Does Each Tablet Really Hold 500 mg?

A quality lab measures the active ingredient in 8 tablets labeled 500 mg. With so few, the sample's standard deviation is itself uncertain, so the test uses the t distribution, whose fatter tails demand stronger evidence before calling the batch off-label.

How it works

The one-sample t-test and the t distribution

In one sentence

For a random sample from a normal population with mean $\mu_0$ , the statistic $t = (\bar x - \mu_0)/(s/\sqrt{n})$ has the $t$ distribution with $n-1$ degrees of freedom. The test rejects $H_0: \mu = \mu_0$ at level $\alpha$ when $|t|$ exceeds the critical value $t^*_{n-1}$ .

General formula

{\htmlData{sym=0}{t}} = \frac{{\htmlData{sym=1}{\bar x}} - {\htmlData{sym=2}{\mu_0}}}{{\htmlData{sym=3}{s}}/\sqrt{\htmlData{sym=4}{{n}}}}, \qquad {\htmlData{sym=5}{\text{df}}} = {\htmlData{sym=4}{n}} - 1

The t statistic is a signal-to-noise ratio: the gap from the target divided by how much an average of this many tablets naturally wobbles.

What the symbols mean

$t$
Test statistic
How many standard errors the sample mean sits from 500 mg.
$\bar x$
Sample mean
The average dose in the tablets tested.
$\mu_0$
Labeled dose
500 mg: the mean the null hypothesis claims.
$s$
Sample standard deviation
How much the tested tablets vary.
$n$
Sample size
How many tablets were tested.
$\text{df}$
Degrees of freedom
n − 1: sets how heavy the t distribution's tails are.

The null hypothesis says the batch's true mean dose is $\mu_0 = 500$ mg. A sample of $n$ tablets gives a mean $\bar x$ and a standard deviation $s$ .
The statistic $t$ counts how many standard errors $s/\sqrt{n}$ the sample mean sits from 500.
If $H_0$ holds and doses are roughly normal, $t$ follows the $t$ distribution with $n-1$ degrees of freedom. The two-sided p-value is the area beyond $\pm t$ .
With few tablets the $t$ curve has heavy tails, so it takes a bigger $|t|$ to reach $p < 0.05$ : 2.36 with 8 tablets, against 1.96 for the normal.

Learn the math

The t-Test for a MeanStats · Lesson 30, Hypothesis Testing Testing a Process Against a SpecEng Stats · Lesson 17, Testing Engineering Claims