112i - Problem Solving with Sample Variation

Lesson 112 — Problem Solving and Consolidation: Sample Variation

Strand: Statistics | Descriptor: AC9M8ST03 | Duration: 45 minutes

Learning Intentions

• To consolidate comparing variation in proportions from random samples of the same size.
• To apply the sample-size–variation relationship to judge real claims.

Success Criteria

1. Calculate and compare sample proportions and ranges across different contexts.
2. Judge whether a claimed difference between samples could be explained by ordinary sampling variation.
3. Work backwards from a proportion to a count, and vice versa.
4. Justify a recommendation for sample size in a real decision.

Warmup

(6 minutes — “believe it or not”, pairs)

Two students each flip a coin (true ) a different number of times.

1. Amelia flips times and gets heads.
2. Noah flips times and gets heads.
3. Both results are further above than the other in one sense — which measure shows that, and which student’s result is more surprising?

Answers: 1. ; 2. ; 3. Amelia’s proportion () is further from as a proportion, but her sample is much smaller — with only flips, a result of heads is not very surprising. Noah’s from flips is a smaller deviation but from a much larger sample, so it is, in one sense, the more noteworthy result. Small samples produce big-looking swings that mean less.

Activities

Activity 1 — Mixed Problem Circuit (16 min)

Stations. Every answer needs the calculation shown, not just the result.

Station A — Calculate and compare.

1. A sample of has successes. Find .
2. A sample of has successes. Find .
3. Explain why Q1 and Q2 give the same despite very different sample sizes, and why the result is more trustworthy as an estimate of a population proportion.

Station B — Ranges.

4. Four samples of size : proportions . Find the range.
5. Four samples of size from the same population: proportions . Find the range.
6. What do Q4 and Q5 together illustrate?

Station C — Working backwards.

7. A sample of gives . Find the number of successes.
8. A sample gives from successes. Find .
9. Two samples of have a combined total of successes across both. If Sample X had , find Sample Y’s proportion.

Station D — Judging claims.

10. A news report says “Poll A found support (); Poll B found support (). Support has clearly grown.” What is wrong with this conclusion?
11. A shop owner tests customers and finds prefer a new snack; a rival tests customers and finds prefer it. Whose result would you trust more as an estimate of all customers’ preference, and why?

Socratic scaffolding for Station D Q10:

Prompt	Purpose
Understand: what is actually being compared?	Two different polls, with very different sample sizes, not necessarily the same population at two different times.
What do we know about small-sample proportions?	They vary widely just by chance, even with no real change underneath.
Devise a plan: could the percentage-point gap be explained without any real change in support?	Yes — Poll A’s small sample alone could easily swing by points or more from ordinary sampling variation.
Carry it out: which poll would you trust more as an estimate, and why?	Poll B — its much larger sample gives a far narrower range of likely variation.
Look back: what should the report have said instead?	That the two polls are not directly comparable without knowing how much each could vary by chance, and that Poll B’s larger sample makes it the more reliable figure.

(Answers: 1. ; 2. ; 3. Same proportion, but draws on four times as much data, so its estimate is less likely to be far from the true population proportion. 4. ; 5. ; 6. That variation between same-size samples shrinks as sample size grows — even though both circuits used the same underlying population. 7. ; 8. ; 9. Sample X has successes, so Sample Y has successes out of , . 10. The two polls have very different sample sizes; Poll A’s small sample could easily have produced by chance even if true support were the same as Poll B’s — the “growth” may not be real. 11. The shop’s rival () — a far larger sample gives a much narrower range of likely sampling variation, so its is more trustworthy as an estimate of the whole customer base, even though it is a less dramatic number.)

Activity 2 — Applied Problem: Judging the Free-throw Shooter (16 min)

Pairs. A single connected problem, worked in stages.

A basketball coach is deciding whether a player is genuinely a free-throw shooter () or has just had a lucky run.

Session 1 ( shots): makes. Session 2 ( shots): makes. Session 3 ( shots): makes.

1. Calculate for each session.
2. Which session’s result deviates furthest from as a proportion?
3. Which session’s result should the coach trust most as evidence of the player’s true long-run rate? Justify using today’s ideas about sample size.
4. The coach wants to decide, after just one practice, whether the player’s true rate has genuinely dropped below . Recommend a minimum number of shots to watch, and justify your recommendation.
5. Write one sentence a sports commentator could honestly say about this player’s shooting, based on all three sessions combined.

Socratic scaffolding (Polya cycle):

Prompt	Purpose
Understand the problem: what is the coach actually trying to find out?	The player’s true, long-run shooting rate — not just what happened in one session.
What do you know from Lesson 111 about small versus large samples?	Small samples (Session 1) swing widely by chance; large samples (Session 3) are more stable estimates.
Devise a plan for Q4	Recommend the largest practical sample, since that minimises the chance of a misleading conclusion; justify with the shrinking-range pattern.
Carry out the plan	Combine all three sessions: makes out of shots, .
Look back	Does the combined estimate sit closer to than any single session? Is it a fairer summary than any one session alone? Why?

