Probability: What Affects Estimates s1

Spring 2013 Math 263 Deb Hughes Hallett

Math 263: Excel Assignment 3 on Sampling Distributions: Hiring Coaches in Football and Basketball

Purpose of Assignment: Sampling Distributions We are about to start statistical inference, which means drawing conclusions about a population from a sample. The central idea is that of a sampling distribution, which involves all the samples of a fixed size from a population. This distribution allows us to decide whether a sample result is surprising. This assignment is intended to get you to think (hard!) about sampling distributions.

The example we consider is the hiring of minority head coaches in universities. You will look at two simulations (computer experiments) of random samples drawn from applicant pools similar to actual pools. These simulations are approximations to sampling distributions. The assignment does not involve much calculation; the purpose is for you to see how to draw conclusions from the data.

We will look at the hiring in two sports:1  Men’s football, FBS (Division I-A), 2007-2008  Women’s basketball, FBS (Division I), 2008-2009

For each sport, you will look at the data in the CoachSimulation.xlsx to decide whether roughly the expected numbers of minority coaches were hired. To do this we must make some assumptions—in particular, that all the candidates for a position are equally qualified. The table below shows the actual data for the head coach positions.

Short-Listed Minority Short-Listed Positions Minorities Candidates Candidates Available Hired Men’s Football, 07-08 74 25 18 2 Women’s Basketball, 08-09 55 21 16 9

The spreadsheet has four pages, two for men and two for women. (One Live and one Frozen page.) For each sport, there are some questions to be answered using the table above, some questions about the Live page, and some questions about the Frozen page.

MEN’S FOOTBALL

Prior Expectations 1. Given the number of minority candidates for men’s football, how many minority coaches would you have expected to be hired for the 18 available positions? (Assume all short-listed candidates were equally qualified.)

1Idea from Tisha Hook and Chris Malone, Winona State University. Based on “Who you Know and Who Knows You: The Hiring Process and Practice of NCAA FBS & FCS Head Coaching Positions” by K. Harrison et al, report for BCA (Black Coaches and Administrators) and “BCA Hiring Report Card for Women’s College Basketball, 2008-2009” by R. Lapchick. 1 Spring 2013 Math 263 Deb Hughes Hallett

2. Would you expect the number of minorities hired to be exactly the number you gave as your answer to #1? Why or why not? 3. If the number of minority coaches hired was roughly as expected from the short-listed applicant pool, is it likely, or unlikely, that two minority coaches were hired? Why?

Men’s Simulation: Live Page of Spreadsheet 4. The candidates are listed on the left. Press F9 (PC) or Apple+ (Mac) to see the sheet recalculate. The big block in the middle shows many samples from the candidate pool. The result of each sample is shown in the yellow column on the right. The result from all the samples is shown in the yellow rows across the top. (a) How many individuals were in each sample? Why was this number chosen? (b) How many samples were taken? (c) What do the numbers represent in the yellow column on the right (Column X)? (d) What do the numbers represent in the yellow rows at the top (Rows 1 and 2)? 5. Compare the numbers in the yellow Row 2 with what you expected in #1. Are they what you expected or not?

Men’s Simulation: Frozen Page of Spreadsheet 6. The data from the live simulation has been “frozen” so it will no longer recalculate, and a bar chart made of the results. Based on this chart, estimate the probability that two minority coaches were hired. 7. Explain in words what the your answer to #6 tells us about whether the actual hiring of minority head coaches in men’s football was as expected.

WOMEN’S BASKETBALL

Prior Expectations 8. Given the number of minority candidates for women’s basketball, how many minority coaches would you have expected to be hired? (Assume all short-listed candidates were equally qualified.) 9. If the number of minority coaches hired was roughly as expected from the short-listed applicant pool, is it likely, or unlikely, that nine minority coaches were hired? Is it more or less likely that 2 minority men were hired?

Women’s Simulation: Live Page 10. How many individuals were in each sample? Why is it different than for the men’s simulation?

Women’s Simulation: Frozen Page 11. Based on the chart made from “frozen” data, estimate the probability that exactly nine minority coaches were hired. 12. Explain in words what the your answer to #11 tells us about whether the actual hiring of minority head coaches in women’s basketball was as expected?