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AP Statistics Unit 3 Visual ReviewA topic-by-topic visual walkthrough of Unit 3: Inference for Categorical Data — confidence intervals and significance tests for one and two proportions, plus chi-square tests.
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TOPIC 3.1
Estimators
A statistic estimates a parameter
We use a sample STATISTIC to estimate an
unknown population PARAMETER.
p̂ estimates p x̄ estimates μ
A good estimator is UNBIASED and has
low variability.
Bias vs. variability
UNBIASED: centered on the true parameter
(no systematic error).
LOW VARIABILITY: estimates cluster tightly.
The ideal estimator is both.
Larger n reduces variability (not bias).
The estimate varies from sample to sample
Different random samples give different p̂ values — the SAMPLING DISTRIBUTION describes that variation (Topic 3.2).
A statistic (p̂, x̄ ) estimates a parameter (p, μ); aim for unbiased, low-variability.
The Review Hub · AP Statistics Unit 3
TOPIC 3.2
Sampling Distribution of p̂
// center & spread of the sample proportion
mean of p̂ = p SD of p̂ = √( p(1−p) / n )
approx. Normal when np ≥ 10 and n(1−p) ≥ 10
The Large Counts condition
Expect at least 10 successes AND 10
failures for the Normal approximation.
np ≥ 10 and n(1−p) ≥ 10
Also need a random sample & the 10% condition.
Worked example
p = 0.4, n = 100:
SD = √(.4·.6/100) = 0.049
np = 40 ≥ 10, n(1−p) = 60 ≥ 10 ✓
So p̂ ≈ Normal(0.4, 0.049).
p̂ is centered at p with SD=√(p(1−p)/n) , ~Normal if np & n(1−p) ≥ 10.
The Review Hub · AP Statistics Unit 3
TOPIC 3.3
Confidence Interval for a Proportion
p̂ ± z*·√( p̂(1−p̂) / n ) (estimate ± margin of error)
Worked example (95%)
p̂ = 0.6, n = 100, z* = 1.96:
ME = 1.96·√(.6·.4/100) = 0.096
CI = 0.6 ± 0.096 = (0.504, 0.696)
z* = 1.645 (90%), 1.96 (95%), 2.576 (99%).
Uses p̂ in the SE (a "one-proportion z-interval").
Check conditions first
• RANDOM sample
• 10%: n ≤ 10% of the population
• LARGE COUNTS: np̂ ≥ 10, n(1−p̂) ≥ 10
Higher confidence → wider interval.
Bigger n → narrower interval.
One-proportion CI: p̂ ± z*√(p̂(1−p̂)/n) .
The Review Hub · AP Statistics Unit 3
TOPIC 3.4
Interpreting a Confidence Interval
Interpret the INTERVAL
"We are 95% confident that the true
proportion of [context] is between
0.504 and 0.696."
Name the parameter & context, with numbers.
The parameter p is fixed — it's the interval that varies.
Interpret the CONFIDENCE LEVEL
"If we took many samples and made a CI
from each, about 95% would capture p."
NOT "95% probability p is in THIS interval."
The METHOD works 95% of the time —
that's what "95% confidence" means.
Use a CI to test a claim
If a claimed value is OUTSIDE the interval, it's not plausible; if INSIDE, the data are consistent with it.
"95% confident p is in (a, b)"; the method captures p 95% of the time .
The Review Hub · AP Statistics Unit 3
TOPIC 3.5
Setting Up a Test for a Proportion
// state the hypotheses about the parameter p
H₀: p = p₀ (the claim / no effect)
Hₐ: p < p₀, p > p₀, or p ≠ p₀
Null vs. alternative
H₀ is the "status quo" — always uses =.
Hₐ is what you're gathering evidence FOR.
Choose Hₐ's direction BEFORE seeing data.
Define p in context (a population proportion).
Significance level α
α is the threshold (often 0.05) — the risk
of a Type I error you'll accept.
