Is a man more likely to be promoted than
a woman
even if they have the same qualifications? We'll consider an experiment investigating
question
from the 1970s where 48 male bank supervisors
were asked whether they would recommend
a particular candidate for promotion
each supervisor was given a resume and
determined whether the candidate should
be promoted or not.
Every one of these 48 resumes
was identical except the 24 labeled as
male candidates
and 24 were labeled as female candidates. Out of these 48 candidates
35 a recommended for promotion and 13
were not.
If men were favored we'd expect to see
higher fraction promotions in the male group
than the female group. If there was no
favoritism
than we'd expect to see about the same
promotion rate in each group.
These two competing ideas are called
hypotheses.
The hypothesis of no difference is
called
the null hypothesis and the other
hypothesis
is called the alternative hypothesis.
Let's take a look at the data.
What we see is that there were more man
recommended for promotion
than women. Seven more, or a difference of
about 29 percent
in promotional rates. So does this prove
that women were discriminated against
No.
No??? But didn't we just determined that if men
were favored,
then we'd expect to see higher fraction
man recommended for promotion
than women? Well, yes, but observing *some* difference
does not automatically mean there was
discrimination.
We wouldn't expect a recommendation
rate in both the male and female group
to be exactly the same. There might be a
difference from chance alone.
So if anything can happen from
randomness,
how do you decide what to conclude from
the data. This is a tough question,
and it's one reason the study of
statistics is so useful.
In this experiment, we should ask how
often would I see data like this
if there was no discrimination? That is,
how uncommon would be to see this large
a difference
in the recommendation rates if there was
no discrimination?
This will help us think about whether
the data are easily explained by chance
or whether we're seen some strong
evidence
that there was discrimination. Now we
have to figure out
what would a result look like that was
from chance alone?
We can answer that question by
reassigning the resumes to the 48
supervisors,
and assuming each banker's response
would be the same as before.
After all, if the bankers weren't
affected by the gender of
the candidate, their responses would be
unchanged. And what's the result? We get a
difference a 4% favoring men.
This is smaller than the twenty-nine
percent difference observed in the real
data.
But one simulation is not itself
convincing.
We want to know how variable the
differences from one sample to another.
So let's run another simulation. This time
we get 4% in favor women.
And another simulation.
A 20% difference in favor men.
We'll run a total of 100 simulations
and plot each simulation's result. Now as
we look at the plot
we ask ourselves, how often do we see a
difference
at least as big as what we actually
observed, which is 29 percent?
In this case pretty rarely.
Only 2 the 100 simulations show this
kind of result.
So now we're left to
think about two possibilities. We know
that it's possible but very unlikely that
if there was no discrimination we
observe a 29% difference from
chance alone.
Alternatively, the data are consistent
with what we'd expect
if there was discrimination against the
female candidates. Because it's so
unlikely to observe a gender promotion
read 29%
if there really was no gender
discrimination, we reject H_0
and conclude there was real gender
discrimination.
We've just covered what is called a
hypothesis test,
which is a formal approach for
evaluating a research question.
We analyzed the data, and we came up with a
randomization approach to assess whether
the difference observed in the data
could reasonably be explained by chance
and conclude there was real gender
discrimination.
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