# Mathematics

EXAM 2

ECON2110: BUSINESS STATISTICS II INSTRUCTOR: ERJON GJOCI

WILLIAM PATERSON UNIVERSITY OF NEW JERSEY COTSAKOS COLLEGE OF BUSINESS

DEPARTMENT OF ECONOMICS, FINANCE, AND GLOBAL BUSINESS

Student Name: _________________________________________________

Due Monday, April 8, 2013 (6pm)

Sections

True/False (24 x 5 points each) = 120 points

Multiple Choice Questions (36 x 5 points each) = 180 points

Problem Solving (4 x 50 points each) = 200 points

Total = 500 points

True/False Questions

1. For an ANOVA test, rejection of the null hypothesis does not identify which treatment means differ significantly. True False

2. In an ANOVA table, k represents the total number of sample observations and n represents the total number of treatments. True False

3. If a confidence interval for the difference between a pair of treatment means includes 0, then we reject the null hypothesis that there is no difference in the pair of treatment means. True False

4. If we want to determine which treatment means differ, we compute a confidence interval for the difference between each pair of means. True False

5. When a blocking effect is included in an ANOVA, the result is a larger error sum of squares. True False

6. When a blocking effect is included in an ANOVA, the analysis is more likely to detect differences in the treatment means. True False

7. In a two-way ANOVA with interaction, there are two factor effects and an interaction effect. True False

8. Interaction between two factors occurs when the effect of one factor on the response variable is the same for any value of another factor. True False

9. If the coefficient of determination is expressed as a percent, its value is between 0% and 100%. True False

10. One assumption underlying linear regression is that the Y values are statistically dependent. This means that in selecting a sample, the Y values chosen, for a particular X value, depend on the Y values for any other X value. True False 11. The least squares technique minimizes the sum of the squares of the vertical distances between the actual Y values and the predicted values of Y. True False 12. The values of a and b in the regression equation are called the regression coefficients. True False

13. The hypothesis to test the slope of a regression equation is H0: α = 0. True False

14. The regression equation is used to estimate a value of the dependent variable Y based on a selected value of the independent variable X. True False

15. In regression analysis, error is defined as ( – Y). True False

16. A confidence interval can be determined for the mean value of Y for a given value of X. True False

17. An example of a dummy variable is “time to product’s first repair” in years. True False

18. The variance inflation factor is used to select or remove independent variables to reduce the effects of multicollinearity in a multiple regression equation. True False

19. In multiple regression analysis, a residual is the difference between the value of an independent variable and its corresponding dependent variable value. True False

20. For a global test of a multiple regression equation, the F-statistic is based on the regression and residual degrees of freedom. True False

21. Interaction occurs when the relationship between an independent variable and a dependent variable is affected by another independent variable. True False

22. In a multiple regression equation with three independent variables, X1, X2, and X3, the interaction term is expressed as (Y)(X1). True False

23. Stepwise regression analysis is a method that assists in selecting the most significant variables for a multiple regression equation. True False

24. Stepwise regression analysis is also called a “backward elimination” method. True False

Multiple Choice Questions

1. When testing for differences between treatment means, the t statistic is based on: A. The treatment degrees of freedom. B. The total degrees of freedom. C. The error degrees of freedom. D. The ratio of treatment and error degrees of freedom.

2. When testing for differences between treatment means, a confidence interval is based on A. the mean square error. B. the standard deviation. C. the sum of squared errors. D. the standard error of the mean.

3. When testing for differences between treatment means, the degrees of freedom for the t statistic are: A. k B. (n – 1) C. (n – k) D. (1/n1 + 1/n2)

4. A manufacturer of automobile transmissions uses two different processes. Management ordered a study of the production costs to see if there is a difference among the two processes. A summary of the findings is shown below.

What is the critical value of F at the 5% level of significance? A. 19.45 B. 3.00 C. 4.41 D. 4.38

5. A manufacturer of automobile transmissions uses two different processes. Management ordered a study of the production costs to see if there is a difference between the two processes. A summary of the findings is shown below.

What is the critical value of F at the 1% level of significance? A. 9.46 B. 8.29 C. 8.18 D. 4.61

6. A manufacturer of automobile transmissions uses three different processes. Management ordered a study of the production costs to see if there is a difference among the three processes. A summary of the findings is shown below.

What are the degrees of freedom for the treatment sum of squares? A. 2 B. 3 C. 10 D. 27

7. A manufacturer of automobile transmissions uses three different processes. Management ordered a study of the production costs to see if there is a difference among the three processes. A summary of the findings is shown below.