Check the SAME conditions as the CI
(random, 10%, Large Counts using p₀).
State H₀: p = p₀ and Hₐ (<, >, ≠) ; pick α; check conditions.
The Review Hub · AP Statistics Unit 3
TOPIC 3.6
Interpreting p-Values
p-value = P(getting a result this extreme, or more, IF H₀ is true)
Small p → reject H₀
p-value ≤ α → REJECT H₀
The data would be unlikely if H₀ were true,
so we have convincing evidence for Hₐ.
"statistically significant"
Small p = surprising data under H₀.
Large p → fail to reject
p-value > α → FAIL TO REJECT H₀
Not enough evidence for Hₐ.
NEVER "accept H₀" or "prove H₀ true."
A p-value is NOT P(H₀ is true).
It assumes H₀ and measures the data's surprise.
p ≤ α → reject H₀ ; p > α → fail to reject (never "accept" H₀).The Review Hub · AP Statistics Unit 3
TOPIC 3.7
Carrying Out a Test for a Proportion
z = ( p̂ − p₀ ) / √( p₀(1−p₀) / n )
Worked example
H₀: p = 0.5, p̂ = 0.58, n = 100:
z = (.58−.5)/√(.5·.5/100)
= .08/.05 = 1.6
Find the p-value = area beyond z on the
Normal curve (double it for ≠).
The 4-step conclusion
1. State H₀/Hₐ & α.
2. Name test & check conditions.
3. Compute z and the p-value.
4. Compare to α → conclude IN CONTEXT.
The SE uses p₀ (the null value), not p̂.
One-proportion z-test: z = (p̂−p₀)/√(p₀(1−p₀)/n) , then find the p-value.
The Review Hub · AP Statistics Unit 3
TOPIC 3.8
Type I & Type II Errors
H₀ is actually TRUE H₀ is actually FALSE
Reject H₀
TYPE I error (prob α)
correct (power)
Fail to reject
correct
TYPE II error (prob β)
Power = 1 − β
POWER (correctly rejecting a false H₀) rises with larger n, larger α, or a bigger true effect. Type I = "false alarm."
Type I = reject a true H₀ (α); Type II = fail to reject a false H₀ (β) ; power = 1−β.The Review Hub · AP Statistics Unit 3
TOPIC 3.9
Difference of Two Proportions
// sampling distribution of p̂₁ − p̂₂
mean = p₁ − p₂
SD = √( p₁(1−p₁)/n₁ + p₂(1−p₂)/n₂ )
Variances add
Since the two samples are independent,
their variances ADD under the root.
Approximately Normal when all four counts
(successes & failures in both) are ≥ 10.
Conditions
• Two INDEPENDENT random samples
• 10% condition for each
• Large Counts in BOTH groups
Basis for the two-proportion CI & test (3.10–3.13).
p̂₁−p̂₂ centered at p₁−p₂; SD=√(p₁(1−p₁)/n₁ + p₂(1−p₂)/n₂) .
The Review Hub · AP Statistics Unit 3
TOPIC 3.10
CI for a Difference of Proportions
(p̂₁ − p̂₂) ± z*·√( p̂₁(1−p̂₁)/n₁ + p̂₂(1−p̂₂)/n₂ )
Uses each p̂ separately (unpooled)
For the INTERVAL, use each sample's own
p̂ in the standard error (no pooling).
The estimate is the DIFFERENCE p̂₁ − p̂₂.
A "two-proportion z-interval."
Same z* values as the one-proportion interval.
Worked example (95%)
p̂₁=.6 (n=100), p̂₂=.5 (n=100):
SE = √(.6·.4/100 + .5·.5/100)
≈ 0.070
CI = 0.1 ± 1.96·.070 = (−.04, .24)
Interval contains 0 → no significant difference.
Two-proportion CI: (p̂₁−p̂₂) ± z*·SE (each p̂ in its own SE).
