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

null hypothesis
μ
alternative hypothesis
μ 1 μ 2
value of X
value of X
Figure 13.8: Graphical illustration of the null and alternative hypotheses assumed by the Student t-test.
The null hypothesis assumes that both groups have the same mean μ, whereas the alternative assumes
that they have different means μ 1 and μ 2 . Notice that it is assumed that the population distributions
are normal, and that, although the alternative hypothesis allows the group to have different means, it
assumes they have the same standard deviation.
.......................................................................................................
Now that we’ve assigned weights to each sample, we calculate the pooled estimate of the variance by
taking the weighted average of the two variance estimates, ˆσ 2 1 and ˆσ 2 2
ˆσ 2 p “ w 1ˆσ 2 1 ` w 2ˆσ 2 2
w 1 ` w 2
Finally, we convert the pooled variance estimate to a pooled standard deviation estimate, by taking the
square root. This gives us the following formula for ˆσ p ,
d
w 1ˆσ 1 2 ˆσ p “ ` w 2ˆσ 2
2
w 1 ` w 2
And if you mentally substitute w 1 “ N 1 ´ 1andw 2 “ N 2 ´ 1 into this equation you get a very ugly
looking formula; a very ugly formula that actually seems to be the “standard” way of describing the pooled
standard deviation estimate. It’s not my favourite way of thinking about pooled standard deviations,
however. 9
13.3.4 The same pooled estimate, described differently
I prefer to think about it like this. Our data set actually corresponds to a set of N observations,
which are sorted into two groups. So let’s use the notation X ik to refer to the grade received by the i-th
student in the k-th tutorial group: that is, X 11 is the grade received by the first student in Anastasia’s
class, X 21 is her second student, and so on. And we have two separate group means ¯X 1 and ¯X 2 ,which
we could “generically” refer to using the notation ¯X k , i.e., the mean grade for the k-th tutorial group. So
far, so good. Now, since every single student falls into one of the two tutorials, and so we can describe
their deviation from the group mean as the difference
X ik ´ ¯X k
9 Yes, I have a “favourite” way of thinking about pooled standard deviation estimates. So what?
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