*Section author: Danielle J. Navarro and David R. Foxcroft*

# The one-sample *t*-test¶

After some thought, I decided that it might not be safe to assume that
the psychology student grades necessarily have the same standard
deviation as the other students in Dr Zeppo’s class. After all, if I’m
entertaining the hypothesis that they don’t have the same mean, then why
should I believe that they absolutely have the same standard deviation?
In view of this, I should really stop assuming that I know the true
value of *σ*. This violates the assumptions of my
*z*-test, so in one sense I’m back to square one. However, it’s
not like I’m completely bereft of options. After all, I’ve still got my
raw data, and those raw data give me an *estimate* of the population
standard deviation, which is 9.52. In other words, while I can’t say
that I know that *σ* = 9.5, I *can* say that
\(\hat\sigma\) = 9.52.

Okay, cool. The obvious thing that you might think to do is run a
*z*-test, but using the estimated standard deviation of 9.52
instead of relying on my assumption that the true standard deviation is
9.5. And you probably wouldn’t be surprised to hear that this would
still give us a significant result. This approach is close, but it’s not
*quite* correct. Because we are now relying on an *estimate* of the
population standard deviation we need to make some adjustment for the
fact that we have some uncertainty about what the true population
standard deviation actually is. Maybe our data are just a fluke …maybe
the true population standard deviation is 11, for instance. But if that
were actually true, and we ran the *z*-test assuming
*σ*=11, then the result would end up being
*non-significant*. That’s a problem, and it’s one we’re going to have to
address.

## Introducing the *t*-test¶

This ambiguity is annoying, and it was resolved in 1908 by a guy called William
Sealy Gosset (Student, 1908), who was
working as a chemist for the Guinness brewery at the time
(Box, 1987). Because Guinness took a dim view
of its employees publishing statistical analysis (apparently they felt it was a
trade secret), he published the work under the pseudonym “A Student” and, to
this day, the full name of the *t*-test is actually **Student’s *t*-test**. The
key thing that Gosset figured out is how we should accommodate the fact that we
aren’t completely sure what the true standard deviation is.[1] The answer is
that it subtly changes the sampling distribution. In the *t*-test our test
statistic, now called a *t*-statistic, is calculated in exactly the same way I
mentioned above. If our null hypothesis is that the true mean is *µ*, but our
sample has mean *X̄* and our estimate of the population standard deviation is
\(\hat{\sigma}\), then our t statistic is:

The only thing that has changed in the equation is that instead of using the
known true value *σ*, we use the estimate \(\hat{\sigma}\). And if this
estimate has been constructed from *N* observations, then the sampling
distribution turns into a *t*-distribution with *N* - 1 **degrees of freedom**
(df). The t distribution is very similar to the normal distribution, but has
“heavier” tails, as discussed earlier in Other useful distributions and illustrated in Fig. 86. Notice,
though, that as df gets larger, the *t*-distribution starts to look identical
to the standard normal distribution. This is as it should be: if you have a
sample size of *N* = 70,000,000 then your “estimate” of the standard deviation
would be pretty much perfect, right? So, you should expect that for large *N*,
the *t*-test would behave exactly the same way as a *z*-test. And that’s
exactly what happens!

## Doing the test in jamovi¶

As you might expect, the mechanics of the *t*-test are almost identical to the
mechanics of the *z*-test. So there’s not much point in going through the
tedious exercise of showing you how to do the calculations using low level
commands. It’s pretty much identical to the calculations that we did earlier,
except that we use the estimated standard deviation and then we test our
hypothesis using the t distribution rather than the normal distribution. And so
instead of going through the calculations in tedious detail for a second time,
I’ll jump straight to showing you how *t*-tests are actually done.
jamovi comes with a dedicated analysis for *t*-tests that is very flexible (it
can run lots of different kinds of *t*-tests). It’s pretty straightforward to
use; all you need to do is specify `Analyses`

→ `T-Tests`

→ ```
One Sample
T-Test
```

, move the variable you are interested in (`X`

) across into the
`Variables`

box, and type in the mean value for the null hypothesis
(`67.5`

) in the `Hypothesis`

→ `Test value`

box. Easy enough (see
Fig. 87, which, amongst other things that we will get to in a
moment, gives you a *t*statistic = 2.25, with 19 degrees of freedom and an
associated *p*-value of 0.036.

Also reported are two other things you might care about: the 95%
confidence interval and a measure of effect size (we’ll talk more about
effect sizes later). So that seems straightforward enough. Now what do
we *do* with this output? Well, since we’re pretending that we actually
care about my toy example, we’re overjoyed to discover that the result
is statistically significant (i.e. *p*-value below 0.05). We could
report the result by saying something like this:

With a mean grade of 72.3, the psychology students scored slightly higher than the average grade of 67.5 (t(19) = 2.25,p< 0.05); the mean difference was 4.80 and the 95% confidence interval was from 0.34 to 9.26.

where *t*(19) is shorthand notation for a *t*-statistic that
has 19 degrees of freedom. That said, it’s often the case that people
don’t report the confidence interval, or do so using a much more
compressed form than I’ve done here. For instance, it’s not uncommon to
see the confidence interval included as part of the stat block after
reporting the mean difference, like this:

*t*(19) = 2.25,

*p*= 0.036, CI

_{95}= [0.34, 9.26]

With that much jargon crammed into half a line, you know it must be really smart.[2]

## Assumptions of the one sample *t*-test¶

Okay, so what assumptions does the one-sample *t*-test make? Well,
since the *t*-test is basically a *z*-test with the
assumption of known standard deviation removed, you shouldn’t be
surprised to see that it makes the same assumptions as the
*z*-test, minus the one about the known standard deviation. That
is

*Normality*. We’re still assuming that the population distribution is normal,[3] and as noted earlier, there are standard tools that you can use to check to see if this assumption is met (section Checking the normality of a sample), and other tests you can do in it’s place if this assumption is violated (section Testing non-normal data with Wilcoxon tests).*Independence*. Once again, we have to assume that the observations in our sample are generated independently of one another. See the earlier discussion about the*z*-test for specifics (section Assumptions of the z-test).

Overall, these two assumptions aren’t terribly unreasonable, and as a
consequence the one-sample *t*-test is pretty widely used in
practice as a way of comparing a sample mean against a hypothesised
population mean.

[1] | Well, sort of. As I understand the history, Gosset only provided a partial solution; the general solution to the problem was provided by Sir Ronald Fisher. |

[2] | More seriously, I tend to think the reverse is true. I get very suspicious of technical reports that fill their results sections with nothing except the numbers. It might just be that I’m an arrogant jerk, but I often feel like an author that makes no attempt to explain and interpret their analysis to the reader either doesn’t understand it themselves, or is being a bit lazy. Your readers are smart, but not infinitely patient. Don’t annoy them if you can help it. |

[3] | A technical comment. In the same way that we can weaken the assumptions of
the z-test so that we’re only talking about the sampling distribution, we
can weaken the t-test assumptions so that we don’t have to assume
normality of the population. However, for the t-test it’s trickier to do
this. As before, we can replace the assumption of population normality with
an assumption that the sampling distribution of X̄ is normal. However,
remember that we’re also relying on a sample estimate of the standard
deviation, and so we also require the sampling distribution of
\(\hat{\sigma}\) to be χ². That makes things nastier, and this version
is rarely used in practice. Fortunately, if the population distribution is
normal, then both of these two assumptions are met. |