Florida Gulf Coast University



Simple Linear Regression problems:

1. In regression analysis if the dependent variable is measured in dollars, the independent variable

a. must also be in dollars

b. must be in some unit of currency

c. can be any units

d. can not be in dollars

2. Regression analysis was applied between sales (in $1,000) and advertising (in $100), and the following regression function was obtained.

[pic] = 80 + 6.2 x

Based on the above estimated regression line, if advertising is $10,000, then the point estimate for sales (in dollars) is

a. $62,080

b. $142,000

c. $700

d. $700,000

3. Regression analysis was applied between sales (in $1000) and advertising (in $100) and the following regression function was obtained.

[pic] = 500 + 4x

Based on the above estimated regression line if advertising is $10,000, then the point estimate for sales (in dollars) is

a. $900

b. $900,000

c. $40,500

d. $505,000

4. A regression analysis between demand (y in 1000 units) and price (x in dollars) resulted in the following equation

[pic] = 9 – 3x

The above equation implies that if the price is increased by $1, the demand is expected to

a. increase by 6 units

b. decrease by 3 units

c. decrease by 6,000 units

d. decrease by 3,000 units

5. If the coefficient of determination is a positive value, then the regression equation

a. must have a positive slope

b. must have a negative slope

c. could have either a positive or a negative slope

d. must have a positive y intercept

6. It is possible for the coefficient of determination to be

a. larger than 1

b. less than one

c. less than zero

d. All of these answers are correct, depending on the situation under consideration.

7. Given below are seven observations collected in a regression study on two variables, x (independent variable) and y (dependent variable). Use Excel to develop a scatter diagram and to compute the least squares estimated regression equation and the coefficient of determination.

x y

2 12

3 9

6 8

7 7

8 6

7 5

9 2

8. Regression analysis is a statistical procedure for developing a mathematical equation that describes how

a. one independent and one or more dependent variables are related

b. several independent and several dependent variables are related

c. one dependent and one or more independent variables are related

d. None of these answers is correct.

9. In regression analysis, the variable that is being predicted is the

a. dependent variable

b. independent variable

c. intervening variable

d. None of these answers is correct.

10. Wageweb conducts survey of salary data and presents summaries on its website. Based on salary data as of October 2007, Wageweb reported that the average annual salary for sales vice presidents was $142,111, with an average annual bonus of $15,432. Assume the following data are a sample of the annual salary and bonus for 10 sales vice presidents. Data are in thousands of dollars:

|Salary ($1000s) |Bonus ($1000s) |

|135 |12 |

|115 |14 |

|146 |16 |

|167 |19 |

|165 |22 |

|176 |24 |

|98 |7 |

|136 |17 |

|163 |18 |

|119 |11 |

a: Develop a scatter diagram for these data with salary as the independent variable

b. What does the scatter diagram developed in (a) indicate about the relationship between salary and bonus?

c. Use the least square method to develop the estimated regression equation

d. Determine and interpret the practical significance of the model

e. Determine the statistical significance of the estimated model

f. Provide and interpret the slope of the estimated regression model

g. predict the bonus for a vice president with an annual salary of $120,000

Answers:

1. In regression analysis if the dependent variable is measured in dollars, the independent variable

ANSWER: c

11. Regression analysis was applied between sales (in $1,000) and advertising (in $100), and the following regression function was obtained.

ANSWER: d

12. Regression analysis was applied between sales (in $1000) and advertising (in $100) and the following regression function was obtained.

ANSWER: b

13. A regression analysis between demand (y in 1000 units) and price (x in dollars) resulted in the following equation

ANSWER: d

14. If the coefficient of determination is a positive value, then the regression equation

ANSWER: c

15. It is possible for the coefficient of determination to be

ANSWER: b

16. Given below are seven observations collected in a regression study on two variables, x (independent variable) and y (dependent variable). Use Excel to develop a scatter diagram and to compute the least squares estimated regression equation and the coefficient of determination.

ANSWER:

[pic]

17. Regression analysis is a statistical procedure for developing a mathematical equation that describes how

ANSWER: c

18. In regression analysis, the variable that is being predicted is the

ANSWER: a

19. Wageweb conducts survey of salary data and presents summaries on its website. Based on salary data as of October 2007, Wageweb reported that the average annual salary for sales vice presidents was $142,111, with an average annual bonus of $15,432. Assume the following data are a sample of the annual salary and bonus for 10 sales vice presidents. Data are in thousands of dollars:

|Salary ($1000s) |Bonus ($1000s) |

|135 |12 |

|115 |14 |

|146 |16 |

|167 |19 |

|165 |22 |

|176 |24 |

|98 |7 |

|136 |17 |

|163 |18 |

|119 |11 |

a: Develop a scatter diagram for these data with salary as the independent variable

b. What does the scatter diagram developed in (a) indicate about the relationship between salary and bonus?

c. Use the least square method to develop the estimated regression equation

d. Determine and interpret the practical significance of the model

e. Determine the statistical significance of the estimated model

f. Provide and interpret the slope of the estimated regression model

g. predict the bonus for a vice president with an annual salary of $120,000

a.

. [pic]

b. there is a positive linear relationship between bonus and salary

as salary increases, bonus increases as well

c.

|SUMMARY OUTPUT | | | | | | |

| | | | | | | |

|Regression Statistics | | | | | |

|Multiple R |0.92479505 | | | | | |

|R Square |0.85524589 | | | | | |

|Adjusted R Square |0.83715163 | | | | | |

|Standard Error |2.08389616 | | | | | |

|Observations |10 | | | | | |

| | | | | | | |

|ANOVA | | | | | | |

|  |df |SS |MS |F* |Significance F* | |

|Regression |1 |205.259 |205.259 |47.26613 |0.000127709 | |

|Total |9 |240 |  |  |  | |

| | | | | | | |

|  |Coefficients |Standard Error |t Stat |P-value |Lower 95% |Upper 95% |

|Intercept |-10.1640754 |3.862297 |-2.63161 |0.030103 |-19.0705472 |-1.2576 |

Salary ($1000s) |0.18425405 |0.0268 |6.875037 |0.000128 |0.122452116 |0.246056 | |

E(bonus) = -10.16 + 0.184 (Sal)

d.

R2= 0.855 which is greater than our benchmark 0.7. therefore there is a strong linear model fit of the data.

SE = 2.08 (or $2080) of 'average' error in the model.

In percentage of an average value of a bonus of $15,432 …see problem description, we have

SE/average bonus = 13.47%

This is less than our benchmark of 15% error..Therefore we can state that due to these 2 indicators, the model indeed has strong practical significance.

e. (has 2 parts)

part a.

Ho: the model has no statistical significance (b1=0)

Ha: the model has statistical significance (b1=/=0)

see ANOVA table for the significance (or p-value) of the F test*

since p-value of F is < alpha of 0.05, reject H0 and conclude Ha. B1=/=0 and the model has statistical significance

part b.

Ho: b1 = 0 (the slope b1 is not significant)

Ha: b1=/= 0 (the slope b1 ia significant)

since p-value** < alpha of 0.05, reject H0 and conclude Ha; slope b1 is statistically significant (meaning Salary is significantly related to Bonus through slope b1)

** see printout

f. for each additional $1000 increase in salary, there is a $184 increase in bonus

g. E(Bonus) = -10.16+ 0.184(120)=

E(bonus)= 11.92

E(Bonus)= $11,920.00

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