146 12 gender_discrimination
Transcript of 146 12 gender_discrimination
![Page 1: 146 12 gender_discrimination](https://reader034.fdocuments.us/reader034/viewer/2022042723/58861b1e1a28abe63e8b7287/html5/thumbnails/1.jpg)
MATH& 146
Lesson 12
Section 2.1
Randomization Case Study:
Gender Discrimination
1
![Page 2: 146 12 gender_discrimination](https://reader034.fdocuments.us/reader034/viewer/2022042723/58861b1e1a28abe63e8b7287/html5/thumbnails/2.jpg)
Example 1
a) Suppose you flip a coin 100 times, getting 51
heads and 49 tails. Would that be evidence of an
unfair coin?
b) Suppose you flip another coin 100 times, getting 90
heads and 10 tails. Would that be evidence of an
unfair coin? (This is about the same likelihood as
winning the Powerball Grand Prize twice in a row.)
c) With 100 flips, what would be the cutoff between a
fair coin and an unfair coin?
2
![Page 3: 146 12 gender_discrimination](https://reader034.fdocuments.us/reader034/viewer/2022042723/58861b1e1a28abe63e8b7287/html5/thumbnails/3.jpg)
Gender Discrimination
We will consider a study investigating gender
discrimination in the 1970s, which is set in the
context of personnel decisions within a bank.
The research question we hope to answer is, "Are
females discriminated against in promotion
decisions made by male managers?"
3
![Page 4: 146 12 gender_discrimination](https://reader034.fdocuments.us/reader034/viewer/2022042723/58861b1e1a28abe63e8b7287/html5/thumbnails/4.jpg)
Gender Discrimination
The participants in this study were 48 male bank
supervisors attending a management institute at
the University of North Carolina in 1972.
They were asked to assume the role of the
personnel director of a bank and were given a
personnel file to judge whether the person should
be promoted to a branch manager position.
4
![Page 5: 146 12 gender_discrimination](https://reader034.fdocuments.us/reader034/viewer/2022042723/58861b1e1a28abe63e8b7287/html5/thumbnails/5.jpg)
Gender Discrimination
The files given to the participants were identical,
except that half of them indicated the candidate
was male and the other half indicated the
candidate was female.
These files were randomly assigned to the
subjects.
5
Crucial point.
![Page 6: 146 12 gender_discrimination](https://reader034.fdocuments.us/reader034/viewer/2022042723/58861b1e1a28abe63e8b7287/html5/thumbnails/6.jpg)
Example 2
Is this an observational study or an experiment?
How does the type of study impact what can be
inferred from the results?
The study is an experiment, as subjects were
randomly assigned a male file or a female file.
Since this is a random experiment, the results can
be used to evaluate a causal relationship between
gender of a candidate and the promotion decision.
6
![Page 7: 146 12 gender_discrimination](https://reader034.fdocuments.us/reader034/viewer/2022042723/58861b1e1a28abe63e8b7287/html5/thumbnails/7.jpg)
Gender Discrimination
For each supervisor we recorded the gender
associated with the assigned file and the
promotion decision. Using the results of the study
summarized below, we would like to evaluate if
females are unfairly discriminated against in
promotion decisions.
7
gender
male female Total
decisionpromoted 21 14 35
not promoted 3 10 13
Total 24 24 48
![Page 8: 146 12 gender_discrimination](https://reader034.fdocuments.us/reader034/viewer/2022042723/58861b1e1a28abe63e8b7287/html5/thumbnails/8.jpg)
Gender Discrimination
In this study, a smaller proportion of females are
promoted than males (0.583 versus 0.875), but it is
unclear whether the difference provides convincing
evidence that females are unfairly discriminated
against.
8
gender
male female Total
decisionpromoted 21 14 35
not promoted 3 10 13
Total 24 24 48
![Page 9: 146 12 gender_discrimination](https://reader034.fdocuments.us/reader034/viewer/2022042723/58861b1e1a28abe63e8b7287/html5/thumbnails/9.jpg)
Example 3
Statisticians are sometimes called upon to
evaluate the strength of evidence. When looking
at the rates of promotion for males and females in
this study, why might we be tempted to
immediately conclude that females are being
discriminated against?
On the other hand, why might be tempted to
conclude that there is no discrimination?
