confint() — Confidence interval

A give-or-take range where the true average probably sits.

mean ± t · (s/√n)

Key idea

• A confidence interval gives a give-or-take range that plausibly holds the true population average. (e.g. (10, 14))
• A 95% CI means this method captures the true average about 95% of the time, over many repeats.
• Higher confidence makes the interval wider, not narrower. (e.g. 95% wider than 90%)
• A wider interval means more uncertainty about where the true average lies. (e.g. (5, 19))
• More data usually shrinks the interval and sharpens the estimate. (e.g. n = 1000)

Definition

$$\bar{x}\pm t_{\alpha /2,n-1}\frac{s}{\sqrt{n}}$$

$\bar{x}$

the sample mean

$t_{\alpha /2,n-1}$

the t critical value for your confidence level

$s$

the sample standard deviation

$n$

the sample size

In plain words

Like guessing 'the candy jar has about 50 sweets, give or take 6.' The interval is your honest fuzzy guess instead of one exact number.

Where you'd use it

• You measure 30 visitors with a sample mean of 171 cm. confint at 95% might return 168 to 174 cm, a give-or-take band that probably contains the true average height.
• A poll of 1000 people finds 52% support. A 95% confidence interval turns that into 49% to 55% — the same as reporting 52% with a ±3 point margin of error.
• From a sample of bottle fills averaging 498 ml, a 95% confint of 495 to 501 ml tells you the true mean fill is plausibly still 500 ml, since 500 sits inside the band.

Worked example 12

From a sample of heights, a 95% interval gives a low-to-high range that probably contains the true average height.

Frequently asked questions

What does a 95% confidence interval mean?

If you repeated the study, 95% of such intervals would contain the true value — not "95% chance it’s in this one".