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# Business Statistics - Nonparametric and Parametric Tests

See attached file for complete description

1) For the problem, you may not make any assumptions concerning normality, etc

The following data represent the number of wins as of March 7, 2005 for the 30 NBA (National Basketball Association) teams

Eastern Conference Western Conference
Atlantic Division Southwest Division
Boston 31 San Antonio 46
Philly 29 Dallas 38
New Jersey 26 Houston 34
New York 25 Memphis 33
Toronto 25 New Orleans 13
Southeast Division Northwest Division
Miami 45 Seattle 41
Washington 33 Denver 30
Orlando 31 Minnesota 30
Charlotte 12 Portland 22
Atlanta 10 Utah 20
Central Division Pacific Division
Detroit 36 Phoenix 45
Cleveland 31 Sacramento 37
Chicago 29 Lakers 29
Indiana 29 Clippers 26
Milwaukee 24 Golden State 18

a) Sportscasters often state that the Western Conference is stronger than the Eastern Conference. Use the appropriate nonparametric
rank test to see if this is true.

b) Similarly, test to see if the 8 teams with the most wins in the Western Division are stronger than the top 8 teams in the Eastern
Division.

2) The following data are given for 20 automobiles.

MPG Horsepower Weight Car's Age Teenage Driver
17 240 4485 2 Yes
21 225 3860 4 No
26 160 2780 1 No
22 225 3520 5 No
20 170 3745 4 No
18 220 3915 4 No
22 184 3390 1 No
19 184 3575 2 No
14 225 4715 3 Yes
20 205 3640 2 No
21 205 3880 1 No
15 185 4230 6 Yes
19 275 4070 4 No
20 180 3495 2 No
22 170 3050 3 No
12 285 5590 3 Yes
20 127 3055 1 Yes
14 270 4660 1 Yes
19 185 3990 2 No
21 250 3620 1 No

a) Develop the regression for this data that attempts to relate the mileage to the other variables
b) Which coefficients are significant?
c) Predict the mileage for a 210 horsepower car weighing 4000 pounds which is 3 years old and driven by a teenager.
d) How much of the variation in mileage is due to the independent variables?

3) I In the analysis for this problem, you may make necessary assumptions such as normality, etc. It has
been stated that the arrival of Southwest Airlines in Philadelphia has had an impact in lowering the prices
of airline tickets in Philadelphia. To test this assumption, the following data was obtained. These data are
the round-trip costs of a coach ticket for travel on March 30 and returning on April 6.

Five cities services by Southwest from Philadelphia were randomly selected. The cost data are:

SW USAIR DELTA AA CONT
HOU 218 204 212 215 203
LAX 178 178 231 231 191
KC 214 216 308 138 188
NASHVILLE 195 182 233 213 152
CHI 158 158 146 178 177

Next, five cities not serviced by Southwest were chosen to represent flights similar to the flights serviced by Southwest
For example, the inclusion of LAX (Los Angeles) resulted in the selection of SFO (San Francisco). The cities selected
here corresponded to a city in the first data set in terms of similar mileage (given below) and direction from Philadelphia.

USAIR DELTA AA CONT
DFW 218 339 218 261
SFO 284 468 295 300
MSP 178 356 189 424
MEMPHIS 191 258 185 265
MILWAULEE 187 188 188 148

LAS 2717 SFO 2878
HOU 1550 DFW 1469
CHI 759 MIL 857
NASH 804 MEMP 1016
K.C. 1129 MSP 1158

a) Test to see if Southwest has had an impact in lowering prices.
b) What other data might be important in testing the impact of Southwest? (The data 2 years ago before Southwest appeared would be helpful)

#### Solution Preview

See attached file for complete solution

1) For the problem, you may not make any assumptions concerning normality, etc

The following data represent the number of wins as of March 7, 2005 for the 30 NBA (National Basketball Association) teams

Eastern Conference Western Conference
Atlantic Division Southwest Division
Boston 31 San Antonio 46
Philly 29 Dallas 38
New Jersey 26 Houston 34
New York 25 Memphis 33
Toronto 25 New Orleans 13
Southeast Division Northwest Division
Miami 45 Seattle 41
Washington 33 Denver 30
Orlando 31 Minnesota 30
Charlotte 12 Portland 22
Atlanta 10 Utah 20
Central Division Pacific Division
Detroit 36 Phoenix 45
Cleveland 31 Sacramento 37
Chicago 29 Lakers 29
Indiana 29 Clippers 26
Milwaukee 24 Golden State 18

a) Sportscasters often state that the Western Conference is stronger than the Eastern Conference. Use the appropriate nonparametric rank test to see if this is true.

b) Similarly, test to see if the 8 teams with the most wins in the Western Division are stronger than the top 8 teams in the Eastern Division.

We will use Mann-Whitney U test for this problem

a) Sportscasters often state that the Western Conference is stronger than the Eastern Conference. Use the appropriate nonparametric rank test to see if this is true.

