Students will be able to:

• Calculate and interpret the mean and standard deviation of the sampling distribution of a sample proportion \(\hat p\)
• Determine if the sampling distribution of \(\hat p\) is approximately Normal
• Calculate and interpret the mean and standard deviation of the sampling distribution for a difference in sample proportions, \(\hat p_1 - \hat p_2\)
• Determine if the sampling distribution of a difference in proportions, \(\hat p_1 - \hat p_2\) , is approximately Normal
• Use a Normal distribution to calculate probabilities involving \(\hat p\) and \(\hat p_1 - \hat p_2\)

One thing that police do is make traffic stops. There are many different reasons that the police might choose to pull someone over and in some cases give the driver a ticket. Is their decision affected by gender? race? socio-economic status? time of day? type of car? Statistics is one way to answer these questions but it can also be used to purposely mislead people about the answers to these questions.

"There are three types of lies - lies, damn lies, and statistics" (attributed to Mark Twain, maybe aprocryphal)

We’re going to look at some Evanston Police Department (EPD) data about traffic stops between 1/1/2020 and 7/23/2020 and try to come up with some truthful answers.

Here is the data, there were 22687 total traffic stops as reported by EPD. There are lots of different questions that we might want to answer based off of this data. As you know, over the past many years Police Departments have been under scrutiny for how they treat people of color, specifically black people. In the Summer of 2020, the murder of George Floyd at the hands of the police was the latest in a string of innocent black lives lost in their interactions with police (https://www.bbc.com/news/world-us-canada-52905408). The ensuing protests and the Black Lives Matter Movement have put police treatment of black Americans in the spotlight. For that reason, we will specifically focus on data about traffic stops of black drivers throughout this lesson. Let's take a look at Data from Evanston. Clearly this could be an issue for any race, or gender, or a lot of other things, but for the purpose of this lesson we'll focus specifically on black vs. non-black drivers in Evanston.

Remember, this is traffic stops by EPD from 1/1/20 - 7/23/20 broken down by race.

See below for questions and instructions.

Below are a few additional questions related to sampling distributions for proportions. These are the sorts of questions that you'll be expected to answer on an AP test. Be sure to check all necessary conditions and show all work. Do your work on paper and upload a scanned image for the questions below. See the example below for what an exemplary response looks like for these types of problems. 

Example problem and solution: Suppose it is known that 21% of all Skittles produced are yellow. We will also assume that every bag of Skittles is a simple random sample from that entire population. Suppose you purchase a bag containing 120 Skittles. What is the probability of that bag containing 30 or more yellow Skittles?

We're coming back to our EPD data. A newspaper article claims that the proportion of traffic stops for black residents in Evanston (p₁) is HIGHER than the proportion of black residents in Evanston (p₂). 

We’re lucky that the Evanston Police Department keeps these sorts of records AND makes them available to the public. We don't have to use sampling in order to estimate the proportion of traffic stops that involve black drivers. But this is a lesson about sampling, SO WE'RE GOING TO PRETEND!

Let’s say we didn’t have all the data that we do have, so we HAD to use random sampling in order to investigate. For the remainder of this lesson, we’re going to pretend that we don’t have the full data, and we’re going to generate random samples in order to simulate real-world sampling.

We’re going to have to use sampling to estimate two different proportions: 

1. The proportion of all traffic stops that are of black drivers

2. The proportion of Evanston residents who are black

There are a total of 72,836 residents in Evanston and 22,687 traffic stops made.

What we really care about here is the DIFFERENCE between our two proportions.

Let p₁  be the true proportion of Evanston black drivers stopped in traffic stops.

Let p₂  be the true proportion of all black residents in Evanston.

Here are the formulas for the standard deviation and the mean of a difference between two proportions. Use the model, the known population proportions, and the sampling sizes you calculated before to carry out sampling for both proportions. Scroll down for detailed instructions.

Here's the real values for p₁ and p₂.

p₁=0.3057

p₂=0.17

Suppose we want to investigate the effectiveness of two potential COVID-19 vaccines. We will call these "Vaccine 1" and "Vaccine 2". 

During Phase 3 trials of the vaccine development process, thousands of volunteers are randomly assigned one of the two vaccines. After 30 days, researchers take blood samples to detect if antibodies are present. The presence of antibodies would indicate that the vaccine has been effective at preventing the individual from experiencing moderate to severe COVID-19 symptoms. The table below shows the results of some randomly selected volunteers from each vaccine trial.