## Sunday, June 30, 2019

### 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.