Colorado State University — Pueblo; Fall 2016
Math 156, Introduction to Statistics
Hypothesis Testing about the Population Mean
Unknow Population Standard Deviation $\sigma$

The Context

This is quite similar to the previous case where but without the [silly] assumption that we know the population standard deviation $\sigma$. So suppose we have some population about which we are interested in the values of a particular numerical measurement. If we pick individuals from the population at random, then we can think of the values of this measurement as a quantitative random variable $X$. We are interested in the population mean, $\mu$, of $X$, and of course we likely don't know the value of the population standard deviation, $\sigma$. We are interested in the question of whether $\mu$ is equal to some specific value $\mu_0$ — maybe the value it used to have before some intervention, or the value someone says it should have — but since the population size $N$ is large, it is impractical simply to go out and collect all the data to compute $\mu$ and see if the null hypothesis $$H_0:\quad \mu=\mu_0$$ is true. Here, $\mu_0$ is always a specific number and we are wondering if the population mean $\mu$, which is out there in the universe but which we have trouble finding out exactly, is equal to that number $\mu_0$.

Perhaps we have reasons to think that if the null hypothesis is not true, then maybe a particular alternative hypothesis $H_a$ might instead be true. As before, the alternative hypothesis can have one of the forms $H_a:\,\mu<\mu_0$, $H_a:\,\mu>\mu_0$, or $H_a:\,\mu≠\mu_0$.

To see which hypothesis, $H_0$ or $H_a$, is supported by the data, we take a sample of size $n$ and compute the sample mean $\overline{X}$. From this, we make the test statistic, called here the $t$-statistic, as $$t=\frac{\overline{X}-\mu_0}{s/\sqrt{n}}$$ where $s$ is the sample standard deviation.

As we have seen before with confidence intervals, the distribution of $t$ will be governed by Student's $t$-Distribution with $df=n-1$ degrees of freedom and mean $0$ if the null hypothesis is true. We therefore compute the probabilty on that version of the $t$-distribution that the test statistic has values like the one we saw, or more in the direction of the alternative hypothesis. This probability is the $p$-value of the test. Here is how we compute it depending upon the particular alternative hypothesis:

where $T$ is the $t$-distribution RV [so we can look up these probabilities in a table or get them with an electronic tool which computes cumulative $t$-distribution probabilities; here is one such].

As before, the $p$-value is the conditional probability

$p=P($one would see values of $\overline{X}$ like what we did see, or more extreme ${}|{}\ H_0$ is true${})$ .
 
[The probability here is for the repeatable experiment of taking SRSs.] Consequently, if $p$ is small, we have reason to reject the null hypothesis, while if $p$ is not small enough, we fail to reject the null hypothesis.

Also as before, with a value $\alpha$ for the significance level, if $p<\alpha$, we say We reject the null hypothesis at the $\alpha$ significance level. A preferred method again, though, is to get the $p$ value and, if it is quite small (as your particular situation determines), you say "We reject the null hypothesis ($p={}$<number>)."