(Answers: 1. Session 1: ; Session 2: ; Session 3: . 2. Session 2 deviates most in raw percentage-point terms ( points below ), but Session 1’s result — despite matching exactly — is the least trustworthy because it comes from so few shots. 3. Session 3 — the largest sample gives the estimate least affected by chance swings. 4. At least – shots; the wider the sample, the narrower the range of results ordinary chance alone could produce, so a genuine drop becomes easier to distinguish from luck. 5. E.g. “Across shots this session, the player made about — slightly below their usual , though this is based on more shots than any single practice and is a fairer summary than any one session alone.“)

Checks for Understanding

(7 minutes — exit ticket, collected)

1. A sample of has successes. Find .
2. Four samples of size give proportions . Find the range.
3. Would you expect a set of four samples of size from the same population to have a larger or smaller range than Q2? Justify.
4. A sample gives from successes. Find .
5. Reasoning. A weather app says “Yesterday out of nearby towns had rain; today, forecasters predict rain in of the region.” Explain why the -out-of- figure is a poor basis for judging the region’s typical rain pattern, using ideas from this unit.

Answers: 1. ; 2. ; 3. Smaller — a larger sample size produces less variation between same-size samples; 4. ; 5. Five towns is a very small sample, so its proportion can swing widely from the true regional rate just by chance; a much larger sample of towns (or a longer run of days) would give a far more reliable estimate.

Common Misconceptions

Misconception	How to pre-empt it
Judging reliability from how “round” or “close to expected” a result looks, ignoring sample size.	Station D and the free-throw problem both reward citing , not just .
Believing a percentage-point gap between two polls/samples always reflects a real difference.	Activity 1 Q10 and exit Q5 — small samples can produce large-looking gaps by chance alone.
Treating the largest single sample’s result as certainly correct.	Reinforce “more reliable”, never “guaranteed correct”.
Averaging proportions instead of combining counts when merging samples of different sizes.	Station C Q9 and the free-throw combined estimate both require combining raw counts first.
Assuming any amount of extra data removes variation entirely.	The free-throw Q4 recommendation should include “reduces” or “narrows”, not “removes”.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A poll of finds . A second poll of from the same population finds . Which is more likely to be close to the true population proportion, and why?

Answer

The poll — its larger sample size makes it less prone to the wide swings that affect small samples, even though its result looks less dramatic.

E2 (AMC Junior style). Two samples of size are combined. Sample 1 has ; Sample 2 has . Find the combined sample’s proportion.

Answer

E3 (Challenge). A population has . Four samples of size give a range of . Four samples of size from the same population give a range of . Roughly what range would you predict for , and what pattern are you extending?

Answer

Each time quadruples (), the range roughly halves (, close to halved). Extending the pattern to (another quadrupling) predicts a range of roughly . (This matches the mathematical fact that sampling variation shrinks with , though Year 8 students are not expected to derive this — noticing the shrinking pattern is the goal.)

E4 (Challenge). A factory’s true defect rate is . A quality inspector checks only items per batch. Explain, using this unit’s ideas, why this inspection plan is likely to be unreliable, and suggest an improvement.

Answer

With such a small sample and a low true rate, most -item checks will find defects purely by chance, giving a false sense of security, while occasional checks might find or and look alarming — neither result reliably reflects the true rate. A much larger sample (e.g. + items per batch) would produce far less erratic proportions and a more trustworthy signal of real changes in the defect rate.

Homework

1. A sample of has successes. Find .
2. Four samples of size : proportions . Find the range.
3. Four samples of size from the same population: proportions . Find the range and compare with Q2.
4. A sample gives from . Find the number of successes.
5. Two samples of combine to a total of successes. Sample P had . Find Sample Q’s proportion.
6. A survey of finds in favour of a new school policy; a survey of finds . Which should the school council trust more when deciding whether the policy has real support, and why?
7. Reasoning. A student claims: “If I only have time to survey people, I should still trust my result as much as someone who surveyed , as long as I’m careful and random.” Explain what is right and what is wrong with this claim.
8. Challenge. A population has . Design an experiment (in words) to test whether variation in really shrinks as grows: state what samples you would take, what you would calculate, and what result would support the claim.

Answers: Q1 — . Q2 — . Q3 — , much smaller than Q2’s , showing less variation at the larger sample size. Q4 — . Q5 — Sample P has successes, so Sample Q has out of , . Q6 — the survey; its much larger sample gives a far narrower range of results that ordinary chance alone could produce, making its estimate more trustworthy. Q7 — right: being careful (random, unbiased) is genuinely necessary, and no sample size fixes a badly chosen sample; wrong: even a perfectly random sample of will vary far more from the true proportion than a random sample of , so the two results do not deserve equal trust as estimates of the population. Q8 — e.g. draw several samples each of sizes , and from the same known population; calculate for each; find the range within each sample-size group; the claim is supported if the ranges shrink as increases from to to .