Homework answers / question archive / SP23: BUSINESS ANALYTICS & MODELING: 20646, Chapter 9 Assignment The IRS wants to develop a method for detecting whether or not individuals have overstated their deductions for charitable contributions on their tax returns

SP23: BUSINESS ANALYTICS & MODELING: 20646, Chapter 9 Assignment The IRS wants to develop a method for detecting whether or not individuals have overstated their deductions for charitable contributions on their tax returns

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SP23: BUSINESS ANALYTICS & MODELING: 20646, Chapter 9 Assignment

The IRS wants to develop a method for detecting whether or not individuals have overstated their deductions for charitable contributions on their tax returns. To assist in this effort, the IRS supplied data found in the table below listing the adjusted gross income (AGI) and charitable contributions for 11 taxpayers whose returns were audited and found to be correct.

AGI (in $1,000s) Charitable Contributions

$54         $4,300

$59         $4,700

$62         $6,429

$68         $7,917

$75         $7,500

$77         $8,500

$84         $12,090

$89         $10,306

$91         $11,920

$99         $12,190

$104       $14,775

(a)

Prepare a scatter plot of the data. Does there appear to be a linear relationship between these variables?

The relationship is highly nonlinear. Charitable contributions decrease and then increase as adjusted gross income increases.

The relationship looks approximately linear. Charitable contributions increase as adjusted gross income increases.   

The relationship looks approximately linear. Charitable contributions are relatively constant as adjusted gross income increases.

The relationship looks approximately linear. Charitable contributions decrease as adjusted gross income increases.

The relationship is highly nonlinear. Charitable contributions increase and then decrease as adjusted gross income increases.

(b)

Develop a simple linear regression model that can be used to predict the level of charitable contributions (in dollars) from a return's AGI (in thousands of dollars). What is the estimated regression equation? (Round your numerical values to four decimal places. Use X1 for AGI (in thousands of dollars) and Y hat for charitable contributions in dollars.)

Y hat =

 

(c)

Interpret the value of R2. (Enter your answer as a percent. Round your answer to two decimal places.)

The value of R2 suggests that

 % of

---Select---

 charitable contributions

---Select---

 adjusted gross income.

(d)

How might the IRS use the regression results to identify returns with unusually high charitable contributions?

The IRS could construct a prediction interval for charitable contributions at each level of adjusted gross income. Charitable contributions below the lower limit would be considered suspect.

The IRS could predict the charitable contributions at each level of adjusted gross income. Charitable contributions that are not equal to the predicted value would be considered suspect.   

The IRS could construct a prediction interval for charitable contributions at each level of adjusted gross income. Charitable contributions between the lower the upper limits would be considered suspect.

The IRS could predict the charitable contributions at each level of adjusted gross income. Charitable contributions that are equal to the predicted value would be considered suspect.

The IRS could construct a prediction interval for charitable contributions at each level of adjusted gross income. Charitable contributions above the upper limit would be considered suspect.

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2.

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RAGSMDA9 9.E.008.

 

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Roger Gallagher owns a used car lot that deals solely in used Corvettes. He wants to develop a regression model to help predict the price he can expect to receive for the cars he owns. He collected the data found in table below describing the mileage, model year, presence of a T-top, and selling price of a number of cars he has sold in recent months. Let Y represent the selling price,

X1

 the mileage (in thousands of miles),

X2

 the model year, and

X3

 the presence (or absence) of a T-top.

Mileage (in thousands of miles) Year       T-top     Price (in dollars)

115         1968       1              $12,375

95           1970       0              $9,500

125         1972       0              $6,500

85           1974       0              $13,450

77           1976       1              $14,125

105         1978       0              $9,800

86           1979       0              $11,750

71           1981       1              $15,000

55           1983       0              $15,000

65           1987       1              $18,000

45           1988       0              $20,800

15           1988       0              $24,000

23           1991       1              $30,400

(a)

If Roger wants to use a simple linear regression function to estimate the selling price of a car, which X variable do you recommend he use?

model year

presence (or absence) of a T-top   

mileage

(b)

Determine the parameter estimates for the regression function represented by:

Y hati = b0 + b1X1i + b2X2i

What is the estimated regression function? (Use X1 for

X1i

 and X2 for

X2i.

