\(X \sim Pois(\lambda = 3) \Rightarrow \mathbb{P}(X = x) = e^{-\lambda} \left(\dfrac{\lambda^x}{x!}\right) = e^{-3} \left(\dfrac{3^x}{x!}\right)\)

• Want to find \(\mathbb{P}[ (X = 2) \,|\, (X \ge 1)]\)

• Using the definition of conditional probability,

\[\mathbb{P}[ (X = 2) \,|\, (X \ge 1)] = \dfrac{\mathbb{P} [ (X = 2) \text{ AND } (X \ge 1) ]}{\mathbb{P}(X \ge 1)} = \dfrac{\mathbb{P}(X = 2)}{\mathbb{P}(X \ge 1)}\]

          \(= \dfrac{\mathbb{P}(X = 2)}{1 - \mathbb{P}(X = 0)} = \dfrac{e^{-3}\left( \dfrac{3^2}{2!}\right)}{1 - \left[e^{-3}\left( \dfrac{3^0}{0!}\right)\right]} \approx \dfrac{0.224}{0.950} = 0.236\)

Question of interest: are females unfairly discriminated against in terms of promotions given by male supervisors?

\[ \mathbb{P}(\text{promote}\,|\,M) = 21/24 = 0.875 \\ \mathbb{P}(\text{promote}\,|\,F) = 14/24 = 0.583 \]

At a first glance, does there appear to be a relationship between promotion and gender?

\(H_0\), Null Hypothesis: "There is nothing going on."

• Promotion and gender are independent.
• No gender discrimination.
• Observed difference in proportions is simply due to chance.

\(H_a\), Alternative Hypothesis: "There is something going on.”

• Promotion and gender are dependent.
• There is gender discrimination.
• Observed difference in proportions is not due to chance.
• \(\mathbb{P}(promote\,|\,M) > \mathbb{P}(promote\,|\,F)\)

\(H_0\) : Defendant is not guilty vs. \(H_a\) : Defendant is guilty

• Assume that the defendant is not guilty.
• Present evidence / collect data.
• Judge the evidence - “Could these data plausibly have happened by chance if the null hypothesis were true?"
• If they were very unlikely to have occurred by chance, then the evidence raises more than a reasonable doubt in our minds about the null hypothesis.

• If the evidence is not strong enough to reject the assumption of innocence, the jury returns with a verdict of not guilty.
  • The jury does not say that the defendant is innocent, just that there is not enough evidence to convict.
• Said statistically: we fail to reject the null hypothesis, or the data is consistent with our model.
  • We never accept the null hypothesis.

The hypothesis test gives us:

\[ \mathbb{P}(\textrm{data} \,|\, H_0) \]

It doesn't give us:

\[ \mathbb{P}(H_0 \,|\, \textrm{data}) \]

• It's also impossible to prove a negative.
  
• It's the same reason we shouldn't say "The defendant is innocent."
  
• We should say "There is not enough evidence to convict the defendant."

What is the null hypothesis?

• \(H_0\): There is no gender discrimination (Gender and promotion are independent)
• \(H_0: \mathbb{P}(promote\,|\,M) = \mathbb{P}(promote\,|\,F)\)

What is the alternative hypothesis?

• \(H_a\): There is gender discrimination (Gender and promotion are not independent)
• \(H_a: \mathbb{P}(promote\,|\,M) > \mathbb{P}(promote\,|\,F)\)

What is our test statistic? (Note that this is a random variable!)

• \(D = \mathbb{P}(promote\,|\,M) - \mathbb{P}(promote\,|\,F)\)

We can compute our observed test statistic:

\[ d_{obs} = \hat{p}_{M} - \hat{p}_{F} \\ d_{obs} = 21/24 - 14/24 \approx .29 \]

1. Shuffle the deck and deal into two piles of twenty four.
   • This mimics the process of each supervision being randomly assigned a male or female file.

2. Compute the proportion that is promoted in each:

   \[ d = \hat{p}_{M} - \hat{p}_{F} \]

3. Repeat steps 1-3 and store each one.

• The proportion of simulated statistics that fell at or above what we observed

• Out of the 10,000 simulations, 247 were greater than 0.2916667.

• Therefore, \(p\)-value = 0.0247. Since this is smaller than \(\alpha = 0.05\), we reject \(H_0\) and based on this data, we have evidence that females are unfairly discriminated against in terms of promotions.