Develop the least squares estimated regression equation

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1. Given are the data for two variables, x and y.

xi

8

13

17

20

22

yi

4

6

10

18

28

a. Compute b1 and b0 (to 3 decimals). b1 b0 Complete the estimated regression equation (to 3 decimals). ŷ = + x

b. Compute the residuals (to 2 decimals).

x1

x2

x3

x4

x5

c.

d. Consider the following three scatter diagrams of the residuals against the independent variable. Which of the following accurately represents the data? 1) http://mindtap.cengagenow.com/ilrn/bca/user/appletImage?dbid=916864703 2) http://mindtap.cengagenow.com/ilrn/bca/user/appletImage?dbid=587799719 3) http://mindtap.cengagenow.com/ilrn/bca/user/appletImage?dbid=115990672 Do the assumptions about the error terms seem to be satisfied?

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2.An important application of regression analysis in accounting is in the estimation of cost. By collecting data on volume and cost and using the least squares method to develop an estimated regression equation relating volume and cost, an accountant can estimate the cost associated with a particular manufacturing volume. Consider the following sample of production volumes and total cost data for a manufacturing operation.

Production Volume (units)

Total Cost ($)

400

4,300

450

5,300

550

5,700

600

6,200

700

6,700

750

7,300

a. Compute b1 and b0 (to 1 decimal). b1 b0 Complete the estimated regression equation (to 1 decimal). ŷ = + x

b. What is the variable cost per unit produced (to 1 decimal)?

c. Compute the coefficient of determination (to 3 decimals). Note: report r2 between 0 and 1. r2 = What percentage of the variation in total cost can be explained by the production volume (to 1 decimal)? %

d. The company's production schedule shows 500 units must be produced next month. What is the estimated total cost for this operation (to the nearest whole number)? $

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3. The commercial division of a real estate firm is conducting a regression analysis of the relationship between x, annual gross rents (in thousands of dollars), and y, selling price (in thousands of dollars) for apartment buildings. Data were collected on several properties recently sold and the following computer output was obtained.

The regression equation is

Y =20.0 + 7.27 X

Predictor

Coef

SE Coef

T

Constant

20.000

3.2213

6.21

X

7.270

1.3624

5.29

Analysis of Variance

SOURCE

DF

SS

Regression

1

41,587.4

Residual Error

7

Total

8

51,984.3

a. How many apartment buildings were in the sample?

b. Write the estimated regression equation (to 2 decimals if necessary). ŷ = + x

c. What is the value of sb1 (to 4 decimals)?

d. Use the F statistic to test the significance of the relationship at a .05 level of significance.

Compute the F test statistic (to 2 decimals).

What is the p-value? p-value is

What is your conclusion?

e. Predict the selling price of an apartment building with gross annual rents of $60,000 (to 1 decimal). $ thousands.

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4. In exercise 5, the following data on x = the number of defective parts found and y = the line speed (feet per minute) for a production process at Brawdy Plastics provided the estimated regression equation = 27.5 – .3x. Excel File: data12-37.xls http://cnow.apps.ng.cengage.com/ilrn/books/aseb06h/images/ch12/aseb06h_ch14_ex37.png

For these data SSE = 16. Develop a 95% confidence interval for the mean number of defective parts for a line speed of 25 feet per minute (to 4 decimals). to

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Given are five observations for two variables, x and y.

xi

1

2

3

4

5

yi

4

5

8

9

13

a. Which of the following scatter diagrams accurately represents the data? 1. http://mindtap.cengagenow.com/ilrn/bca/user/appletImage?dbid=1861290724 2. http://mindtap.cengagenow.com/ilrn/bca/user/appletImage?dbid=2006501144 3. http://mindtap.cengagenow.com/ilrn/bca/user/appletImage?dbid=1379434553

b. What does the scatter diagram indicate about the relationship between the two variables?

c. Develop the estimated regression equation by computing the the slope and the y intercept of the estimated regression line (to 1 decimal). ŷ = + x

d. Use the estimated regression equation to predict the value of y when x = 4 (to 1 decimal). ŷ =

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6. Consider the following data on production volume (x) and total cost (y) for a manufacturing operation. Excel File: data12-29.xls http://cnow.apps.ng.cengage.com/ilrn/books/aseb06h/images/ch12/14.21.jpg The estimated regression equation is ŷ = 1246.67 + 7.6x. Use α = .05 to test whether the production volume is significantly related to the total cost. Complete the ANOVA table. Enter all values with nearest whole number, except the F test statistic (to 2 decimals).

