak add-in, execute the following steps.

1. On the File tab, click Options.
2. Under Add-ins, select Analysis ToolPak and click on the Go button.
3. Check Analysis ToolPak and click on OK.
4. On the Data tab, in the Analysis group, you can now click on Data Analysis.

For Regression Analysis:
1. On the Data tab, in the Analysis group, click Data Analysis.
2. Select Regression and click OK.
3. Select the Y Range. This is the predictor variable (also called dependent variable).
4. Select the X Range. These are the explanatory variables (also called independent variables). These columns must be adjacent to each other.
5. Check Labels.
6. Click in the suitable place where you have to place Output
7. Click OK.

SUMMARY OUTPUT ( For first model- 2nd order polynomial regression model)
Regression Statistics
Multiple R	0.923474
R Square	0.852805
Adjusted R Square	0.850487
Standard Error	1943.583
Observations	130
ANOVA
	df	SS	MS	F	Significance F
Regression	2	2779502517	1.39E+09	367.9009	1.45E-53
Residual	127	479744406.7	3777515
Total	129	3259246924
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	13200	502.2526242	26.28159	3.36E-53	12206.13	14193.87
x	21.33548	2.90133732	7.353669	2.1E-11	15.59425	27.0767
x^2	0.000129	0.003659451	0.035117	0.972041	-0.00711	0.00737

SUMMARY OUTPUT ( For second model- 3rd order polynomial regression model)
Regression Statistics
Multiple R	0.98494705
R Square	0.97012069
Adjusted R Square	0.96940928
Standard Error	879.141175
Observations	130
ANOVA
	df	SS	MS	F	Significance F
Regression	3	3161862884	1053954295	1363.655	7.88E-96
Residual	126	97384039.99	772889.2063
Total	129	3259246924
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	7982.15143	326.5670721	24.44260954	1.18E-49	7335.885	8628.418
x	97.2287876	3.655808079	26.59570346	1.58E-53	89.99405	104.4635
x^2	-0.2294098	0.010451853	-21.9492012	7.05E-45	-0.25009	-0.20873
x^3	0.00018732	8.42189E-06	22.24220369	1.86E-45	0.000171	0.000204

SUMMARY OUTPUT ( For third model- 4th order polynomial regression model)
Regression Statistics
Multiple R	0.9850385
R Square	0.9703009
Adjusted R Square	0.9693505
Standard Error	879.98537
Observations	130
ANOVA
	df	SS	MS	F	Significance F
Regression	4	3162450143	790612536	1020.97	2.17E-94
Residual	125	96796781.35	774374.251
Total	129	3259246924
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	7731.8684	435.2606123	17.7637676	5.38E-36	6870.434	8593.303
x	102.68768	7.258442377	14.1473437	1.02E-27	88.32232	117.053
x^2	-0.2584327	0.034930886	-7.3984014	1.76E-11	-0.32757	-0.1893
x^3	0.0002422	6.3593E-05	3.80878687	0.000218	0.000116	0.000368
x^4	-3.369E-08	3.86867E-08	-0.8708418	0.38551	-1.1E-07	4.29E-08

Summarized summary:

Model	R2	R2adj	se
Second-Order Model	85.28%	85.05%	1943.583
Third-Order Model	97.01%	96.94%	879.141
Fourth-Order Model	97.03%	96.94%	879.985

The model with largest R2adj value is the best model.
According to R2adj, the second-order model is clearly worse than either the third or fourth-order models, however, results are not so clear concerning whether the third or fourth-order model is best.

c)

The fourth-order model explains an insignificant amount of variation in total weekly cost compared to the third-order model. Therefore, the fourth-order term in the model is not needed and a simpler model is preferred.

d)

	Coefficients
Intercept	7982.151431
x	97.22878756		125	400	700
x^2	-0.22940982		15625	160000	490000
x^3	0.000187321		1953125	64000000	343000000
			fitted value
			16917.08	22156.67	27882.754
beta 1	97.22878756		change (MCF)
beta2*(2)	-0.45881964		48.65703	3.615235	51.417594
beta3*(3)	0.000561964		48.66	3.62	51.42