Writing Math:
You can write math by using

$LATEX$ script, which is 'math code' that is surrounded by dollar signs. This works here, in Rmarkdown files, and in our Quarto files!

• Greek letters:
  $\alpha$,
  $\beta$,
  $\sigma$,
  $\mu$
• Superscripts:
  $x^2$ and subscripts
  $p_i$
• Variables:
  $x$,
  $y$, and equations
  $P(X<5)=.05$

Inference using Randomization

Consider the sex discrimination case study in chapter 11.1.
https://openintro-ims.netlify.app/foundations-randomization.html#caseStudySexDiscrimination

Work with your assigned partner to answer the questions below.

1. Is this an observational study or an experiment? How does the type of study impact what can be inferred from the results? (Evan)

This is an experiment because there is an experimental treatment (identified sex of candidates), and all potentially confounding variables are kept constant. As a result, it is possible to determine causality. Any significant discrepancy between promotions of males and females is caused from discrimination by the supervisors. Confounding variables are accounted for, since all else in employee files are identical in the experiment.

2. What is the hypothesis being tested? (in words with no statistical jargon) (Anusha & Meghan)

• Null hypothesis: Individuals that identify as females are equally likely as males to be promoted to branch managers by male supervisors at banks.
• Alternative hypothesis: Individuals that identify as females are less likely than males to be promoted to branch manager by male supervisors at banks.

3. How did they get the proportions 0.583 and 0.875? Be specific (your answer should contain a fraction). (Abbey & Kenji)
   They looked at the amount of males promoted (21) out of the total males (24) which gave the proportion .875

• 21/24 = .875
  Then, they took the amount of females promoted (14) out of the total amount of females (24) which gives the proportion .583

• 14/24=.583

4. Briefly explain the process of randomization in this case study (Matthew)

All 48 bank supervisors were given a personnel file that were identical besides the gender that the candidate identified as. Half of the bank supervisors were given personnel files that listed male, and half female. After the experiment was complete the results were then randomized to see if the bank supervisors would have promoted or not prometed regardless of the gender of the applicant.

5. What is the difference in promotion rates between the two simulated groups in Table 11.2 ? How does this compare to the observed difference 29.2% from the actual study (Eden & Sarah)

   In the simulated groups, 17 out of 24 (70.83%) women were promoted, and 18 out of 24 (75.00%) men were promoted. By subtracting the fraction of women promoted by the fraction of men promoted we detemined that men were 4.17% more likely to get promoted than women.
   The simulated and observed values differ by 600.24%. This value was obtained by using the percent difference formula in which 4.17 is subtracted from 29.2, and that total is divided by 4.17. Such a large difference between the simulation, which assumes no bias, and the actual observations show that there was a hiring bias based on gender.

6. Let

   $p_m$ be the proportion of males who were promoted, and
   $p_f$ the proportion of females who were promoted. Write the null and alternative hypotheses as a mathematical statement comparing these two parameters
   $p_m$ and
   $p_f$ (Gunner & Meriam)
   Null Hypothesis: There is no difference among the propotion of males promoted and propotion of females promoted.
   
   $P_m-P_f=0$
   Alternative: There is a difference between the proportion of males who were promoted and the proportion of females who were promoted.
   
   $P_m-P_f\ne 0$

7. The book states "It appears that a difference of at least 29.2% under the null hypothesis would only happen about 2% of the time according to Figure 11.4." Where did that number come from? (Sean & Jake)

   There was only 2 out of the 100 "shuffles" which had a difference under the null hypothesis of 29.2% or greater.

8. What conclusion can we make about this study? What is our evidence? (Raquel & Ryan)

We can conclude from this study that there is discrimination against promoting females by male supervisors.

Simulations show that there is no difference between promoting men and women. They were both equally likely to be promoted. The 29.2% bias towards promoting men observed had a roughly 2% chance of occurring.