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Chapter 12: Q52E. (page 746)

Consider the following data that fit the quadratic modelE(y)=尾0+尾1x+尾2x2:

a. Construct a scatterplot for this data. Give the prediction equation and calculate R2based on the model above.

b. Interpret the value ofR2.

c. Justify whether the overall model is significant at the 1% significance level if the data result into a p-value of 0.000514.

Short Answer

Expert verified

a. Scatter plot, prediction equation is y=8.541667-2.25357x+0.386905x2and value of R2calculated here is 0.9516

b. The value ofR2here is 0.9516 which is a high value denoting that almost 95% of the variation in the variables is explained by the model. This means that the model is a good fit for the data.

c. At 1% significance level, it can be concluded that 尾1鈮�尾2鈮�0

Step by step solution

01

Scatterplot for the data

X

Y

0

8

1

7.3

2

5.6

3

5.9

4

5.2

5

6.5

6

8.8

7

12.1

SUMMARY OUTPUT

















Regression Statistics








Multiple R

0.975509








R Square

0.951618








Adjusted R Square

0.932265








Standard Error

0.586556








Observations

8

















ANOVA









df

SS

MS

F

Significance F




Regression

2

33.83476

16.91738

49.17163

0.000515




Residual

5

1.720238

0.344048






Total

7

35.555













Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Lower 95.0%

Upper 95.0%

Intercept

8.541667

0.49366

17.30272

1.18E-05

7.272673

9.810661

7.272673

9.810661

X

-2.25357

0.329452

-6.84036

0.001019

-3.10046

-1.40669

-3.10046

-1.40669

X^2

0.386905

0.045254

8.549672

0.000361

0.270576

0.503233

0.270576

0.503233

On solving the second-order model equation in excel, the output generated is attached here. The prediction equation, therefore isy=8.541667-2.25357x+0.386905x2

The value ofR2 calculated here is 0.9516.

02

Interpretation of R2

The value ofR2 here is 0.9516 which is a high value denoting that almost 95% of the variation in the variables is explained by the model. This means that the model is a good fit for the data.

03

Goodness of fit

H0:尾1=尾2=0

Ha:At least one of the parameters尾1 or尾2 is non zero

Here, F test statistic=SSEn-(k+1)=1.7202385=0.3440

H0is rejected if p-value < 伪. For 伪=0.01, since 0.000514 < 0.01

Sufficient evidence to rejectH0 at 95% confidence interval.

Therefore,尾1鈮�尾2鈮�0.

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Most popular questions from this chapter

Suppose you have developed a regression model to explain the relationship between y and x1, x2, and x3. The ranges of the variables you observed were as follows: 10 鈮� y 鈮� 100, 5 鈮� x1 鈮� 55, 0.5 鈮� x2 鈮� 1, and 1,000 鈮� x3 鈮� 2,000. Will the error of prediction be smaller when you use the least squares equation to predict y when x1 = 30, x2 = 0.6, and x3 = 1,300, or when x1 = 60, x2 = 0.4, and x3 = 900? Why?

Suppose you fit the quadratic model E(y)=尾0+尾1x+尾2x2to a set of n = 20 data points and found R2=0.91, SSyy=29.94, and SSE = 2.63.

a. Is there sufficient evidence to indicate that the model contributes information for predicting y? Test using a = .05.

b. What null and alternative hypotheses would you test to determine whether upward curvature exists?

c. What null and alternative hypotheses would you test to determine whether downward curvature exists?

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)=尾0+尾1x1+尾2x2+尾3x1x2

where

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

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

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

a. In terms of the 尾鈥檚 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 尾鈥檚 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: 尾3 = 0 yielded a p-value of 0.027. Using 伪 = .05, interpret this result.

g. The researchers reported that the estimated values of both 尾2 and 尾3 are negative. Consequently, they conclude that 鈥渉igh-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.鈥� Do you agree?

Consider a multiple regression model for a response y, with one quantitative independent variable x1 and one qualitative variable at three levels.

a. Write a first-order model that relates the mean response E(y) to the quantitative independent variable.

b. Add the main effect terms for the qualitative independent variable to the model of part a. Specify the coding scheme you use.

c. Add terms to the model of part b to allow for interaction between the quantitative and qualitative independent variables.

d. Under what circumstances will the response lines of the model in part c be parallel?

e. Under what circumstances will the model in part c have only one response line?

Reality TV and cosmetic surgery. Refer to the Body Image: An International Journal of Research (March 2010) study of the impact of reality TV shows on a college student鈥檚 decision to undergo cosmetic surgery, Exercise 12.17 (p. 725). Recall that the data for the study (simulated based on statistics reported in the journal article) are saved in the file. Consider the interaction model, , where y = desire to have cosmetic surgery (25-point scale), = {1 if male, 0 if female}, and = impression of reality TV (7-point scale). The model was fit to the data and the resulting SPSS printout appears below.

a.Give the least squares prediction equation.

b.Find the predicted level of desire (y) for a male college student with an impression-of-reality-TV-scale score of 5.

c.Conduct a test of overall model adequacy. Use a= 0.10.

d.Give a practical interpretation of R2a.

e.Give a practical interpretation of s.

f.Conduct a test (at a = 0.10) to determine if gender (x1) and impression of reality TV show (x4) interact in the prediction of level of desire for cosmetic surgery (y).
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