It doesn't really matter which one you're calling male versus female.

So let's just say this is our male pile, and this is our female pile.

The next step is going to be to determine how many files were promoted in each pile.

Which means we need to count the number of number cards in each pile.

Among the males, I'm counting one, two, three,

four, five, six, seven, eight,

nine, ten, 11, 12, 13, 14, 15, 17.

So we have 17 out of 24 males promoted.

Which should leave about 18 out of 24 females promoted.

In the next step we need to calculate the proportions and take the difference and

note that on our dot plot.

And we would repeat this many, many times to build a simulation distribution.

So how do we ultimately make a decision?

If the results from the simulations look like the data,

then we decide that the difference between the proportions of promoted files,

between males and females, was due to chance.

And that promotion and gender are independent.

If, on the other hand, the results from the simulations do not look like the data,

then we decide that the observed difference in the promotion rates

was unlikely to have happened just by chance, and

that it can be attributed to an actual effect of gender.

In other words, we conclude that these data provide evidence of

a dependency between promotion decisions, and gender.

If we repeat the simulation many times, and record the simulated differences in

proportions of males and females promoted, we can build a distribution like this one.

For example, here we have a dot plot of the distribution of

the simulated differences, and promotion rates based on a hundred simulations.

While we showed earlier how to simulate this experiment using playing cards,

we should note that the task of the simulation is best left up to computation.

It's faster and less prone to errors.

The distribution is centered at zero which we can also think about as the null value,

since according to the null hypothesis,

there should be no difference between the proportion rates of males and females.

Yielding a difference of zero.

We can see from the distribution of the simulated differences in promotion rates,

that it is very rare to get a difference as high as 30%,

the observed difference from the original data.

If in fact gender does not play a part in promotion decisions.

The low likelihood of this event, or a difference even more extreme,

suggests that promotion decisions may not be independent of gender, and so

we would reject the null hypothesis.

Our conclusion is then that these data show convincing evidence of an association

between gender and promotion decisions made by male bank supervisors.