Writing the data together

Eastern Conference Western Conference

Miami 45 E San Antonio 46 W
Detroit 36 E Phoenix 45 W
Washington 33 E Seattle 41 W
Boston 31 E Dallas 38 W
Orlando 31 E Sacramento 37 W
Cleveland 31 E Houston 34 W
Philly 29 E Memphis 33 W
Chicago 29 E Denver 30 W
Indiana 29 E Minnesota 30 W
New Jersey 26 E Lakers 29 W
New York 25 E Clippers 26 W
Toronto 25 E Portland 22 W
Milwaukee 24 E Utah 20 W
Charlotte 12 E Golden State 18 W
Atlanta 10 E New Orleans 13 W

And On arranging all the data in ascending order and ranking the number of wins in order from lowest to highest

S No Team No of wins E/W Rank Modified rank after breaking the tie
1 Atlanta 10 E 1 1
2 Charlotte 12 E 2 2
3 New Orleans 13 W 3 3
4 Golden State 18 W 4 4
5 Utah 20 W 5 5
6 Portland 22 W 6 6
7 Milwaukee 24 E 7 7
8 New York 25 E tie 8 8.5
9 Toronto 25 E tie 8 8.5
10 New Jersey 26 E tie 10 10.5
11 Clippers 26 W tie 10 10.5
12 Philly 29 E tie 12 13.5
13 Chicago 29 E tie 12 13.5
14 Indiana 29 E tie 12 13.5
15 Lakers 29 W tie 12 13.5
16 Denver 30 W tie 16 16.5
17 Minnesota 30 W tie 16 16.5
18 Boston 31 E tie 18 19
19 Orlando 31 E tie 18 19
20 Cleveland 31 E tie 18 19
21 Washington 33 E tie 18 21.5
22 Memphis 33 W tie 18 21.5
23 Houston 34 W 23 23
24 Detroit 36 E 24 24
25 Sacramento 37 W 25 25
26 Dallas 38 W 26 26
27 Seattle 41 W 27 27
28 Miami 45 E tie 28 28.5
29 Phoenix 45 W tie 28 28.5
30 San Antonio 46 W 30 30

Tie is handled by calculating the average of the serial no

eg
Serial No Rank
8 8
9 8
Average =(8+9)/2= 8.5

Serial No Rank
12 12
13 12
14 12
15 12

Average =(12+13+14+15)/2= 13.5

Separating the ranks of Eastern Conference and Western Conference

Team E/W Modified rank after breaking the tie
Atlanta E 1 209
Charlotte E 2 256
Milwaukee E 7
New York E 8.5
Toronto E 8.5
New Jersey E 10.5
Philly E 13.5
Chicago E 13.5
Indiana E 13.5
Boston E 19
Orlando E 19
Cleveland E 19
Washington E 21.5
Detroit E 24
Miami E 28.5

New Orleans W 3
Golden State W 4
Utah W 5
Portland W 6
Clippers W 10.5
Lakers W 13.5
Denver W 16.5
Minnesota W 16.5
Memphis W 21.5
Houston W 23
Sacramento W 25
Dallas W 26
Seattle W 27
Phoenix W 28.5
San Antonio W 30

Eastern Conference
Team No of wins Modified rank breaking the tie
Atlanta 10 1
Charlotte 12 2
Milwaukee 24 7
New York 25 8.5
Toronto 25 8.5
New Jersey 26 10.5
Philly 29 13.5
Chicago 29 13.5
Indiana 29 13.5
Boston 31 19
Orlando 31 19
Cleveland 31 19
Washington 33 21.5
Detroit 36 24
Miami 45 28.5
Total ranks= 209
R1

Western Conference
Team No of wins Modified rank breaking the tie
New Orleans 13 3
Golden State 18 4
Utah 20 5
Portland 22 6
Clippers 26 10.5
Lakers 29 13.5
Denver 30 16.5
Minnesota 30 16.5
Memphis 33 21.5
Houston 34 23
Sacramento 37 25
Dallas 38 26
Seattle 41 27
Phoenix 45 28.5
San Antonio 46 30
Total ranks= 256
R2

n1=no of teams in Eastern Conference= 15
n2=no of teams in Western Conference= 15
R1=sum of ranks of Eastern Conference= 209
R2=sum of ranks of Western Conference= 256

Calculating the U statistic

U= n1*n2+1/2*(n2*(n2+1))-R2= 89 =15*15+ 0.5*15*(15+1)-256

Testing the hypothesis:

Null Hypothesis: Ho: &#956;1=&#956;2
There is no difference between the mean no of wins of Western Conference and Eastern Conference
Alternative Hypothesis: H1: &#956;1<&#956;2
Mean no of wins of Western Conference is greater than the mean no of wins of Eastern Conference
Since we are finding out whether one of the means is greater than the other
this is a two tailed test.
Let us test the hypothesis at a significance level of &#945;= 0.05
No of tails= 1
From the tables:
n1= 15
n2= 15
U= 89
level of significance= 0.172842 P (one-tailed)
The two samples are not significantly different (P >= 0.05, one-tailed test).

Thus we would accept the null hypothesis
There is no difference between the mean no of wins of Western Conference and Eastern Conference

b) Similarly, test to see if the 8 teams with the most wins in the Western Division are stronger than the top 8 teams in the Eastern Division.

Writing the data together for the top 8 teams

Eastern Conference Western Conference

Miami 45 E San Antonio 46 W
Detroit 36 E Phoenix 45 W
Washington 33 E Seattle ...

#### Solution Summary

The solution provides answers to three questions- one on Mann-Whitney U test, another on regression and the third one on Testing differences between means with dependent samples.

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