What are the degrees of freedom for the error sum of squares? A. 3 B. 10 C. 27 D. 30

8. A manufacturer of automobile transmissions uses three different processes. Management ordered a study of the production costs to see if there is a difference among the three processes. A summary of the findings is shown below.

What are the total degrees of freedom? A. 27 B. 28 C. 29 D. 30

9. The college of business was interested in comparing the attendance for three different class times for a business statistics class. The data follow.

What is the blocking variable? A. Day. B. Class time. C. Tuesday. D. 8:00 am class.

10. The college of business was interested in comparing the attendance for three different class times for a business statistics class. The data follow.

What is the treatment variable? A. Day. B. Class time. C. Tuesday. D. 8:00 am class.

11. The college of business was interested in comparing the attendance for three different class times for a business statistics class. The data follow.

What are the block and treatment degrees of freedom? A. 5 and 3. B. 5 and 5. C. 4 and 2. D. 3 and 15.

12. The college of business was interested in comparing the attendance for three different class times for a business statistics class. The data follow.

What is the critical F statistic for testing the hypothesis of equal treatment means at the 0.05 significance level? A. 1.96. B. 6.94. C. 3.84. D. 4.46.

13. A sales manager for an advertising agency believes that there is a relationship between the number of contacts that a sales person makes and the amount of the sales dollars earned. What is the dependent variable? A. Salesperson B. Number of contacts C. Amount of sales dollars D. Sales manager

14. A sales manager for an advertising agency believes that there is a relationship between the number of contacts that a sales person makes and the amount of the sales dollars earned. What is the independent variable? A. Salesperson B. Number of contacts C. Amount of sales D. Sales manager

15. A sales manager for an advertising agency believes that there is a relationship between the number of contacts that a sales person makes and the amount of the sales dollars earned. A regression analysis shows the following results:

What is the Y-intercept of the linear equation? A. -12.201 B. 2.195 C. -1.860 D. 12.505

16. A sales manager for an advertising agency believes that there is a relationship between the number of contacts that a sales person makes and the amount of the sales dollars earned. A regression analysis shows the following results:

What is the slope of the linear equation? A. -12.201 B. 2.195 C. -1.860 D. 12.505

17. A sales manager for an advertising agency believes that there is a relationship between the number of contacts that a sales person makes and the amount of the sales dollars earned. A regression analysis shows the following results:

What is the standard error of the slope? A. -0.176 B. 6.560 C. -12.201 D. 12.505 18. A sales manager for an advertising agency believes that there is a relationship between the number of contacts that a sales person makes and the amount of the sales dollars earned. A regression analysis shows the following results:

What is the decision regarding the hypothesis that the slope equals zero? A. Fail to reject the null hypothesis B. Fail to reject the alternative hypothesis C. Reject the null hypothesis D. Reject the alternative hypothesis

19. A sales manager for an advertising agency believes that there is a relationship between the number of contacts that a sales person makes and the amount of the sales dollars earned. A regression ANOVA shows the following results:

What is the value of the standard error of estimate? A. 9.310 B. 8.778 C. 8.328 D. 86.68 20. A sales manager for an advertising agency believes that there is a relationship between the number of contacts that a sales person makes and the amount of the sales dollars earned. A regression ANOVA shows the following results:

What is the value of the coefficient of correlation? A. 0.6317 B. 0.9754 C. 0.9513 D. 9.3104

21. A sales manager for an advertising agency believes that there is a relationship between the number of contacts that a sales person makes and the amount of the sales dollars earned. A regression ANOVA shows the following results:

What is the value of the coefficient of determination? A. 9.3104 B. 0.9754 C. 0.6319 D. 0.9513

22. A sales manager for an advertising agency believes that there is a relationship between the number of contacts that a sales person makes and the amount of the sales dollars earned. A regression analysis shows the following results

= 33.4. = 2814.4. The 95% confidence interval for 30 calls is A. 55.8, 51.5 B. 51.4, 55.9 C. 46.7, 60.6 D. 31.1, 76.2

23. A sales manager for an advertising agency believes that there is a relationship between the number of contacts that a sales person makes and the amount of the sales dollars earned. A regression analysis shows the following results:

= 33.4. = 2814.4. The 95% prediction interval for a particular person making 30 calls is A. 55.8, 51.5 B. 51.4, 55.9 C. 46.7, 60.6 D. 31.1, 76.2

24. A sales manager for an advertising agency believes that there is a relationship between the number of contacts that a sales person makes and the amount of the sales dollars earned. A regression analysis shows the following results:

What is the regression equation? A. = 2.195 – 12.201X B. = -12.201 + 2.195X C. = 12.201 + 2.195X D. = 2.1946 + 12.201X

