IB Maths AI HLSample Mean DistributionsPaper 1 & 2~7 min read
Confidence Interval for the Mean
A single sample mean x̄ is just a point estimate — one guess at the true population mean μ. A confidence interval upgrades that to a range the true mean is likely to sit in, written a < μ < b. Your GDC does all the arithmetic — the real skill is choosing the right interval: a z-interval when the population variance σ2 is known, a t-interval when it isn’t (the usual exam case). Then you read off the bounds and, often, comment on a claimed value of μ.
📘 What you need to know
Point estimate: a sample mean x̄ is a single best guess; a confidence interval gives a likely range for μ.
Confidence level = probability that the interval contains μ — not that μ falls in a fixed interval.
z-interval: use when the population variance σ2 is known.
t-interval: use when the population variance is unknown — it uses the unbiased estimate sn−12.
Population must be normal (or n large enough for the CLT).
Interpreting a claim: value inside the interval → claim supported; outside → reject the claim.
What a confidence interval means
You can never pin down μ exactly from a sample. A 95% confidence interval is built so that, if you repeated the sampling many times, about 95 of every 100 intervals would capture the true mean.
🤔 Why “the interval contains μ”, not “μ is in the interval”?
The population mean μ is a fixed (if unknown) number — it doesn’t move, so it makes no sense to give it a probability. What varies is the interval, which changes with every sample. So the 95% refers to the process: 95% of the intervals it produces will contain μ. Out of 100 intervals you’d expect ~95 hits — though all 100, or (very rarely) none, is possible.
Width of the interval: two things change it — the confidence level (raise it → wider, because you demand more certainty) and the sample size (raise it → narrower, because more data sharpens the estimate).
z-interval or t-interval?
The one decision that matters: do you know the population variance σ2? That single fact picks the interval type and what you type into the GDC.
σ² known
z-interval
Enter the population σ, the confidence level, and either the raw data or x̄ with n.
σ² unknown
t-interval
Uses the unbiased estimate sn−1. Enter the confidence level, and raw data or x̄, sn−1, n.
Which interval? Follow the variance
In most exam questions you’re given a sample SD, so σ² is unknown → t-interval.
🧭 Recipe — a t-interval on the GDC
Spot that σ2 is unknown (you’re given a sample SD, not the population’s) → t-interval.
Get the unbiased variancesn−12: if given the sample variance sn2, multiply by nn−1 (the booklet formula).
Enter the confidence level, x̄, sn−1, and n into the t-interval function.
Read off the lower and upper bounds and write a < μ < b (round as asked).
🧠 Memory aid — “know σ? z. Don’t? t.”
z needs the true population σ; the moment you only have a sample standard deviation, switch to t. The t-interval quietly uses the unbiased estimate sn−12 = nn−1sn2 — the same nn−1 correction from the unbiased-estimates topic.
WE 1
A 95% t-interval (Cara’s burgers)
Burger weights are normally distributed. Cara samples 12 burgers: mean 293 g, sample standard deviation 5.5 g. Find a 95% confidence interval for the population mean, to 4 sf.
σ² unknown → t-interval; first get s₍ₙ₋₁₎²s₍ₙ₋₁₎² = 12/11 × 5.5² = 33enter into GDC t-intervallevel 0.95, x̄ = 293, s₍ₙ₋₁₎ = √33, n = 12Lower = 289.35… Upper = 296.64…289.4 < μ < 296.6 (4sf)“sample SD 5.5” is s₍ₙ₎ — correct it to s₍ₙ₋₁₎ before the interval.
WE 2
Commenting on a claim
The butcher claims the burgers weigh 300 g. Comment on this claim using the interval from WE 1.
compare 300 with 289.4 < μ < 296.6300 > 296.6 → outside the intervalconclusionreject the claim300 g lies above the interval, so the butcher’s claim is not supported.
WE 3
A z-interval (variance known)
A machine fills bottles with volume known to have σ = 4 ml. A sample of 25 bottles has mean 503 ml. Find a 95% confidence interval for the mean fill volume, to 4 sf.
σ known → z-intervalσ = 4, level 0.95, x̄ = 503, n = 25read off GDC z-intervalLower = 501.43… Upper = 504.56…501.4 < μ < 504.6 (4sf)because σ is the population value, use z — no n/(n−1) correction needed.
WE 4
Effect of confidence level on width
For Cara’s burgers (x̄ = 293, sn−1 = √33, n = 12), find a 90% confidence interval and compare its width with the 95% interval from WE 1.
t-interval at level 0.90Lower = 290.02… Upper = 295.97…290.0 < μ < 296.0 (4sf)compare widths90%: ≈ 5.96 vs 95%: ≈ 7.30lower confidence → narrower interval, as expected.
WE 5
Effect of sample size on width
Suppose Cara had sampled 48 burgers (instead of 12) with the same x̄ = 293 and sn−1 = √33. Find the 95% interval and compare its width with WE 1.
t-interval, level 0.95, n = 48Lower = 291.33… Upper = 294.66…291.3 < μ < 294.7 (4sf)compare widthsn = 48: ≈ 3.34 vs n = 12: ≈ 7.304× the sample → roughly half the width; bigger samples sharpen the estimate.
💡 Top tips
Know σ → z; don’t → t. A given sample SD means population variance is unknown.
Correct the variance first for a t-interval: sn−12 = nn−1sn2.
Interpreting a claim: inside → supported; outside → rejected.
Word it carefully: the interval contains μ with given probability — don’t say μ “has a probability” of lying inside.
Check normality / CLT applies before trusting the interval.
⚠ Common mistakes
Using a z-interval when only a sample SD is given — that’s a t-interval situation.
Feeding sn straight in without the nn−1 correction when the GDC wants sn−1.
Saying “μ has a 95% chance of being in the interval” — it’s the interval that contains μ.
Thinking a wider interval is “better” — it just reflects higher confidence or a smaller sample.
Mixing up the bounds or rounding before reading both off the GDC.
Forgetting to comment on the claim when asked — state inside/outside and the conclusion.
That wraps up the Combinations of Normal Distributions & Sample Mean Distributions unit! You can now combine normal variables, build the sample mean distribution X̄ ∼ N(μ, σ2/n), invoke the Central Limit Theorem for non-normal populations, and construct and interpret confidence intervals for μ. Next this feeds directly into hypothesis testing, where the same z- and t-machinery is used to test claims about a population mean.
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