Three Small Sample
Hypothesis Testing Problems on Means
This post presents
three problems concerning hypothesis tests on means for small samples. The topics covered here include – (1)
comparing SAT scores for the original and expanded Big Ten conference, (2)
comparing Zillow rent estimates to rent requests, and (3) comparing number of
touchdowns for the first QB chosen and the second QB chosen in the NFL draft.
This post contains
questions and links to answers published in other posts.
Question One: Once upon a time the Big Ten consisted of 10
schools. Four new schools Rutgers, University of Maryland,
Pennsylvania State, and University of Nebraska entered the conference in recent
years.
What
did the entry of these four schools do to the mean of the 25^{th}
percentile of the Verbal SAT score in the Big Ten?
Conduct
a hypothesis test for a difference in the mean for Verbal SAT at the 25^{th}
percentile between the two groups.
How
does the existence of Northwestern the outlier impact the results presented
here?
Big Ten Verbal and Math SAT Averages


Original Big Ten Schools



School

Verbal SAT 25th Percentile

1

Ohio State

540

2

University of Michigan

620

3

Michigan State

420

4

University of Minnesota

550

5

University of Iowa

540

6

Purdue

520

7

Indiana University

520

8

Northwestern

690

9

University of Illinois

560

10

University of Wisconsin

530

New Big Ten Schools


1

Rutgers

520

2

University of Maryland

580

3

Penn State

530

4

University of Nebraska

490

Answer to Big Ten
Problem:
Expanded Versus Old Big Ten Problem: http://www.dailymathproblem.com/2016/05/newbigtenversusoldbigtensmall.html
Question Two: The table below has data on requested and estimated rents
on home in Venice California.
Requested and Estimated Rents for 3+ bed
room houses in Venice California



Requested Rent

Zillow
Rent
estimate

Difference Rent  Estimate

% Difference

1

3798

4000

202

5.3%

2

4550

5100

550

12.1%

3

4600

4700

100

2.2%

4

4895

4600

295

6.0%

5

5395

5300

95

1.8%

6

5900

4600

1300

22.0%

7

6500

5100

1400

21.5%

8

6800

5000

1800

26.5%

9

6500

6500

0

0.0%

10

6500

7100

600

9.2%

11

6995

7900

905

12.9%

12

7750

6000

1750

22.6%

13

7950

6900

1050

13.2%

14

7950

4600

3350

42.1%

15

8000

4800

3200

40.0%

16

8000

5200

2800

35.0%

17

8500

6400

2100

24.7%

18

8500

8500

0

0.0%

19

11500

6900

4600

40.0%

20

12500

12000

500

4.0%

21

12500

6600

5900

47.2%

22

15000

7600

7400

49.3%

Source
of data is
Compare the requested rents to the Zillow estimates?
Conduct a classical hypothesis test that the difference
between requested and Zillow estimate rent was zero?
Conduct the nonparametric Wilcoxon rent for differences
between requested and Zillow estimate
rents
Answers to rent
problems:
The descriptive statistics and the result of the classical
hypothesis test is presented here:
The results of the Wilcoxon Test is presented here:
Question Three: The table below contains information on the number of career
touchdown passes made by the first QB selected and the second QB selected in
every NFL draft spanning the 19702002 period. (I did not include
QBs chosen after 2002 because a large number of such QBs have not yet completed
their careers and I wanted to focus on QBs that had completed or had almost
completed their career.)
What
is the average, median, minimum, 25^{th} percentile, 75^{th}
percentile, maximum and the standard deviation of career touchdown passes for
the first QB choice and the second QB choice?
Conduct
appropriate hypothesis tests.
What
does your statistical analysis suggest about the relative value of the first QB
chosen compared to the second QB chosen in the NFL draft?
What
are the potential implications regarding the risk of trading up in order to get
the first QB in the draft?
Data:
Touchdowns for First QB Picked and Second
QB Picked


Year

First QB Chosen

# of Touchdowns

Second QB Chosen

# of Touchdowns

2002

David Carr

65

Joey Harrington

79

2001

Michael Vick

128

Drew Brees

363

2000

Chad Pennington

102

Giovanni Carmazzi

0

1999

Tim Couch

64

Donavan McNabb

234

1998

Peyton Manning

491

Ryan Leaf

14

1997

Jim Druckenmiller

1

Jake Plummer

161

1996

Tony Banks

77

Bobby Hoying

11

1995

Steve McNair

174

Kerry C ollins

208

1994

Heath Shuler

15

Trent Dilfer

113

1993

Drew Bledsoe

251

Rick Mirer

50

1992

David Klinger

16

Tommy Maddox

48

1991

Dan McGwire

2

Todd Marinovich

8

1990

Jeff George

154

Andrew Ware

5

1989

Troy Aikman

165

Mike Elkins

0

1988

Chris Chandler

170

Don Mcpherson

0

1987

Vinny Testaverde

275

Kelly Stoufer

7

1986

Jim Everett

203

Chuck Long

19

1985

Randall Cunningham

207

Frank Reich

40

1984

Boomer Essiason

247

Jeff Hostetler

94

1983

John Elway

300

Todd Blackedge

29

1982

Art Schlichter

3

Jim McMahon

100

1981

Rich Campbell

3

Neil Lomax

136

1980

Marc Wilson

86

Mark Malone

60

1979

Jack Thompson

33

Phil Simms

199

1978

Doug Williams

100

Matt Cavanaugh

28

1977

Steve Pisarklewicz

3

Tommy Kramer

159

1976

Richard Todd

124

Mike Kruczek

0

1875

Steve Bartkowski

156

Steve Grogan

182

1974

Danny White

155

David Jaynes

0

1973

Bert Jones

124

Gary Huff

16

1972

Jerry Tagge

3

John Reaves

17

1971

Jim Plunkett

164

Archie Manning

125

1970

Terry Bradshaw

212

Mike Phipps

55

I have written too many posts on NFL draft issues.
Here are some links to my analysis on this topic.
This post has descriptive statistics related to the table
above:
This post uses the football data to illustrate three tests –
the classical test on differences between means, the paired ttest, and the
Wilcoxon rank sum test.
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