Learning Statistics with R - A tutorial for psychology students and other beginners, 2018a

Anastasia’s students
Bernadette’s students
Frequency
0 1 2 3 4 5 6 7
Frequency
0 1 2 3 4 5 6 7
50 60 70 80 90 100
Grade
(a)
50 60 70 80 90 100
Grade
(b)
Figure 13.6: Histograms showing the overall distribution of grades for students in Anastasia’s class
(panel a) and in Bernadette’s class (panel b). Inspection of these histograms suggests that the students
in Anastasia’s class may be getting slightly better grades on average, though they also seem a little more
variable.
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will be clearer when we come to talk about the paired samples t-test later on. For now, let’s just point
out that if we have an experimental design where participants are randomly allocated to one of two
groups, and we want to compare the two groups’ mean performance on some outcome measure, then an
independent samples t-test (rather than a paired samples t-test) is what we’re after.
Okay, so let’s let μ 1 denote the true population mean for group 1 (e.g., Anastasia’s students), and
μ 2 will be the true population mean for group 2 (e.g., Bernadette’s students), 8 and as usual we’ll let ¯X 1
and ¯X 2 denote the observed sample means for both of these groups. Our null hypothesis states that the
two population means are identical (μ 1 “ μ 2 ) and the alternative to this is that they are not (μ 1 ‰ μ 2 ).
Written in mathematical-ese, this is...
H 0 : μ 1 “ μ 2
H 1 : μ 1 ‰ μ 2
To construct a hypothesis test that handles this scenario, we start by noting that if the null hypothesis
is true, then the difference between the population means is exactly zero, μ 1 ´ μ 2 “ 0 As a consequence,
a diagnostic test statistic will be based on the difference between the two sample means. Because if the
null hypothesis is true, then we’d expect ¯X 1 ´ ¯X 2 to be pretty close to zero. However, just like we saw
with our one-sample tests (i.e., the one-sample z-test and the one-sample t-test) we have to be precise
8 A funny question almost always pops up at this point: what the heck is the population being referred to in this case?
Is it the set of students actually taking Dr Harpo’s class (all 33 of them)? The set of people who might take the class (an
unknown number) of them? Or something else? Does it matter which of these we pick? It’s traditional in an introductory
behavioural stats class to mumble a lot at this point, but since I get asked this question every year by my students, I’ll
give a brief answer. Technically yes, it does matter: if you change your definition of what the “real world” population
actually is, then the sampling distribution of your observed mean ¯X changes too. The t-test relies on an assumption that
the observations are sampled at random from an infinitely large population; and to the extent that real life isn’t like that,
then the t-test can be wrong. In practice, however, this isn’t usually a big deal: even though the assumption is almost
always wrong, it doesn’t lead to a lot of pathological behaviour from the test, so we tend to just ignore it.
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