One sample is never the whole truth: Suppose a factory wants the true mean lifetime of all the millions of bulbs it makes. It can't test every bulb, so it tests a small sample and gets a sample mean, say 1180 hours.
From one number to a range
The sample mean wobbles
If it tested a different sample it would get a slightly different mean — 1175, 1192, … It wobbles around the real value but rarely hits it exactly.
A confidence interval gives a range
It turns that single wobbly number into an honest range of believable values for the true mean, like "between 1136 and 1224 hours".
The next question
So how wide should that range be, and how do we build it?
What '95% confidence' actually means: A 95% confidence interval is built by a method that, used over and over on fresh samples, captures the true mean about 95% of the time.
How to say it, and how it is built
- We write a single interval like (a, b) and say: "we are 95% confident the true population mean lies between a and b."
- The interval is centred on the sample mean x̄ and reaches out a margin of error on each side.
Why t and not z, and why s_{n-1}: We almost never know the true population standard deviation, so we estimate it from the sample. The unbiased estimate is sn-1 — the GDC's 'sample standard deviation' that divides by n − 1, not n.
Why the t-distribution: Because we're estimating the spread too, the right distribution is the t-distribution with n − 1 degrees of freedom (slightly wider than the normal to allow for that extra uncertainty). On the GDC you never look t up by hand — the t-interval menu does it all.
IB-style question — build a 95% interval from summary stats
A botanist measures the heights of a random sample of 8 seedlings of a new variety. The sample mean is x̄ = 24.5 cm with unbiased standard deviation sn-1 = 2.8 cm.
Find a 95% confidence interval for the mean height of all seedlings of this variety.
Step by step
- Write the formula. The centre is the sample mean; the margin uses t with df = n − 1.
- On the GDC choose the t-interval and enter summary stats: x̄ = 24.5, sn-1 = 2.8, n = 8, C-Level = 0.95.
- Margin of error = t* × sn-1/√n.
- Interval = centre ± margin.
Final answer
The 95% confidence interval is (22.2 cm, 26.8 cm) (3 s.f.). We are 95% confident the true mean seedling height is between about 22.2 and 26.8 cm.
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A wider net is safer but vaguer: An interval is only useful if you can read it. Picture two coffee machines.
Machine A
- A 95% interval of (245 mL, 251 mL) for its mean fill
- Tight and informative
Machine B
- (230 mL, 266 mL)
- So wide it barely says anything useful
A narrow interval is a precise estimate; a wide one is vague. Two things control the width: how big your sample is and how confident you insist on being. Which way does each one push the width?
The two levers on width: Look at the margin t*·s_{n-1}/√n. The sample mean x̄ stays the centre; only the reach on each side changes.
Bigger sample n
- You divide by a larger √n, so the margin SHRINKS
- → narrower interval (more data = more precision)
Higher confidence (90% → 95% → 99%)
- t* gets bigger, so the margin GROWS
- → wider interval (to be more certain of trapping μ, you must cast a wider net)
IB-style question — quadruple the sample
A drinks company estimates the mean fill volume of its cans. From a sample, x̄ = 500 mL with sn-1 = 20 mL.
(a) Find the 95% interval when n = 16.
(b) Find the 95% interval when n = 64, and comment on the effect of the larger sample.
Step by step
- (a) df = 16 − 1 = 15. GDC t-interval with x̄ = 500, s = 20, n = 16, 95%.
- So the n = 16 interval, to 3 s.f.
- (b) df = 64 − 1 = 63. Same x̄ and s, but n = 64.
- So the n = 64 interval.
Final answer
(a) (489, 511) mL. (b) (495, 505) mL. Quadrupling n (16 → 64) roughly halves the margin (because √n doubled), so the interval is about half as wide — the larger sample gives a more precise estimate of the true mean fill.
IB-style question — raise the confidence level
A teacher takes a random sample of n = 25 test scores: x̄ = 72 marks, sn-1 = 10 marks.
Find the 90% and the 99% confidence intervals for the mean score, and state which is wider and why.
Step by step
- df = 25 − 1 = 24 for both. 90% first (GDC t-interval, C-Level 0.90).
- So the 90% interval.
- Now 99% (C-Level 0.99): the t-value is larger, so the margin grows.
- So the 99% interval.
Final answer
90%: (68.6, 75.4); 99%: (66.4, 77.6). The 99% interval is wider — demanding more confidence (a 99% capture rate) forces a larger t*, so the net must reach further to be that sure of containing the true mean.