Junior rugby league survey: reading a two-way table honestly
The Tāwhirimātea junior rugby league club has 420 registered players aged 8 to 17. The committee emailed a survey to every family asking which training night the player prefers (Tuesday, Wednesday or Thursday) and which age grade the player is in. 180 forms came back.
Results:
| Grade | Tuesday | Wednesday | Thursday | Total |
|---|---|---|---|---|
| Under 11 | 28 | 34 | 18 | 80 |
| 11–13 | 12 | 22 | 26 | 60 |
| 14+ | 22 | 8 | 10 | 40 |
| Total | 62 | 64 | 54 | 180 |
The club president writes on the noticeboard: "Wednesday wins with 36% - we'll move every grade to Wednesday nights." The 14+ coach objects. The club also wants to run a prize draw among the 180 who replied, and to estimate Thursday preference across all 420 players for an inter-club competition entry.
1. Evaluate the president's claim that "Wednesday wins". Use the table to support or challenge it.
Read the worked answer
Wednesday has 64 of 180 responses, which is 35.6% - the largest single night, so the number is right, but the conclusion is not.
Three problems:
- 36% is not a win in any decision sense. 116 of 180 players (64%) prefer a different night. The club would be moving the majority to a night they did not choose.
- The lead is an artefact of group size. Wednesday is the top choice only for Under 11s (34/80 = 42.5%). For 11–13 it is second (22/60 = 36.7%, behind Thursday's 43.3%). For 14+ it is last with 8/40 = 20%, against 55% for Tuesday. The Under 11 group is simply the biggest in the sample, so it pulls the total.
- Non-response. Only 180 of 420 families replied (42.9%), so the sample may not represent the club.
Better judgement: choose per grade (Under 11 Wednesday, 11–13 Thursday, 14+ Tuesday), or hold a second vote with a higher response rate. This is Simpson's-paradox-style reasoning - an overall percentage hiding a reversed pattern within groups.
2. The prize draw is made by pulling one reply form at random from the 180. Find: (a) P(Thursday, given the reply is from the 11–13 grade); (b) P(the reply is from the 14+ grade, given it chose Tuesday); (c) P(Tuesday or Thursday).
Read the worked answer
Conditional probability means restricting the denominator to the given group - not dividing by 180 each time.
(a) Thursday given 11–13 Denominator = the 11–13 row total = 60. Thursday count in that row = 26. P = 26/60 = 0.433 → 43.3%
(b) 14+ given Tuesday Denominator = the Tuesday column total = 62. 14+ count in that column = 22. P = 22/62 = 0.355 → 35.5%
Note the symmetry trap: P(14+ | Tuesday) = 22/62 is not the same as P(Tuesday | 14+) = 22/40 = 55%. The given condition decides the denominator.
(c) Tuesday or Thursday These are mutually exclusive, so add: 62 + 54 = 116 of 180. P = 116/180 = 0.644 → 64.4%
Since the draw is genuinely random from the returned forms only, these probabilities describe the respondents, not all 420 players.
3. Estimate how many of the 420 players prefer Thursday, and give two specific reasons this estimate could be off by substantially more than a handful of players.
Read the worked answer
Estimate Thursday share among respondents = 54/180 = 0.30 Estimated Thursday players = 0.30 × 420 = 126 players
Why it could be well out
- Non-response bias. Only 180 of 420 families replied. Families missed by an email, or who had already decided to leave the club, are unlikely to reply, and their preferences may cluster. If the true reply rate within a grade differs from its share of the club, the 30% moves. For example, if the 14+ grade is actually 120 of the 420 players rather than the 40 who replied, and 14+ Thursday support is only 25%, the estimate falls.
- The sample's grade mix may not match the club's. Respondents were 80 / 60 / 40 across grades. If the real club is, say, 200 / 130 / 90, weighting each grade's Thursday rate (22.5%, 43.3%, 25.0%) by the real mix gives 0.476×22.5 + 0.310×43.3 + 0.214×25.0 ≈ 29.6% - close here, but nothing in the survey guarantees that, and a club with proportionally more 11–13 players would push the estimate up.
The honest answer to give the competition organiser is "about 126, but treat it as a rough figure and say it is based on 43% of members".