Source of Variation

Sum of Squares

Degrees of Freedom

Mean Square

F

Regression

Error

Total

What is your conclusion?

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7. In exercise 20, data on x = weight (pounds) and y = price ($) for ten road-racing bikes provided the estimated regression equation = 28574 -1439x (Bicycling website, March 8, 2012). For these data SSE = 7,102,922.54 and SST = 52,120,800. Use the F test to determine whether the weight for a bike and the price are related at the .05 level of significance. Click on the webfile logo to reference the data.

http://cnow.apps.ng.cengage.com/ilrn/books/aseb06h/images/webfiles/webfile.png

Calculate the value of the test statistic (to 1 decimal).

The p-value is .

What is your conclusion?

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8. Brawdy Plastics, Inc., produces plastic seat belt retainers for General Motors at the Brawdy Plastics plant in Buffalo, New York. After final assembly and painting, the parts are placed on a conveyor belt that moves the parts past a final inspection station. How fast the parts move past the final inspection station depends upon the line speed of the conveyor belt (feet per minute). Although faster line speeds are desirable, management is concerned that increasing the line speed too much may not provide enough time for inspectors to identify which parts are actually defective. To test this theory, Brawdy Plastics conducted an experiment in which the same batch of parts, with a known number of defective parts, was inspected using a variety of line speeds. The following data were collected. Excel file: data12-05.xls http://cnow.apps.ng.cengage.com/ilrn/books/aseb06h/images/ch12/aseb06h_ch14_ex05.png

If required, enter negative values as negative numbers.

a. Select a scatter diagram with the line speed as the independent variable. http://cnow.apps.ng.cengage.com/ilrn/books/aseb06h/images/ch12/aseb06h_graph_14-5.png

b. What does the scatter diagram developed in part (a) indicate about the relationship between the two variables?

c. Use the least squares method to develop the estimated regression equation (to 1 decimal). ŷ = + x

(continued on next page)

d. Predict the number of defective parts found for a line speed of 25 feet per minute.

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9. A sales manager collected the following data on annual sales and years of experience.

Salesperson

Years of Experience

Annual Sales ($1000s)

1

1

80

2

3

97

3

4

92

4

4

102

5

6

103

6

8

111

7

10

119

8

10

123

9

11

117

10

13

136

a. Which of the following scatter diagrams accurately represents the data? http://mindtap.cengagenow.com/ilrn/bca/user/appletImage?dbid=990422330 http://mindtap.cengagenow.com/ilrn/bca/user/appletImage?dbid=1173210639 http://mindtap.cengagenow.com/ilrn/bca/user/appletImage?dbid=1200279338

b. Develop an estimated regression equation that can be used to predict annual sales given the years of experience. Compute b1 and b0 (to 3 decimals). b1 b0 Complete the estimated regression equation. = + x

c. Use the estimated regression equation to predict annual sales for a salesperson with 12 years of experience (to 2 decimals). $ thousand

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10. One of the biggest changes in higher education in recent years has been the growth of online universities. The Online Education Database is an independent organization whose mission is to build a comprehensive list of the top accredited online colleges. The following table shows the retention rate (%) and the graduation rate (%) for 29 online colleges (Online Education Database website, January 2009). Click on the webfile logo to reference the data.

http://cnow.apps.ng.cengage.com/ilrn/books/aseb06h/images/webfiles/webfile.png

http://cnow.apps.ng.cengage.com/ilrn/books/aseb06h/images/ch12/aseb06h_ch14_57.png

a. Consider the three scatter diagrams below. http://cnow.apps.ng.cengage.com/ilrn/books/aseb06h/images/ch12/aseb06h_ch14_57a.png

Select a scatter diagram with retention rate as the independent variable.

What does the scatter diagram indicate about the relationship between the two variables?

b. Develop the estimated regression equation (to 4 decimals).

ŷ = + RR(%)

c. Test for a significant relationship. Use α = .05.

The p-value is .

Conclusion:

d. Did the estimated regression equation provide a good fit?

e. Suppose you were the president of South University. After reviewing the results, would you have any concerns about the performance of your university as compared to other online universities?

f. Suppose you were the president of the University of Phoenix. After reviewing the results, would you have any concerns about the performance of your university as compared to other online universities?

