Soccer Standings: Write the Season Report

Use random samples to compare two divisions without overstating the evidence.

Season Standings

A large youth soccer league has North and South divisions. Every team plays 20 matches. Use the random samples to evaluate possible statements for a season report.

The reporter randomly selects 24 teams from each division. These box plots show their final season points. Higher points indicate a better season record.

Which headline best summarizes both the centers and the middle-half spreads of the samples?

A. North has a higher typical point total and a smaller IQR.B. North has a lower typical point total and a smaller IQR.C. North has a higher typical point total and a larger IQR.D. North has a lower typical point total and a larger IQR.

Calculate each sample's IQR. North: points; South: points.

Select every statement that uses the season-point evidence appropriately.

An editor wants to print, “South is the higher-scoring division because its sample maximum is 50 points.” What is the best revision?

A. Compare sample sizes; equal sample sizes require equal typical standings.B. Compare the medians; one extreme team does not describe typical standings.C. Compare the ranges; the larger range always means higher typical standings.D. Compare the minima; the lowest team always determines typical standings.

The league will randomly choose one team from each division for an interview. Which expectation is most reasonable before the teams are chosen?

A. The North team must have more points because its sample median is higher.B. The South team must have more points because its sample range is larger.C. The teams must have equal points because both samples contain 24 teams.D. The North team may have more points, but either team's total could be higher.

Player Minutes

The reporter independently selects six players at random from all registered players in each division. Each value is a player's average minutes per match this season. Matches last 80 minutes.

Random Samples of Player Minutes
Division	Minutes per match
North	38, 44, 50, 50, 56, 62
South	48, 52, 56, 56, 60, 64

Find the sample mean for each division. North: minutes; South: minutes.

Calculate the South sample's mean absolute deviation (MAD). Average the absolute distances from its mean.

South sample MAD: minutes.

The North sample has a MAD of 6 minutes. Which sentence correctly compares player-to-player variation in these samples?

A. Both vary equally because each sample contains six players.B. North varies less because its sample mean is 6 minutes lower.C. North varies more because its average distance from its mean is larger.D. South varies more because its sample mean is 6 minutes higher.

Select every statement that the player-minute samples support.

10.

A player hopes to join a division where players typically average at least 54 minutes per match, with less player-to-player variation. Which advice best uses these samples?

A. Prefer North: its larger MAD guarantees every player at least 54 minutes.B. Consider South: its higher mean and smaller MAD fit, but guarantee nothing.C. Consider either: equal sample sizes show identical playing-time patterns.D. Prefer North: its lower mean and larger MAD better match both goals.

Check the Evidence

For a separate study, the reporter randomly samples 30 players from each position in each division. Compare season assists within the same position.

Season Assists: Sample Summaries
Position	North mean	South mean	North MAD	South MAD
Midfielder	7	9	4	4
Forward	8	10	1	1
Defender	3	3	1	2

11.

For which position is the evidence of a difference in typical assists strongest relative to player-to-player spread?

A. All positions: equal sample sizes make the evidence equally strong.B. Midfielder: its mean difference is largest compared with its MADs.C. Forward: its mean difference is largest compared with its MADs.D. Defender: its mean difference is largest compared with its MADs.

12.

For midfielders, express the difference between the sample means as a multiple of their common MAD.

The mean difference is times the common MAD.

13.

The editor sees equal defender means and plans to leave defenders out of the comparison. Which note identifies information the editor would miss?

A. The South sample shows more variation in assists despite the equal means.B. The South sample shows a higher typical assist total despite the equal means.C. The North sample shows more variation in assists despite its smaller MAD.D. The North sample shows a higher typical assist total despite the equal means.

14.

A reporter writes, “South players average more minutes because South coaches are more supportive.” Use the player-minute sample means and MADs to write a cautious comparison. Then explain why these random samples do not establish the reporter's explanation.