3.2.1 Justification of the one-sample $t$-test

The general concept of conducting an hypothesis test is to

1. 1.

   Calculate a test statistic;

2. 2.

   Compare the test statistic to a sampling distribution, either using a critical value or a $p$-value.

This is a valid method for testing all sorts of parameters from different statistical models. The test statistic and sampling distribution change according to the parameter being tested. You will see further examples throughout the course.

We give a justification for the use of both the $t$ statistic and Students-$t$ distribution. First recall the null hypothesis,

Since an unbiased estimator of $\mu$ is the sample mean $\overline{X}$, it seems sensible to compare the observed sample mean $\overline{x}$ to the hypothesised population mean ${\mu }_0$. The question is then:

When $\mu ={\mu }_0$, how far from ${\mu }_0$ can we reasonably expect $\overline{x}$ to be?

To answer this we need to look at the sampling distribution of the estimator $\overline{X}$. This will show us, for a given sample size $n$, how much sampling variability we can expect in the sample mean. We start with the case where the population variance ${\sigma }^2$ is known, since in this case we can use the Central Limit Theorem to help prove the sampling distribution of the test statistic.

This is an exact result if $X_1,\mathrm{\dots },X_n$ are Normal random variables, and so holds for all $n$. Otherwise the limiting result follows by application of the Central Limit Theorem.

If $H_0$ is true, we can adjust this to,

By the usual standardisation technique,

This then gives us both a test statistic $Z$ and its sampling distribution but only if we know the population variance ${\sigma }^2$.

When the population variance is unknown, it is replaced by the sample variance $s^2$, but this is itself a realisation of a random variable $S^2$ (different samples of size $n$ will each have a different sample variance). Deriving the sampling distribution of

is therefore considerably harder, as it is the ratio of two related random variables $\overline{X}$ and $S^2$. In fact, the $t$-distribution was first tabulated as the sampling distribution for this test statistic, by William S. Gossett (work published in 1908) as part of his work in quality control for the Guinness brewery in Dublin. Gossett found that the Normal distribution was no longer an appropriate sampling distribution for $T$, due to the extra uncertainty introduced when estimating $S$ (especially for small samples). His derivation of the $t$-distribution had three steps:

1. 1.

   Showing that the sampling distribution of $S$ is ${\chi }_n^2$;

2. 2.

   Showing that $S$ is independent of $\overline{X}$;

3. 3.

   Showing that the ratio of a Normal random variable to the square root of a ${\chi }_n^2$ random variable follows a $t_{n-1}$-distribution.

The entire proof takes up a page of A4 paper. To cover this thoroughly is beyond the scope of this course.