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11. Consider the data set below. Excel File: data12-33.xls http://cnow.apps.ng.cengage.com/ilrn/books/aseb06h/images/ch12/1433_data.jpg The estimated regression equation is ŷ = 30.33 - 1.88x.

a. Estimate the standard deviation of ŷ p when x = 3 (to 3 decimals).

b. Develop a 95% confidence interval for the expected value of y when x = 3 (to 2 decimals). ( , )

c. Estimate the standard deviation of an individual value of y when x = 3 (to 2 decimals).

d. Develop a 95% prediction interval for y when x = 3 (to 2 decimals). ( , )

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12. A large city hospital conducted a study to investigate the relationship between the number of unauthorized days that employees are absent per year and the distance (miles) between home and work for the employees. A sample of 10 employees was selected and the following data were collected. Excel file: data12-13.xls http://cnow.apps.ng.cengage.com/ilrn/books/aseb06h/images/ch12/aseb06h_ch14_ex13.png

If required, enter negative values as negative numbers.

a. Select a scatter diagram for these data. http://cnow.apps.ng.cengage.com/ilrn/books/aseb06h/images/ch12/aseb06h_graph_14-13.png

Does a linear relationship appear reasonable?

b. Develop the least squares estimated regression equation that relates the distance to work to the number of days absent (to 4 decimals). ŷ = + x

c. Predict the number of days absent for an employee who lives 5 miles from the hospital (round to nearest whole number). days

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13. In exercise 7 a sales manager collected the following data on y = annual sales and x = years of experience. The estimated regression equation for these data is ŷ = 80 + 4x. Click on the webfile logo to reference the data.

http://cnow.apps.ng.cengage.com/ilrn/books/aseb06h/images/webfiles/webfile.png http://cnow.apps.ng.cengage.com/ilrn/books/aseb06h/images/ch12/14.19.a.jpg

a. Compute SST, SSR, and SSE.

SSE

SST

SSR

b.

c. Compute the coefficient of determination r2. Round your answer to a whole percentage. % Does this least squares line provide a good fit?

d. What is the value of the sample correlation coefficient (to 2 decimals)?

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14. Given are five observations for two variables, x and y.

xi

1

2

3

4

5

yi

4

7

6

12

15

The estimated regression equation is ŷ = 0.7 + 2.7x.

a. Compute the mean square error using the following equation (to 3 decimals). http://cnow.apps.ng.cengage.com/ilrn/books/aseb06h/images/ch12/14-mse.jpg

b. Compute the standard error of the estimate using the following equation (to 3 decimals). http://cnow.apps.ng.cengage.com/ilrn/books/aseb06h/images/ch12/14.x4.jpg

c. Compute the estimated standard deviation b1 using the following equation (to 3 decimals). http://cnow.apps.ng.cengage.com/ilrn/books/aseb06h/images/ch12/14.x3.jpg

d. Use the t test to test the following hypotheses (α = .05): http://cnow.apps.ng.cengage.com/ilrn/books/aseb06h/images/ch12/14.23.a.jpg Compute the value of the t test statistic (to 2 decimals). The p-value is What is your conclusion?

e. Use the F test to test the hypotheses in part (d) at a .05 level of significance. Complete the F table below. Calculate the Sum of Squares (to 1 decimal), the Mean Squares (to 3 decimals), and the F ratio (to 2 decimals).

Source of Variation

Degrees of Freedom

Sum of Squares

Mean Square

F

Regression

Error

Total

f. What is the p-value? What is your conclusion, based on this F test?

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15. Consider the following data for two variables, x and y. Excel File: data12-51.xls http://cnow.apps.ng.cengage.com/ilrn/books/aseb06h/images/ch12/aseb06h_ch14_51.gif

a. Consider the three scatter diagrams below. http://cnow.apps.ng.cengage.com/ilrn/books/aseb06h/images/ch12/aseb06h_ch14_51a.png

Which scatter diagram accurately represents the data?

Does the scatter diagram indicate any influential observations?

b. Compute the standardized residuals for these data (to 2 decimals, if necessary). Enter negative values as negative numbers.

Observation 1

Observation 2

Observation 3

Observation 4

Observation 5

Observation 6

Observation 7

Observation 8

Do the data include any outliers?