James Jesse, the head of the Bureau of Animal Husbandry, perceives that several agencies received large increases in appropriations last year because they encouraged interest groups to testify for them before the House Appropriations Committee. The accompanying sample data for five agencies similar to Jesse's were gathered. Using regression analysis, calculate a regression equation. Does a relationship exist? The Bureau of Animal Husbandry could pressure 15 groups to testify for it. What percentage increase in appropriation would this number of groups predict? Place a 90% confidence interval around that estimate. How large an increase is each additional interest group worth?
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Interest Groups Testifying
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Percentage Increase in Appropriation
|
|
25
|
22
|
|
14
|
17
|
|
7
|
8
|
|
18
|
19
|
Caretakers, a local nonprofit organization, operates the concession stands for the NewlandBaseballPark, home of the Newland Nuggets (a minor league baseball team). Doing so permits Caretakers to raise funds to operate its soup kitchen. Caretakers is concerned with waste in the concessions area. Too many precooked hot dogs are left over after a game. Heinz Canine, Caretakers' research analyst, feels that hot dog consumption (and other concession sales) can be predicted by the number of advance sale tickets purchased for a game (attendance is usually twice advance sales). Heinz gathers the accompanying data for a sample of 10 days. Run the regression for Heinz and tell him whether his hypothesis is correct. Predict how many hot dogs will be sold if 1,000 advance tickets are sold.
Assumptions of Linear Regression
Graph each set of data shown in the accompanying table. Determine whether the relationships are linear or nonlinear. If nonlinear, state what type of relationship it is.
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No. 1
|
No. 2
|
No. 3
|
No. 4
|
|
X
|
Y
|
X
|
Y
|
X
|
Y
|
X
|
Y
|
|
2
|
1
|
2
|
1
|
1
|
5
|
1
|
15
|
|
5
|
3
|
5
|
3
|
4
|
8
|
2
|
11
|
|
7
|
6
|
8
|
5
|
7
|
11
|
4
|
6
|
|
9
|
10
|
11
|
9
|
10
|
14
|
6
|
4
|
|
14
|
15
|
14
|
14
|
13
|
17
|
9
|
3
|
|
17
|
16
|
|
|
16
|
20
|
15
|
2
|
The Intercity Bus Company is concerned with the high cost of fuel. The company believes that fuel is being wasted because drivers are exceeding the speed limit. A series of tests (N = 85) shows the following relationship between bus speed (X) and miles per gallon of fuel (Y) for speeds between 35 and 85 miles per hour (mph):
Multiple Regression
If Janice Position-Classification, personnel officer for the Bureau of Forms, can forecast agency separations 6 months from now, she can plan recruitment efforts to replace these people. Janice believes that separations 6 months from now are determined by the number of agency people passed over for promotion (X1), the number of agency people 64 years old or older (X2), and the ratio of government salaries to private sector salaries (X3). Using regression, Janice finds the following:
Y^ = 27.4 + .35X1 + .54X2 - 271X3
sb1 = .0031 sb2 = 0.136 sb3 = 263 Sy|x = 54
R2 = .89 Adj. R2 = .85 N = 214
Write a one-page memo explaining the results, and then forecast the number of separations if 418 people are passed over for promotion, 327 people are 64 years old or older, and government salaries equal those in the private sector.
The McKeesport Fire Department wants to know how likely it is that a truck pump will fail. The fire chief, George Pyro (no relation), thinks pump failure is a function of age (X1) and water hardness (X2 measured on a scale of 1 to 10). The department statistician runs a regression on a dummy variable (coded 1 for failure and 0 for no failure) for 217 pumps. She finds the following:
Y^ = .14 + .01X1 + .05X2
sb1 = .0002 sb2 = .025 Sy|x = .04
R2 = .93 Adj. R2 = .92
Write a memo explaining what the regression means. If the average water hardness is 3 and the chief would like to replace any pump with a probability of failing of .80 or more, at what age should pumps be replaced?