t.test() — One-sample t-test

Checks if your group’s average matches a number you guessed.

t.test(x, mu = 0)

Key idea

• A one-sample t-test checks whether your sample's average (the mean) differs from a guessed value called mu.
• A small p-value (below 0.05) is evidence of a real difference from mu. (e.g. p = 0.01)
• Example: testing if average test scores differ from a target of 70. (e.g. mean = 74, mu = 70)
• It works best when the data are roughly bell-shaped (called normal).
• A large p-value means you cannot rule out that the true average equals mu. (e.g. p = 0.40)

Definition

$$t=\frac{\bar{x}-{\mu }_0}{s/\sqrt{n}}$$

$\bar{x}$

the sample mean

${\mu }_0$

the hypothesised mean

$s$

the sample standard deviation

$n$

the sample size

$t$

the test statistic

$$t=\frac{\bar{x}-\mu }{s/\sqrt{n}}=\frac{5.38-5}{0.554/\sqrt{5}}=1.53$$

Where you'd use it

• You measure 20 visitors and get a mean height of 172 cm, and wonder if the true average is really 170. t.test(x, mu = 170) gives a p-value; if it comes out around 0.30, the 2 cm gap is easily ordinary wobble, so you cannot say the average differs from 170.
• A bag of chips should weigh 50 g. You weigh 15 bags averaging 48 g; a t.test against mu = 50 with a p-value under 0.05 would be real evidence the bags are underfilled.
• A machine is supposed to pour 500 ml. Testing a sample of bottles with t.test(x, mu = 500) tells you whether any drift from 500 is a genuine problem or just normal variation.

Frequently asked questions

What does the p-value tell me?

How surprising the data would be if the null were true. Small p (< 0.05) is evidence against it.

Which t-test do I need?

One-sample (vs a number), two-sample (two groups), or paired (matched pairs).