25. Which of the following is a characteristic of the F-distribution? A. Normally distributed B. Positively skewed C. Negatively skewed D. Equal to the t-distribution

26. In a regression analysis, three independent variables are used in the equation based on a sample of forty observations. What are the degrees of freedom associated with the F-statistic? A. 3 and 39 B. 4 and 40 C. 3 and 36 D. 2 and 39

27. Which statistic is used to test hypotheses about individual regression coefficients? A. t-statistic B. z-statistic C. (chi-square statistic) D. F

28. Which statistic is used to test a global hypothesis about a multiple regression equation? A. t-statistic B. z-statistic C. (chi-square statistic) D. F

29. The coefficient of determination measures the proportion of A. explained variation relative to total variation. B. variation due to the relationship among variables. C. error variation relative to total variation. D. variation due to regression.

30. What happens as the scatter of data values about the regression plane increases? A. Standard error of estimate increases B. R2 increases C. (1 – R2) decreases D. Error sum of squares decreases

31. All other things being held constant, what is the change in the dependent variable for a unit change in the first independent variable for the multiple regression equation: Ŷ = 5.2 + 6.3X1 – 7.1 X2? A. -7.1 B. +6.3 C. +5.2 D. +4.4

32. The best example of a null hypothesis for a global test of a multiple regression model is: A. H0: β1 = β2 = β3 = β4 = 0 B. H0: µ1 = µ2 = µ3 = µ4 = 0 C. H0: β1 = 0 D. If F is greater than 20.00 then reject.

33. The best example of an alternate hypothesis for a global test of a multiple regression model is: A. H1: β1 = β2 = β3 = β4 = 0 B. H1: β1 ≠ β2 ≠ β3 ≠ β4 ≠ 0 C. H1: Not all the β’s are equal to 0 D. If F is less than 20.00 then fail to reject.

34. The best example of a null hypothesis for testing an individual regression coefficient is: A. H0: β1 = β2 = β3 = β4 = 0 B. H0: µ1 = µ2 = µ3 = µ4 = 0 C. H0: β1 = 0 D. H0: β1 ≠ 0

35. In multiple regression analysis, residuals (Y – Ŷ) are used to: A. Provide a global test of a multiple regression model. B. Evaluate multicollinearity. C. Evaluate homoscedasticity. D. Compare two regression coefficients.

36. In multiple regression analysis, residuals (Y – Ŷ) are used to: A. Provide a global test of a multiple regression model. B. Evaluate the assumption of linearity. C. Calculate the variance inflation factor. D. Compare two regression coefficients.

Problem Solving Questions

1. (50 points) A company compared the variance of salaries for employees who have been employed for 5 years or less with employees who have been employed for 10 years or more. They randomly selected 21 employees with 5 years or less experience and 15 employees with 10 years or more experience. The standard deviation for the group with 5 years or less experience was $2,225; the standard deviation for the group with 10 years or more experience is $1,875.

a) What is the F test statistic for the hypothesis test?

b) Using the 0.05 significance level, what is the F critical value for the hypothesis test? c) Using the 0.05 significance level, what is the decision regarding the null hypothesis?

2. (50 points) A bottle cap manufacturer with four machines and six operators’ wants to see if variation in production is due to the machines and/or the operators. ANOVA table follows.

a) What is the critical value of F for the machine treatment effect at the 1% level of significance?

b) What is the critical value of F for the operator block effect at the 1% level of significance? c) What is the mean square for machines? d) What is the mean square for operators? e) What is the mean square for error? f) What is the computed value of F for the machines? g) What is the computed value of F for the operators? h) Test the hypothesis that all operators are equally productive. State your decision in terms of the null hypothesis.

3. (50 points) A company wants to study the effect of an employee’s length of employment on their number of workdays absent. The results of the regression analysis follow.

What is the slope of the linear equation?

What is the Y intercept of the linear equation?

What is the least squares equation?

What is the meaning of a negative slope?

What is the standard error of estimate?

4. (50 points) Twenty-one executives in a large corporation were randomly selected for a study to determine the effect of several factors on annual salary (expressed in $000’s). The factors selected were age, seniority, years of college, number of company divisions they had been exposed to and the level of their responsibility. A regression analysis was performed using a popular spreadsheet program with the following regression output:

Write out the multiple regression equation.

Which independent variable has the most significant effect on annual salary?

What proportion of the total variation in salary is accounted for by the set of independent variables?

Test the hypothesis that the regression coefficient for age is equal to 0 at the 0.05 significance level. Report the degrees of freedom, the test statistic, the critical value, and your decision.