IB Maths AI HLHypothesis Testing for Population ParametersPaper 1 & 2~8 min read
Type I & Type II Errors
Every hypothesis test can get it wrong in two ways. A Type I error rejects a true H0 (a “false positive”); a Type II error accepts a false H0 (a “false negative”). The neat part: the critical region fixes these probabilities, so you can calculate them — usually with the same binomial or Poisson skills you already have.
📘 What you need to know
Type I = reject H0 when it is true — “false positive”. Type II = accept H0 when it is false — “false negative”.
P(Type II) = P(not in critical region | actual parameter) — you must be given the true population parameter.
After the test: rejected H0 → only a Type I was possible; accepted H0 → only a Type II was possible.
Trade-off: shrinking one error grows the other. Only a bigger sample shrinks both.
The four outcomes
A test compares reality (is H0 actually true?) with your conclusion (did you reject it?). Two combinations are correct; two are errors.
Reality × Conclusion
Type I sits top-left (true but rejected); Type II sits bottom-right (false but accepted).
reject true H₀Type I“False positive”. Court: found guilty though innocent.
accept false H₀Type II“False negative”. Court: found innocent though guilty.
🧠 “One — guilty too soon”
Type I is the eager error: you reject (convict) when you shouldn’t. Type II is the lazy error: you fail to reject (let off) when you should. Roman I before II = reject before accept.
Calculating the probabilities
Both probabilities are decided before the sample is taken — they come from the critical region and the significance level α%.
Type I error
P(Type I) = P(in critical region | H0 true)
continuous → = α% | discrete → ≤ α%
Not in the booklet — know the conditional set-up ✗
Type II error
P(Type II) = P(not in critical region | actual parameter)
Needs the TRUE population value — given in the question ✗
🧭 Recipe — finding P(Type I) and P(Type II)
Get the critical region from the rejection rule (e.g. “reject if hits > 77” → region is X > 77).
Type I: compute the probability of landing in that region using the null parameter — P(in region | H0).
Type II: compute the probability of landing outside the region using the actual parameter — P(not in region | true value).
Round to 3 sf and state clearly which error each value is.
🤔 Why is discrete P(Type I) only “≤ α%”?
For a continuous distribution you can place the boundary exactly where the tail area equals α%, so P(Type I) = α% precisely. With a discrete distribution (binomial/Poisson) X only takes whole values, so the region jumps in steps and the tail probability lands just under α% — never exactly on it. That’s why it’s ≤ α%.
After the test & the trade-off
Once you’ve run the test, only one error is even possible — depending on your decision.
Rejected H0? You could only have made a Type I error. Accepted H0? You could only have made a Type II error. A favourite exam twist: “state which error could not have occurred.”
Change
P(Type I)
P(Type II)
Lower α%
down ↓
up ↑
Raise α%
up ↑
down ↓
Bigger sample
down ↓
down ↓
🧠 Only one lever moves both
Adjusting α% is a see-saw — push one error down, the other pops up. A larger sample size is the only move that lowers both at once.
Worked examples
All five use Lucy’s axe-throwing test. X ~ B(100, p) is the number of hits in 100 throws. Right-hand baseline is 70%, so H0: p = 0.7, H1: p > 0.7. She rejects H0 if hits > 77.
WE 1
State the critical region
From Lucy’s rejection rule, write down the critical region for X.
reject H₀ if the axe hits more than 77 timescritical region: X > 77, i.e. 78 ≤ X ≤ 100X is discrete, so “more than 77” starts at 78.
WE 2
Find P(Type I error)
Find the probability of a Type I error for Lucy’s test.
P(Type I) = P(in critical region | H₀ true)= P(X > 77 | p = 0.7) = P(78 ≤ X ≤ 100 | 0.7)= 0.04786…P(Type I) = 0.0479 (3sf)
WE 3
Compare with the significance level
The test is one-tailed at 5%. Comment on the size of the Type I probability you found.
binomial is discrete → P(Type I) ≤ α%0.0479 ≤ 0.05 ✓consistent — the discrete region sits just under 5%, not exactly on it.
WE 4
Find P(Type II error)
Lucy actually hits the target 80% of the time with her left hand. Find the probability of a Type II error.
P(Type II) = P(not in critical region | true p = 0.8)= P(X ≤ 77 | p = 0.8) = P(0 ≤ X ≤ 77 | 0.8)= 0.2610…P(Type II) = 0.261 (3sf)
WE 5
Which error could have occurred?
Lucy runs the test and gets 75 hits, so she does not reject H0. State which error she could have made, and which she could not.
75 is not > 77 → outside critical region → accept H₀could only have made a Type II errora Type I is impossible once H₀ is accepted — that needs a rejection.
💡 Top tips
Type I uses the null parameter, Type II uses the actual one — never mix them up.
“More than 77” → 78 to 100 for a discrete variable; be careful with strict vs non-strict inequalities.
Type II needs a given true value — if none is supplied, you cannot find it.
Errors are set before sampling: they describe the test design, not one particular result.
After the test: reject → Type I possible; accept → Type II possible. State the impossible one if asked.
Only a bigger sample lowers both error probabilities together.
⚠ Common mistakes
Swapping the conditions — using p = 0.7 for Type II or the true p for Type I.
Claiming P(Type I) = α% for a binomial/Poisson test — it’s only ≤ α% (discrete).
Trying to find P(Type II) without the true parameter — impossible.
Saying both errors can happen after a test — only the one matching your decision can.
Thinking lowering α% is free — it raises the Type II probability.
That wraps up Hypothesis Testing for Population Parameters. Every test in this unit shares one through-line: write the hypotheses (define the parameter, pick the tail) → get a p-value or critical region → compare with α → conclude in context, tentatively. You’ve now done it for the mean (z/t), proportion (binomial), rate (Poisson) and correlation (regression t-test) — and seen how Type I & II errors measure the chance the whole process is wrong. Master the through-line and the rest is just choosing the right distribution.
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