 Round your numerical values to one decimal place.)

Y hati =

 

Does

X2

 help to explain the selling price of the cars if

X1

 is also in the model?

The

---Select---

 value

---Select---

 when the X2 variable is added to a model containing X1. This suggests that model year

---Select---

 significantly help explain the selling price of cars after mileage is already included in the model.

What might be the reason for this?

These two variables could be highly correlated and therefore they are providing similar information.

These two variables could be highly correlated and therefore they are providing very different information.   

These two variables are uncorrelated and therefore they are providing very different information.

These two variables are uncorrelated and therefore they are providing similar information.

(c)

Use the binary variable

X3i

 to indicate whether or not each car in the sample has a T-top (

X3i = 0

 indicates a car does not have a T-Top,

X3i = 1

 indicates a car has a T-Top). Determine the parameter estimates for the regression function represented by:

Y hati = b0 + b1X1i + b3X3i

Does

X3

 help to explain the selling price of the cars if

X1

 is also in the model? Explain.

The

---Select---

 value

---Select---

 when the X3 variable is added to a model containing X1. This suggests that the presence (or absence) of a T-top

---Select---

 significantly help explain the selling price of cars after mileage is already included in the model.

(d)

According to the previous model, on average, how much does a T-top add to the value of a car (in dollars)? (Round your answer to the nearest dollar.)

$

(e)

Determine the parameter estimates for the regression function represented by:

Y hati = b0 + b1X1i + b2X2i + b3X3i

What is the estimated regression function? (Use X1 for

X1i,

 X2 for

X2i,

 and X3 for

X3i.

 Round your numerical values to one decimal place.)

Y hati =

 

(f)

Of all the regression functions considered here, which do you recommend Roger use?

 

Y hati = b0 + b1X1i

 

Y hati = b0 + b2X2i

   

 

Y hati = b0 + b2X3i

 

Y hati = b0 + b1X1i + b2X2i

 

Y hati = b0 + b1X1i + b3X3i

 

Y hati = b0 + b1X1i + b2X2i + b3X3i

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3.

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RAGSMDA9 9.E.012.

 

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The O-rings in the booster rockets on the space shuttle are designed to expand when heated to seal different chambers of the rocket so that solid rocket fuel is not ignited prematurely. According to engineering specifications, the O-rings expand by some amount, say at least 5%, in order to ensure a safe launch. Hypothetical data on the amount of O-ring expansion and the atmospheric temperature in Fahrenheit at the time of several different launches are given in the table below.

Temperature     % O-ring Expansion

93           22.2

88           20.9

87           20.7

81           19.6

73           18.8

72           19.1

68           17.4

64           16.1

55           15.4

(a)

Prepare a scatter plot of the data. Does there appear to be a linear relationship between these variables?

Yes, there does appear to be a linear relationship.

No, there appears to be no relationship.   

No, there appears to be a nonlinear relationship.

(b)

Obtain a simple linear regression model to estimate the amount of O-ring expansion as a function of atmospheric temperature. What is the estimated regression function? (Let X1 represent the temperature in Fahrenheit and Y represent the percentage of O-ring expansion. Round your numerical values to four decimal places.)

Y hat =

 

(c)

Interpret the R2 statistic for the model you obtained. (Enter your answer as a percent. Round your answer to two decimal places.)

The R2 statistic indicates that approximately

 % of

---Select---

 the percentage of O-ring expansion

---Select---

 by temperature.

(d)

Suppose that NASA officials are considering launching a space shuttle when the temperature is 26 degrees. What amount of O-ring expansion (in %) should they expect at this temperature according to your model? (Round your answer to two decimal places.)

 %

(e)

On the basis of your analysis of these data, would you recommend that the shuttle be launched if the temperature is 26 degrees? Why or why not?

Yes, our estimate in part (d) is above 5% expansion.

No, our estimate in part (d) is an example of extrapolation and we cannot be sure the model will be accurate for temperatures this low.   

No, our estimate in part (d) is below 5% expansion.

Yes, the scatter plot shows a linear relationship between temperature and O-ring expansion.

