146 12 gender_discrimination online

MATH& 146

Lesson 12

Section 2.1

Randomization Case Study:

Gender Discrimination

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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 all

heads. Would that be evidence of an unfair coin?

(This is about the same likelihood as winning the

Powerball Grand Prize four times in a row.)

c) With 100 flips, what would be the cutoff between a

fair coin and an unfair coin?

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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?"

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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.

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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.

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Crucial point.

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.

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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.

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decision

promoted not promoted Total

male 21 3 24

gender female 14 10 24

Total 35 13 48


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.

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decision


male 21 3 24


Total 35 13 48

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?

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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.

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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.

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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.

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decision


male 21 3 24


Total 35 13 48

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.

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decision


male 21 3 24


Total 35 13 48

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.

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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.

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Hypothesis Test

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.

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If Alternative Is True

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.

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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.

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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.

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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.

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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.

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larger group smaller group Total

red 24

black 24

Total 35 13 48

Example 4

What is the difference in the proportion rates

between red and black in the larger group (our

simulation for those promoted).

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larger group smaller group Total

red 24

black 24

Total 35 13 48

# red in larger group # black in larger group red black

24 24 24

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?

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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.

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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.

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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.

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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.

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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?

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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.

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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.

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146 12 gender_discrimination online

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