# Statistical Techniques in Business and Economics

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Statistical Techniques in Business and Economics

Problem 88 (Ch3)

Refer to the Baseball 2005 data, which reports information on the 30 major league teams for the 2005 baseball season.

Select the variable team salary and find the mean, median, and the standard deviation.

Select the variable that refers to the age the stadium was built. (Hint: Subtract the year in which the stadium was built from the current year to find the stadium age and work with that variable.) Find the mean, median, and the standard deviation.

Select the variable that refers to the seating capacity of the stadium. Find the mean, median, and the standard deviation.

Problem 56 (Ch 5)

Assume the likelihood that any flight on Northwest Airlines arrives within 15 minutes of the scheduled time is .90. We select four flights from yesterday for study.

a. What is the likelihood all four of the selected flights arrived within 15 minutes of the scheduled time?

b. What is the likelihood that none of the selected flights arrived within 15 minutes of the scheduled time?

c. What is the likelihood at least one of the selected flights did not arrive within 15 minutes of the scheduled time?

Problem 64 (Ch.64)

An internal study by the Technology Services department at Lahey Electronics revealed company employees receive an average of two emails per hour. Assume the arrival of these emails is approximated by the Poisson distribution.

a. What is the probability Linda Lahey, company president, received exactly 1 email between 4 P.M. and 5 P.M. yesterday?

b. What is the probability she received 5 or more email during the same period?

c. What is the probability she did not receive any email during the period?

Problem 18 (Ch 14)

Suppose that the sales manager of a large automotive parts distributor wants to estimate as early as April the total annual sales of a region. On the basis of regional sales, the total sales for the company can also be estimated. If, based on past experience, it is found that the April estimates of annual sales are reasonably accurate, then in future years the April forecast could be used to revise production schedules and maintain the correct inventory at the retail outlets.

Several factors appear to be related to sales, including the number of retail outlets in the region stocking the company's parts, the number of automobiles in the region registered as of April 1, and the total personal income for the first quarter of the year. Five independent variables were finally selected as being the most important (according to the sales manager). Then the data were gathered for a recent year. The total annual sales for that year for each region were also recorded. Note in the following table that for region 1 there were 1,739 retail outlets stocking the company's automotive parts, there were 9,270,000 registered automobiles in the region as of April 1 and so on. The sales for that year were $37,702,000.

Annual Sales ($ millions), Y Number of Retail Outlets, X1 Number of Automobiles Registered (millions), X2 Personal Income ($ billions), X3 Average Age of Automobiles (years), X4 Number of Supervisors, X5

37.702

1,739

9.27

85.4

3.5

9.0

24.196

1,221

5.86

60.7

5.0

5.0

32.055

1,846

8.81

68.1

4.4

7.0

3.611

120

3.81

20.2

4.0

5.0

17.625

1,096

10.31

33.8

3.5

7.0

45.919

2,290

11.62

95.1

4.1

13.0

29.600

1,687

8.96

69.3

4.1

15.0

8.114

241

6.28

16.3

5.9

11.0

20.116

649

7.77

34.9

5.5

16.0

12.994

1,427

10.92

15.1

4.1

10.0

Consider the following correlation matrix. Which single variable has the strongest correlation with the dependent variable? The correlations between the independent variables outlets and income and between cars and outlets are fairly strong. Could this be a problem? What is this condition called?

Ã? sales

Outlets

Cars

Income

Age

Outlets

0.899

Ã? Ã? Ã? Ã?

Cars

0.605

0.775

Ã? Ã? Ã?

Income

0.964

0.825

0.409

Ã? Ã?

Age

â?'0.323

â?'0.489

â?'0.447

â?'0.349

Ã?

Bosses

0.286

0.183

0.395

0.155

0.291

The output for all five variables is on the following page. What percent of the variation is explained by the regression equation?

The regression equation is

Sales = -19.7 - 0.00063 outlets + 1.74 cars + 0.410 income

+ 2.04 age - 0.034 bosses

Predictor Coef StDev t-ratio

Constant -19.672 5.422 -3.63

Outlets 0.000629 0.002638 0.24

Cars 1.7399 0.5530 3.15

Income 0.40994 0.04385 9.35

Age 2.0357 0.8779 2.32

Bosses -0.0344 0.1880 -0.18

Analysis of Variance

SOURCE DF SS MS

Regression 5 1593.81 318.76

Error 4 9.08 2.27

Total 9 1602.89

Conduct a global test of hypothesis to determine whether any of the regression coefficients are not zero. Use the .05 significance level.

Conduct a test of hypothesis on each of the independent variables. Would you consider eliminating outlets and bosses? Use the .05 significance level.

The regression has been rerun below with outlets and bosses eliminated. Compute the coefficient of determination. How much has R2 changed from the previous analysis?

The regression equation is

Sales = -18.9 + 1.61 cars + 0.400 income + 1.96 age

Predictor Coef StDev t-ratio

Constant -18.924 3.636 -5.20

Cars 1.6129 0.1979 8.15

Income 0.40031 0.01569 25.52

Age 1.9637 0.5846 3.36

Analysis of Variance

SOURCE DF SS MS

Regression 3 1593.66 531.22

Error 6 9.23 1.54

Total 9 1602.89

Following is a histogram and a stem-and-leaf chart of the residuals. Does the normality assumption appear reasonable?

Histogram of residual N = 10 Stem-and-leaf of residual N = 10

Leaf Unit = 0.10

Midpoint Count

-1.5 1 * 1 -1 7

-1.0 1 * 2 -1 2

-0.5 2 ** 2 -0

-0.0 2 ** 5 -0 440

0.5 2 ** 5 0 24

1.0 1 * 3 0 68

1.5 1 * 1 1

1 1 7

Following is a plot of the fitted values of Y (i.e., ?) and the residuals. Do you see any violations of the assumptions?

Problem 22 (Ch 17)

Banner Mattress and Furniture Company wishes to study the number of credit applications received per day for the last 300 days. The information is reported on the next page.

Number of Credit Applications Frequency (Number of Days)

0

50

1

77

2

81

3

48

4

31

5 or more

13

To interpret, there were 50 days on which no credit applications were received, 77 days on which only one application was received, and so on. Would it be reasonable to conclude that the population distribution is Poisson with a mean of 2.0? Use the .05 significance level. Hint: To find the expected frequencies use the Poisson distribution with a mean of 2.0. Find the probability of exactly one success given a Poisson distribution with a mean of 2.0. Multiply this probability by 300 to find the expected frequency for the number of days in which there was exactly one application. Determine the expected frequency for the other days in a similar manner.

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