The Review Hub · AP Statistics Unit 3
TOPIC 3.11
Interpreting a Difference Interval
Does the interval contain 0?
A confidence interval for p₁ − p₂ tests whether the two proportions differ.
0 inside → no significant difference · 0 outside → significant difference
Entirely positive
If the whole interval is > 0, we're
confident p₁ > p₂.
Entirely negative → p₁ < p₂.
The sign of the difference tells the direction.
Interpret in context
"We are 95% confident the true difference
in proportions (group1 − group2) is
between __ and __."
Always define which group is #1.
If a difference interval contains 0, the proportions aren't significantly different .
The Review Hub · AP Statistics Unit 3
TOPIC 3.12
Setting Up a Two-Proportion Test
H₀: p₁ = p₂ (i.e. p₁ − p₂ = 0)
Hₐ: p₁ ≠ p₂, p₁ > p₂, or p₁ < p₂
POOL the proportion for the test
Since H₀ says the proportions are EQUAL, combine both samples into one pooled estimate:
p̂_c = (x₁ + x₂) / (n₁ + n₂) (total successes ÷ total sample)
Use p̂_c in the standard error for the test (unlike the interval, which stays unpooled).
Check: two independent random samples + Large Counts in both, using p̂_c.
H₀: p₁=p₂; for the test POOL: p̂_c=(x₁+x₂)/(n₁+n₂) .
The Review Hub · AP Statistics Unit 3
TOPIC 3.13
Carrying Out a Two-Proportion Test
z = (p̂₁ − p̂₂) / √( p̂_c(1−p̂_c)(1/n₁ + 1/n₂) )
Pooled standard error
The test statistic z uses the POOLED p̂_c
in both factors of the SE.
Compute z, then find the p-value from the
Normal curve (double it for ≠).
Same 4-step process; conclude in context.
Decision & conclusion
p-value ≤ α → reject H₀
"There is convincing evidence that the
two proportions differ (in context)."
Otherwise, "not enough evidence."
Interval & test agree at matching levels.
z = (p̂₁−p̂₂)/√(p̂_c(1−p̂_c)(1/n₁+1/n₂)) — pooled SE, then the p-value.The Review Hub · AP Statistics Unit 3
TOPIC 3.14
Setting Up a Chi-Square Test
expected count = (row total × column total) / grand total
used for 2+ categories / two categorical variables
Three chi-square tests
• GOODNESS OF FIT: one variable vs. a
claimed distribution.
• HOMOGENEITY: same distribution across
several populations?
• INDEPENDENCE: two variables related?
Hypotheses & conditions
H₀: no association / distributions equal.
Hₐ: there IS an association / a difference.
Condition: every EXPECTED count ≥ 5.
df = (rows − 1)(columns − 1).
Random sample(s) required, as always.
Chi-square: expected = (row×col)/total ; need every expected ≥ 5.
The Review Hub · AP Statistics Unit 3
TOPIC 3.15
Carrying Out a Chi-Square Test
χ² = Σ ( observed − expected )² / expected
Bigger χ² → stronger evidence
Each cell adds (O−E)²/E. A large total χ²
means observed counts are FAR from
what H₀ expects.
Find the p-value from the χ² curve with
the correct df (always right-tailed).
Worked snippet
One cell: O = 30, E = 25.
(30 − 25)² / 25 = 1.0
Sum this over all cells to get χ².
Reject H₀ if p ≤ α → conclude association
exists / distributions differ (in context).
χ² = Σ(O−E)²/E — larger χ² & small p → reject H₀ (right-tailed).The Review Hub · AP Statistics Unit 3
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How to use the visual review
Spend 30 seconds per slide before clicking next. Look at the diagram, then ask yourself: "Could I state this condition, interpret this interval, or explain this p-value from memory?"
Use the fullscreen button () on desktop for the best experience. Use arrow keys to navigate. Tap "Show all slides" to jump around.
This is great for review the night before the exam — fast, visual, and covers every idea you need to recognize in Unit 3.