10.2: The One-sample t-test

Let's say that after some thought, we have decided that it might not be safe to assume that the psychology student grades 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:

 

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. So, we could just type this new number into the syntax we used previously, and out would come the answer. 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 adjustments 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.

(#fig:ttesthyp_onesample)Graphical illustration of the null and alternative hypotheses assumed by the (two sided) one sample t-test. Note the similarity to the z-test. The null hypothesis is that the population mean μ is equal to some specified value μ₀, and the alternative hypothesis is that it is not. Like the z-test, we assume that the data are normally distributed; but we do not assume that the population standard deviation σ is known in advance.

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 (see 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. 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 ¯X and our estimate of the population standard deviation is \(\hat{\sigma}\), then our t statistic is:

\(\ t = {{\bar{X}-\mu} \over \hat{\sigma}/ \sqrt{N} }\)

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 Section ???.6 and illustrated in Figure 10.5. 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!

Figure 13.5: The t distribution with 2 degrees of freedom (left) and 10 degrees of freedom (right), with a standard normal distribution (i.e., mean 0 and std dev 1) plotted as dotted lines for comparison purposes. Notice that the t distribution has heavier tails (higher kurtosis) than the normal distribution; this effect is quite exaggerated when the degrees of freedom are very small, but negligible for larger values. In other words, for large df the t distribution is essentially identical to a normal distribution.

Doing the Test in SPSS

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 syntax.

SPSS has a handy one-sample t-test function. Let's use our zeppo.sav data set and use the Analyze\Compare Means\One-Sample T test... menu item:

Once that dialog is opened, you'll see:

In this dialog, you'll want to move the grades variable to the column under Test Variable(s): and in the Test Value area you will want to enter the value you are comparing our data set against. In this case, the test value is 67.5, the mean of non-psychology students:

Now, click OK and SPSS will output the following:

Easy enough. Now let's go through the output.

Reading this output from top to bottom, you can see it’s trying to lead you through the data analysis process. The first block of output shows your sample size N, the mean of our sample, the standard deviation, and the standard error of the mean. The second block gives the results of the t-test showing the t-score (2.255), degrees of freedom (19), One-sided p-value (.018), Two-sided p-value (.036), the mean difference between our scores and the hypothesized value, and finally, the 95% confidence interval.

The last block shows effect sizes that we'll discuss in detail a bit later in this chapter.

So, what does all that mean? Well, it appears that since our two-sided p-value is less than .05 we can reject the null hypothesis that psychology students score the same as other students.

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<.05); the 95% confidence interval is [67.8, 76.8].

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, like this:

t(19)=2.25, p<.05, CI95=[67.8,76.8]

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

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
• Independence. Once again, we have to assume that the observations in our sample are generated independently of one another.

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 hypothesized population mean.