No, the scatter plot shows no relationship between temperature and O-ring expansion.

No, the scatter plot shows a nonlinear between temperature and O-ring expansion.

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4.

[–/4.8 Points]

 

DETAILS

RAGSMDA9 9.E.013.

 

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An analyst for an investment company wants to develop a regression model to predict the annual rate of return for a stock based on the price-earnings (PE) ratio of the stock and a measure of the stock's risk. The data found in below were collected for a random sample of stocks.

PE Ratio                Risk        Return

7.3          0.8          7.8

11           1.1          13.2

8.6          0.9          9.1

11.1        1              11.1

11.5        1.5          12.3

12.1        1.1          13

12.4        1              11.5

12.4        1.1          14.3

12.9        1.4          15

13.3        1.2          16.9

(a)

Prepare scatter plots for each independent variable versus the dependent variable. What type of model do these scatter plots suggest might be appropriate for the data?

The scatter plots suggest that there is no relationship between the either of the two independent variables and the dependent variable.

The scatter plots suggest that a quadratic relationship might be appropriate for both variables.   

The scatter plots suggest that a linear relationship might be appropriate for both variables.

(b)

Let

Y =

 Return,

X1 =

 PE Ratio, and

X2 =

 Risk. Obtain the regression results for the following regression model:

Y hati = b0i + b1X1i + b2X2i

Report the estimated regression equation you found. (Enter your answer using

X1

 for

X1i

 and

X2

 for

X2i.

 Round your numerical values to five decimal places.)

Y hati =

 

Interpret the value of

R2

 for this model. (Express R2 as a percent. Round your answer to two decimal places.)

The value of R2 suggests that

 % of variation in

---Select---

 can be accounted for using this model.

(c)

Obtain the regression results for the following regression model:

Y hati = b0 + b1X1i + b2X2i + b3X3i + b4X4i

where

X3i = X1i2

 and

X4i = X2i2.

Report the estimated regression equation you found. (Enter your answer using

X1

 for

X1i,

 

X2

 for

X2i,

 

X3

 for

X3i,

 and

X4

 for

X4i.

 Round your numerical values to five decimal places.)

Y hati =

 

Interpret the value of

R2

 for this model. (Express R2 as a percent. Round your answer to two decimal places.)

The value of R2 suggests that

 % of variation in

---Select---

 can be accounted for using this model.

(d)

Which of the previous two models would you recommend that the analyst use?

We should recommend the first model since it has fewer variables and is simpler.

We should recommend the first model since the calculations are easier to perform.   

We should recommend the first model since it has the highest adjusted R2.

We should recommend the second model since it has the highest adjusted R2.

We should recommend the second model since it has more variables and is more complicated.

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5.

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DETAILS

RAGSMDA9 9.E.017.

 

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Creative Confectioners is planning to introduce a new brownie. A small-scale "taste test" was conducted to assess consumers' preferences (Y) with regard to moisture content

(X1)

 and sweetness

(X2).

 Data from the taste test may be found in the table below.

Moisture             Sweetness          Preference

2.8          4.5          68

3.1          4.3          72

3.9          5.2          71

4.4          5.4          74

4.4          6.1          73

5.0          6.4          77

4.8          7.8          79

5.6          8.2          80

6.3          7.9          81

7.0          8.3          84

7.7          8.9          87

7.7          10.0        84

8.5          10.1        83

8.8          10.7        80

9.4          11.0        75

9.6          12.1        73

(a)

Prepare a scatter plot of moisture content versus preference. What type of relationship does your plot suggest?

linear

quadratic   

(b)

Prepare a scatter plot of sweetness versus preference. What type of relationship does your plot suggest?

linear

quadratic   

(c)

Estimate the parameters for the following regression function:

Y hati = b0 + b1X1i + b2X1i2 + b3X2i + b4X2i2

What is the estimated regression function? (Enter your answer using

X1

 for

X1i

 and

X2

 for

X2i.

 Round your numerical values to four decimal places.)

Y hati =

 

(d)

Using the estimated regression function in part (c), what is the expected preference rating of a brownie recipe with a moisture content of 7 and a sweetness rating of 9.5? (Round your answer to the nearest integer.)