9
![Page 10: 146 12 gender_discrimination](https://reader034.fdocuments.us/reader034/viewer/2022042723/58861b1e1a28abe63e8b7287/html5/thumbnails/10.jpg)
Gender Discrimination
The large difference in promotion rates (58.3% for
females versus 87.5% for males) suggest there
might be discrimination against women in
promotion decisions. However, we cannot yet be
sure if the observed difference represents
discrimination or is just from random chance.
10
![Page 11: 146 12 gender_discrimination](https://reader034.fdocuments.us/reader034/viewer/2022042723/58861b1e1a28abe63e8b7287/html5/thumbnails/11.jpg)
Gender Discrimination
Generally there is a little bit of fluctuation in sample
data, and we wouldn't expect the sample
proportions to be exactly equal, even if the truth
was that the promotion decisions were
independent of gender.
11
![Page 12: 146 12 gender_discrimination](https://reader034.fdocuments.us/reader034/viewer/2022042723/58861b1e1a28abe63e8b7287/html5/thumbnails/12.jpg)
Point Estimates
The table below shows there were 7 fewer
promotions in the female group than in the male
group, a difference in promotion rates of 7/24, or
29.2%. This observed difference is what we call a
point estimate of the true effect.
12
gender
male female Total
decisionpromoted 21 14 35
not promoted 3 10 13
Total 24 24 48
![Page 13: 146 12 gender_discrimination](https://reader034.fdocuments.us/reader034/viewer/2022042723/58861b1e1a28abe63e8b7287/html5/thumbnails/13.jpg)
Point Estimates
The point estimate (the statistic from the sample
that is used to answer the research question) of
the difference is large, but the sample size for the
study is small, making it unclear if this observed
difference represents discrimination or whether it is
simply due to chance.
13
gender
male female Total
decisionpromoted 21 14 35
not promoted 3 10 13
Total 24 24 48
![Page 14: 146 12 gender_discrimination](https://reader034.fdocuments.us/reader034/viewer/2022042723/58861b1e1a28abe63e8b7287/html5/thumbnails/14.jpg)
Hypotheses
We label these two competing claims (random
chance or nonrandom discrimination), H0 and HA:
H0: Null hypothesis. The variables gender and
decision are independent. They have no
relationship, and the observed difference
between the proportion of males and females
who were promoted, 29.2%, was due to
chance.
14
![Page 15: 146 12 gender_discrimination](https://reader034.fdocuments.us/reader034/viewer/2022042723/58861b1e1a28abe63e8b7287/html5/thumbnails/15.jpg)
Hypotheses
HA: Alternative hypothesis. The variables
gender and decision are not independent. The
difference in promotion rates of 29.2% was not
due to chance, and equally qualified females
are less likely to be promoted than males.
15
![Page 16: 146 12 gender_discrimination](https://reader034.fdocuments.us/reader034/viewer/2022042723/58861b1e1a28abe63e8b7287/html5/thumbnails/16.jpg)
Hypothesis Tests
These hypotheses are part of what is called a
hypothesis test. A hypothesis test is a statistical
technique used to evaluate competing claims
using data.
Often times, the null hypothesis takes a stance of
no difference or no effect. If the null hypothesis
and the data notably disagree, then we will reject
the null hypothesis in favor of the alternative
hypothesis.
16
![Page 17: 146 12 gender_discrimination](https://reader034.fdocuments.us/reader034/viewer/2022042723/58861b1e1a28abe63e8b7287/html5/thumbnails/17.jpg)
Hypothesis Tests
We will choose between these two competing
claims by assessing if the data conflict so much
with H0 that the null hypothesis cannot be deemed
reasonable.
If this is the case, and the data support HA, then
we will reject the notion of independence and
conclude that these data provide strong evidence
of discrimination.
17
![Page 18: 146 12 gender_discrimination](https://reader034.fdocuments.us/reader034/viewer/2022042723/58861b1e1a28abe63e8b7287/html5/thumbnails/18.jpg)
Simulating the Study
In the study, 35 bank supervisors recommended
promotion and 13 did not.
Suppose the bankers' decisions were independent
of gender. Then, if we conducted the experiment
again with a different random assignment of files,
differences in promotion rates would be based only
on random fluctuation.
18
![Page 19: 146 12 gender_discrimination](https://reader034.fdocuments.us/reader034/viewer/2022042723/58861b1e1a28abe63e8b7287/html5/thumbnails/19.jpg)
Simulating the Study
We can actually perform this randomization, which
will simulate what would have happened if the
bankers' decisions had been independent of
gender but we had distributed the files differently.
