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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’s 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: Write a first-order model relating to

  1. Two quantitative independent variables.
  2. Four quantitative independent variables.
  3. Five quantitative independent variables.

Buy-side vs. sell-side analysts’ earnings forecasts. Refer to the Financial Analysts Journal (July/August 2008) comparison of earnings forecasts of buy-side and sell-side analysts, Exercise 2.86 (p. 112). The Harvard Business School professors used regression to model the relative optimism (y) of the analysts’ 3-month horizon forecasts. One of the independent variables used to model forecast optimism was the dummy variable x = {1 if the analyst worked for a buy-side firm, 0 if the analyst worked for a sell-side firm}.

a) Write the equation of the model for E(y) as a function of type of firm.

b) Interpret the value ofβ0in the model, part a.

c) The professors write that the value ofβ1in the model, part a, “represents the mean difference in relative forecast optimism between buy-side and sell-side analysts.” Do you agree?

d) The professors also argue that “if buy-side analysts make less optimistic forecasts than their sell-side counterparts, the [estimated value ofβ1] will be negative.” Do you agree?

Question: Personality traits and job performance. Refer to the Journal of Applied Psychology (Jan. 2011) study of the relationship between task performance and conscientiousness, Exercise 12.54 (p. 747). Recall that the researchers used a quadratic model to relate y = task performance score (measured on a 30-point scale) to x1 = conscientiousness score (measured on a scale of -3 to +3). In addition, the researchers included job complexity in the model, where x2 = {1 if highly complex job, 0 if not}. The complete model took the form

E(y)=β0+β1x1+β2x12+β3x2+β4x1x2+β5x12x2herex2=1E(y)=β0+β1x1+β2x12+β3(1)+β4x1(1)+β5(1)2(1)E(y)=(β0+β3)+(β1+β4)x1+(β2+β5)(x1)2

a. For jobs that are not highly complex, write the equation of the model for E1y2 as a function of x1. (Substitute x2 = 0 into the equation.)

b. Refer to part a. What do each of the b’s represent in the model?

c. For highly complex jobs, write the equation of the model for E(y) as a function of x1. (Substitute x2 = 1 into the equation.)

d. Refer to part c. What do each of the b’s represent in the model?

e. Does the model support the researchers’ theory that the curvilinear relationship between task performance score (y) and conscientiousness score (x1) depends on job complexity (x2)? Explain.

Question: Manipulating rates of return with stock splits. Some firms have been accused of using stock splits to manipulate their stock prices before being acquired by another firm. An article in Financial Management (Winter 2008) investigated the impact of stock splits on long-run stock performance for acquiring firms. A simplified version of the model fit by the researchers follows:

E(y)=β0+β1x1+β2x2+β3x1x2

where

y = Firm’s 3-year buy-and-hold return rate (%)

x1 = {1 if stock split prior to acquisition, 0 if not}

x2 = {1 if firm’s discretionary accrual is high, 0 if discretionary accrual is low}

a. In terms of the β’s in the model, what is the mean buy and- hold return rate (BAR) for a firm with no stock split and a high discretionary accrual (DA)?

b. In terms of the β’s in the model, what is the mean BAR for a firm with no stock split and a low DA?

c. For firms with no stock split, find the difference between the mean BAR for firms with high and low DA. (Hint: Use your answers to parts a and b.)

d. Repeat part c for firms with a stock split.

e. Note that the differences, parts c and d, are not the same. Explain why this illustrates the notion of interaction between x1 and x2.

f. A test for H0: β3 = 0 yielded a p-value of 0.027. Using α = .05, interpret this result.

g. The researchers reported that the estimated values of both β2 and β3 are negative. Consequently, they conclude that “high-DA acquirers perform worse compared with low-DA acquirers. Moreover, the underperformance is even greater if high-DA acquirers have a stock split before acquisition.” Do you agree?

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  2. Interpret the’s in the model, part a.
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