Monday, March 13, 2017

Three Small Sample Hypothesis Testing Problems on Means

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 25th 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 25th 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:





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 1970-2002 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, 25th percentile, 75th 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 t-test, and the Wilcoxon rank sum test.



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