19
![Page 20: 146 12 gender_discrimination](https://reader034.fdocuments.us/reader034/viewer/2022042723/58861b1e1a28abe63e8b7287/html5/thumbnails/20.jpg)
Simulating the Study
Using a deck of cards, count out 24 red and 24
black (or remove 2 red and 2 black). We will use
these cards to represent the 24 males and 24
females of the study.
20
![Page 21: 146 12 gender_discrimination](https://reader034.fdocuments.us/reader034/viewer/2022042723/58861b1e1a28abe63e8b7287/html5/thumbnails/21.jpg)
Simulating the Study
Now shuffle your set of 48 cards, then deal out a
sample of 13 cards. You should now have two
groups of cards, one with 35 cards and one with
13 cards. Count the number of red and black
cards in each group and complete the following
table.
21
red black Total
larger group 35
smaller group 13
Total 24 24 48
![Page 22: 146 12 gender_discrimination](https://reader034.fdocuments.us/reader034/viewer/2022042723/58861b1e1a28abe63e8b7287/html5/thumbnails/22.jpg)
Example 4
What is the difference in the proportion rates
between red and black in the larger group (our
simulation for those promoted).
22
# red in larger group # black in larger group red black
24 24 24
red black Total
larger group 35
smaller group 13
Total 24 24 48
![Page 23: 146 12 gender_discrimination](https://reader034.fdocuments.us/reader034/viewer/2022042723/58861b1e1a28abe63e8b7287/html5/thumbnails/23.jpg)
Example 5
How many of you observed a difference of at least
29.2%?
Under our simulation, would you consider a
difference of at least 29.2% an event that occurs
often, sometimes, rarely, or never?
23
![Page 24: 146 12 gender_discrimination](https://reader034.fdocuments.us/reader034/viewer/2022042723/58861b1e1a28abe63e8b7287/html5/thumbnails/24.jpg)
Checking for Independence
We each computed one possible difference under
the null hypothesis, which represents one
difference due to chance.
While we could physically deal out cards over and
over again, it is much more efficient to perform this
simulation using a computer.
24
![Page 25: 146 12 gender_discrimination](https://reader034.fdocuments.us/reader034/viewer/2022042723/58861b1e1a28abe63e8b7287/html5/thumbnails/25.jpg)
Checking for Independence
Repeating the simulation on a computer, we might
get another difference due to chance of –0.042 (for
example), and another of 0.208, and so on until we
repeat the simulation enough times that we have a
good idea of what represents the distribution of
differences from chance alone.
25
![Page 26: 146 12 gender_discrimination](https://reader034.fdocuments.us/reader034/viewer/2022042723/58861b1e1a28abe63e8b7287/html5/thumbnails/26.jpg)
Checking for Independence
The figure below shows a plot of the differences
found from 100 simulations, where each dot
represents a simulated difference between the
proportions of male and female files recommended
for promotion.
26
![Page 27: 146 12 gender_discrimination](https://reader034.fdocuments.us/reader034/viewer/2022042723/58861b1e1a28abe63e8b7287/html5/thumbnails/27.jpg)
Checking for Independence
Because we simulated differences in a way that
made no distinction between men and women, it
makes sense that the differences from chance
alone fall around zero with some random
fluctuation for each simulation.
27
![Page 28: 146 12 gender_discrimination](https://reader034.fdocuments.us/reader034/viewer/2022042723/58861b1e1a28abe63e8b7287/html5/thumbnails/28.jpg)
Example 6
Based on the graph below, how often would you
observe a difference of at least 29.2% (0.292)?
Often, sometimes, rarely, or never?
28
![Page 29: 146 12 gender_discrimination](https://reader034.fdocuments.us/reader034/viewer/2022042723/58861b1e1a28abe63e8b7287/html5/thumbnails/29.jpg)
Making a Conclusion
When we conduct formal studies, we reject a null
hypothesis if the (randomly selected) data strongly
conflict with that hypothesis. In our analysis, we
determined that there was only a 2% probability
(p-value) of obtaining a sample where 29.2%
more males than females get promoted by chance
alone.
29
![Page 30: 146 12 gender_discrimination](https://reader034.fdocuments.us/reader034/viewer/2022042723/58861b1e1a28abe63e8b7287/html5/thumbnails/30.jpg)
Making a Conclusion
Because this probability is so low, we are forced to
conclude that the data provide strong evidence of
gender discrimination against women by the
supervisors.
In this case, we would reject the null hypothesis in
favor of the alternative.
30