TEST 2 KEY FALL 2007

POPULATION

SALES

A)

(1000's)

($1,000)

106

2607

96

2099

108

2945

49

1050

72

1266

52

883

54

1647

70

1841

51

1908

79

2130

110

2688

110

2703

F)

NULL:
THE MODEL IS NOT SIGNIFICANT

ALTERNATIVE:
THE MODEL IS

BEST FIT IS MEASURED BY THE LEAST
SQUARES CRITERION WHICH

SIGNIFICANT

MINIMIZES THE DISTANCE THE POINTS
ARE OFF THE LINE

F TEST

FCAL =

34.03828729

SUMMARY OUTPUT

FCRIT =

4.964602701

SINCE
FCAL > FCRIT REJECT THE NULL

Regression
Statistics

THE MODEL
IS SIGNIFICANT

Multiple R

0.879161

R Square

0.772925

P TEST

Adjusted R Square

0.750217

PCAL =

0.000165099

Standard Error

340.2563

PCRIT =

0.05

Observations

12

SINCE PCAL
< PRCIT REJECT THE NULL

THE MODEL IS
SIGNIFICANT

ANOVA

df

SS

MS

F

Significance F

THERE IS A SIGNIFICANT LINEAR

Regression

1

3940760

3940760

34.03829

0.000165

RELATIONSHIP
BETWEEN POPULATION

Residual

10

1157743

115774.3

AND SALES.

Total

11

5098503

G)

Coefficients

Standard Error

t Stat

P-value

Lower 95%

Upper 95%

Lower 95.0%

Upper 95.0%

NULL:BETA
1 IS 0

Intercept

80.50921

340.1664

0.236676

0.817686

-677.429

838.4472

-677.429

838.4472

ALTERNATIVE:
BETA 1 IS NOT 0

X Variable 1

23.82538

4.08372

5.834234

0.000165

14.72628

32.92448

14.72628

32.92448

T TEST

TCAL =

5.834234079

RESIDUAL OUTPUT

B)

R SQUARED =

0.772925

TCRIT =

2.228138842

THIS MEANS THAT 77.3% OF THE
VARIATION IN SALES

SINCE TCAL
> TCRIT REJECT THE NULL

Observation

Predicted Y

Residuals

IS EXPLAINED BY THE VARIATION IN POPULATION

BETA 1 IS
NOT ZERO

1

2606

1.00042

2

2367.746

-268.746

C)

RESIDUAL STANDARD ERROR IS =

340.2563

P TEST

3

2653.65

291.3497

THE PROBLEM IS THAT THIS
APPROXIMATION OF THE

PCAL =

0.000165099

4

1247.953

-197.953

POPULATION STANDARD DEVIATION IS
RELATIVE SO

PCRIT =

0.05

5

1795.937

-529.937

WE HAVE NO GOOD IDEA HOW GOOD OR BAD
IT IS

SINCE PCAL
< PRCIT REJECT THE NULL

6

1319.429

-436.429

BETA 1 IS
NOT ZERO

7

1367.08

279.9202

D)

THE FITTED REGRESSION MODEL IS

8

1748.286

92.71413

THERE IS A SIGNIFICANT LINEAR

9

1295.604

612.3964

SALES =

80.50921

+'

23.82538

POPULATION

RELATIONSHIP BETWEEN POPULATION
AND

10

1962.714

167.2857

SALES.

11

2701.301

-13.3011

E)

THE INTERCEPT IS

80.50921

12

2701.301

1.698897

THIS MEANS WHEN POPULATION IS ZERO
SALES WILL

H)

BE

$80509.21.

CONFIDENCE
INTERVAL BUILT AROUND

BETA 1 (THE
SLOPE)

THE SLOPE IS

23.82538

THIS MEANS FOR EACH ADDITIONAL 1000
POPULATION

BETA 1 =

23.82538

SALES WILL INCREASE BY $23825.38.

T =

2.228139

SE(BETA 1)
=

4.08372

I)

residuals vs fitted

LOWER

14.72628

UPPER

32.92448

ZERO IS NOT
IN INTERVAL, REJECT THE NULL

THAT BETA 1
IS 0.  IN FACT NOW WE CAN SAY

THAT WE ARE
95% CONFIDENT THAT THE TRUE

POPULATION
SLOPE IS BETWEEN 14.72 AND 32,92.

THERE IS A
SIGNIFICAN POSITIVE
RELATIONSHIP

BETWEEN
POPULATION AND SALES.

LOOKING FOR RANDOM, SEE NO
DISTINGUISHABLE PATTERN, ONLY 3 NEGATIVE POINTS

MAY BE DUE TO ONLY 12 POINTS

I) cont

actual vs predicted

histogram of residuals

min

-529.937

max

612.3964

range

1142.333

width

228.4666

start

end

frequency

-529.937

-301.47

2

-301.47

-73.0034

2

-73.0034

155.4632

4

155.4632

383.9298

3

383.9298

612.3964

1

looking for
a straight line , this looks good

EXTRA
CREDIT

XSTAR

6200

(SHOULD HAVE BEEN 62000 BUT A ZERO

WAS MISSING ON TEST)

TO MAKE
UNIT AGREE

6.2

XBAR

79.75

XSTAR -XBAR

-73.55

X

X-XBAR

SQUARED

106

26.25

689.0625

96

16.25

264.0625

108

28.25

798.0625

49

-30.75

945.5625

looking for normal, this is
approximately normal

72

-7.75

60.0625

52

-27.75

770.0625

54

-25.75

663.0625

70

-9.75

95.0625

51

-28.75

826.5625

79

-0.75

0.5625

110

30.25

915.0625

110

30.25

915.0625

SE(MEAN
RESPONSE)

316.0104

SUM

0

6942.25

T

2.228139

YHAT

228.2266

LOWER

-475.888

UPPER

932.3416

WITH 95% CONFIDENCE WE CAN SAY THAT

A POPULATION OF 6200 WILL YIELD
BETWEEN A SALES LOSS OF($475,000) AND SALES OF $932.34