>>We discussed the general setup – the philosophy – behind hypothesis testing. Now let’s
visit some of the details in how to implement a hypothesis test for a test for the population
mean of a single population. Recall from our discussion of confidence intervals for a population
mean that we could encounter two situations: one where sigma, the standard deviation of
the population, is known, and the other where sigma is not known. We said that it would
be rare that sigma would be known – that we wouldn’t know the population mean, but
we would somehow know the dispersion of the population. However, let’s first discuss
the case where sigma is known for pedagogical purposes, as it makes for an easy transition
into hypothesis testing.
Also, like our discussion of confidence intervals, when sigma is known, there are three cases
that we may encounter. The first case is where the population is known to be normally distributed
and the sample size is small. The second case is where we don’t know anything about the
population distribution, but we collect a sufficiently large sample size such that the
central limit theorem kicks in. In either of those two cases, we use the Z distribution
to identify the rejection region, and the test statistic is a z score. In the third
case, when the population distribution is non-normal and when the sample size is small
– that is, we don’t think the central limit theorem will help us – we use nonparametric
techniques that we’ll discuss in a later unit.
Say we’ve got a set of hypotheses about a population mean: the null hypothesis is
that the mean is equal to mu_naught, and the alternative hypothesis is that the mean is
not equal to mu_naught. This is a two-tailed test, as we have strict equality in the null,
and strict inequality in the alternative. We want to reject the null if the evidence
lies significantly above or significantly below the mean, therefore the rejection region
of the size alpha is dispersed in both tails. And recall that we said that the test statistic
essentially represents our evidence. So what is the test statistic? When we are testing
about a population mean when sigma is known, the test statistic is a Z score. Test statistic
z_naught or z_zero is measured as the difference between y-bar and mu_naught in the numerator
and sigma over root n in the denominator. That is, how different is y-bar from the hypothesized
mean in standard deviation units. Remember, a z score represents how different a realization
of a random variable is from the mean of the distribution in standard deviation units.
That’s what we’re using here: how different is our sample evidence from the hypothesized
value in standard deviation units. If y-bar is “far enough” from mu_naught, the test
statistic will lie in the rejection region. How cool is that? We’re taking the notion
of a z score and applying it to measure the distance from the sample to the hypothesized
value.
So we’re accounting for variability and sample size here with the denominator, sigma
over root n. Recall that sigma over root n is the standard deviation of the sampling
distribution of y-bar. We’re explicitly accounting for variability and sample size
when we’re determining how far apart the sample mean is from the mu_naught. And confidence
comes into play with the size of the rejection region. So we’re considering variability,
sample size, and confidence when we draw a conclusion about a hypothesis.
Remember, we can’t just make a decision about a hypothesis based on y-bar alone. It’s
a random variable. We need to account for how disperse the data are: if the data are
really disperse, y-bar could be quite a bit different from mu_naught, and we might still
be convinced that the mean is indeed mu_naught. That is, the higher the variability, the more
possibilities there are for y-bar to take on values that are different from mu_0. Putting
sigma over root n in the denominator helps us account for this variability. Also, the
larger the n, the better we feel that y-bar is representing the real mean in the population.
So if y-bar is really different from mu_naught, maybe mu_naught isn’t the real mean and
we should be rejecting the null.
So, to recap, if the test statistic lies beyond the critical value – that is, if the test
statistic lies in the rejection region – then we can conclude that the sample mean lies
too far from the hypothesized mean for the two to be equal, therefore we reject the null
hypothesis. Here, “too far” is defined by our confidence. If the test statistic doesn’t
lie in the rejection region, then we conclude that we do not reject the null hypothesis
based on the evidence. Note that we never say “accept the null.” The best we can
do is “fail to reject the null.” It’s a pessimistic perspective, but there’s too
much variability in sampling for us to use language like “accept the null.”
So, how do we determine if the test statistic lies in the rejection region? There are two
approaches that we can use to draw a conclusion about a set of hypotheses: the critical value
approach, and the p-value approach. We’ll discuss both of these in upcoming videos.