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Chapter 12: Q.65E (page 749)

Question: Orange juice demand study. A chilled orange juice warehousing operation in New York City was experiencing too many out-of-stock situations with its 96-ounce containers. To better understand current and future demand for this product, the company examined the last 40 days of sales, which are shown in the table below. One of the company鈥檚 objectives is to model demand, y, as a function of sale day, x (where x = 1, 2, 3, c, 40).

  1. Construct a scatterplot for these data.
  2. Does it appear that a second-order model might better explain the variation in demand than a first-order model? Explain.
  3. Fit a first-order model to these data.
  4. Fit a second-order model to these data.
  5. Compare the results in parts c and d and decide which model better explains variation in demand. Justify your choice.

Short Answer

Expert verified

Answers:

  1. Scatterplot
  2. Looking at the data, a second-order model equation might be a better fit for the data as it can be seen that there is an upward curvature kind of relationship between x and y.
  3. The first-order model equation can be written asy^=2802.362+122.9507x
  4. The second-order model equation for y on x isy^=4944.221-189.029x+7.462924x2
  5. The R2value for the first-order model equation is around 35% while for the second-order model equation the value of was around 50%. value gives an estimate about if the model is a better fit for the data. A higher value for a model denotes that the model is a better fit for the data. Since the value of is higher for the second-order equation, the second-order model equation is a better fit for the data.

Step by step solution

01

Scatterplot

Demand for containers, y

Sale day, x

X2

4581

1

1

4239

2

4

2754

3

9

4501

4

16

4016

5

25

4680

6

36

4950

7

49

3303

8

64

2367

9

81

3055

10

100

4248

11

121

5067

12

144

5201

13

169

5133

14

196

4211

15

225

3195

16

256

5760

17

289

5661

18

324

6102

19

361

6099

20

400

5902

21

441

2295

22

484

2682

23

529

5787

24

576

3339

25

625

3798

26

676

2007

27

729

6282

28

784

3267

29

841

4779

30

900

9000

31

961

9531

32

1024

3915

33

1089

8964

34

1156

6984

35

1225

6660

36

1296

6921

37

1369

10005

38

1444

10153

39

1521

11520

40

1600

02

Model fit for the data

Looking at the data, a second-order model equation might be a better fit for the data as it can be seen that there is an upward curvature kind of relationship between x and y.

03

First-order model equation

Excel summary output

SUMMARY OUTPUT

















Regression Statistics








Multiple R

0.613045








R Square

0.375824








Adjusted R Square

0.359398








Standard Error

1876.566








Observations

40

















ANOVA









df

SS

MS

F

Significance F




Regression

1

80572885

80572885

22.88027

2.61E-05




Residual

38

1.34E+08

3521501






Total

39

2.14E+08













Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Lower 95.0%

Upper 95.0%

Intercept

2802.362

604.7266

4.634096

4.13E-05

1578.156

4026.567

1578.156

4026.567

Sale day, x

122.9507

25.70397

4.783332

2.61E-05

70.91568

174.9856

70.91568

174.9856

The first-order model equation can be written asy^=2802.362+122.9507x

04

Second-order model equation

Excel summary output

SUMMARY OUTPUT

















Regression Statistics








Multiple R

0.723292








R Square

0.523151








Adjusted R Square

0.497376








Standard Error

1662.232








Observations

40

















ANOVA









df

SS

MS

F

Significance F




Regression

2

1.12E+08

56079162

20.29636

1.12E-06




Residual

37

1.02E+08

2763016






Total

39

2.14E+08













Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Lower 95.0%

Upper 95.0%

Intercept

4944.221

829.6005

5.959761

7.12E-07

3263.291

6625.151

3263.291

6625.151

Sale day, x

-183.029

93.31859

-1.96134

0.057398

-372.111

6.052169

-372.111

6.052169


7.462924

2.207279

3.381051

0.001716

2.990552

11.9353

2.990552

11.9353

The second-order model equation for y on x isy^=4944.221-189.029x+7.462924x2

05

Comparison between the models

The R2value for the first-order model equation is around 35% while for the second-order model equation the value of R2was around 50%. R2value gives an estimate about if the model is a better fit for the data. The higher valueR2 for a model denotes that the model is a better fit for the data. Since the value of is higher for the second-order equation, the second-order model equation is a better fit for the data.

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Most popular questions from this chapter

Question:How is the number of degrees of freedom available for estimating 蟽2(the variance of蔚 ) related to the number of independent variables in a regression model?

It is desired to relate E(y) to a quantitative variable x1and a qualitative variable at three levels.

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Question: There are six independent variables, x1, x2, x3, x4, x5, and x6, that might be useful in predicting a response y. A total of n = 50 observations is available, and it is decided to employ stepwise regression to help in selecting the independent variables that appear to be useful. The software fits all possible one-variable models of the form

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Question: Accuracy of software effort estimates. Periodically, software engineers must provide estimates of their effort in developing new software. In the Journal of Empirical Software Engineering (Vol. 9, 2004), multiple regression was used to predict the accuracy of these effort estimates. The dependent variable, defined as the relative error in estimating effort, y = (Actual effort - Estimated effort)/ (Actual effort) was determined for each in a sample of n = 49 software development tasks. Eight independent variables were evaluated as potential predictors of relative error using stepwise regression. Each of these was formulated as a dummy variable, as shown in the table.

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a. In step 1 of the stepwise regression, how many different one-variable models are fit to the data?

b. In step 1, the variable x1 is selected as the best one- variable predictor. How is this determined?

c. In step 2 of the stepwise regression, how many different two-variable models (where x1 is one of the variables) are fit to the data?

d. The only two variables selected for entry into the stepwise regression model were x1 and x8. The stepwise regression yielded the following prediction equation:

Give a practical interpretation of the 尾 estimates multiplied by x1 and x8.

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a.E(y)=尾0+尾1x1+尾2x2b.E(y)=尾0+尾1x1c.E(y)=尾0+尾1x1+尾2x12d.E(y)=尾0+尾1x1+尾2x2+尾3x1x2e.E(y)=尾0+尾1x1+尾2x2+尾3x1x2+尾4x21+尾5x22
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