1)Given the following regression output,

  Predictor

Coef

SE Coef

T

P

  Constant

84.998

1.863

45.61

0.000

       X₁

2.391

1.200

1.99

0.051

       X₂

–0.4086

0.1717

–2.38

0.020

  Analysis of Variance

  Source

DF

SS

MS

F

P

  Regression

2

77.907

38.954

4.14

0.021

  Residual Error

62

583.693

9.414

  Total

64

661.600

(a)

Write the regression equation. 

(b)

If X₁ is 4 and X₂ is 11, what is the value of the dependent variable? 

(c)

How large is the sample? How many independent variables are there?

(d-1)

State the decision rule for 0.05 significance level: H₀: β₁ = β₂ = 0; H₁: Not all β's are 0. 

(d-2)

Compute the value of the F statistic. 

(d-3)

What is the conclusion? Use the 0.05 significance level.

(e-1)

State the decision rule for 0.05 significance level

(e-2)

Compute the value of the test statistic. 

(e-3)

Which variable would you consider eliminating?

(f)

Outline a strategy for deleting independent variables in this case.

2. 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.

(a)

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? 

(b)

The following regression equation was obtained using the five independent variables. 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

(c)

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

(d)

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

(e)

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

The regression equation is sales = −18.9 + 1.61 cars + 0.400 income + 1.96 age

(f)

Following is a histogram of the residuals. Does the normality assumption appear reasonable?

  The normality assumption appears  reasonable.

(g)

Following is a plot of the fitted values of Y (i.e., 

3. The director of special events for Sun City believed that the amount of money spent on fireworks displays on the 4th of July was predictive of attendance at the Fall Festival held in October. She gathered the following data to test her suspicion.

Determine the regression equation. 

Complete the given table. 

Complete the ANOVA table. 

Is the amount spent on fireworks related to attendance at the Fall Festival?

In a particular chi-square goodness-of-fit test, there are six categories and 500 observations. Use the .01 significance level.

(a)

How many degrees of freedom are there?

(b)

What is the critical value of chi-square? 

Advertising expenses are a significant component of the cost of goods sold. Listed below is a frequency distribution showing the advertising expenditures for 60 manufacturing companies located in the Southwest. The mean expense is $52.0 million and the standard deviation is $11.32 million. Is it reasonable to conclude the sample data are from a population that follows a normal probability distribution?

Advertising Expense
($ Million)

Number
of Companies

  25 up to 35

5

  35 up to 45

10

  45 up to 55

21

  55 up to 65

16

  65 up to 75

8

Total

60

State the decision rule. Use the .05 significance level. 

H₀: The population of advertising expenses follows a normal distribution.

H₁: The population of advertising expenses does not follow a normal distribution.

Compute the value of chi-square. 

What is your decision regarding H₀?

There are four entrances to the Government Center Building in downtown Philadelphia. The building maintenance supervisor would like to know if the entrances are equally utilized. To investigate, 400 people were observed entering the building. The number using each entrance is reported below. At the .01 significance level, is there a difference in the use of the four entrances? 

  Entrance

Frequency

  Main Street

140

  Broad Street

120

  Cherry Street

90

  Walnut Street

50

     Total

400

 H₀: There is no difference in the use of the four entrances.

 H₁: There is a difference in the use of the four entrances.

Banner Mattress and Furniture Company wishes to study the number of credit applications
received per day for the last 300 days.

Number of Credit

Frequency

Applications

(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. 

H₀: Distribution with Poisson with µ = 2.

H₁: Distribution is not Poisson with µ = 2