SWBAT (students will be able to): Define/recognize/use/describe the following: 

• Parameter vs. Statistic
• Sampling Distribution
• Distribution of a Sample
• Distribution of a Population

SWBAT Evaluate claims using sampling distributions

A statistic is a number that describes some characteristic of a sample.

A parameter is a number that describes some characteristic of a population. 

The table below shows three commonly used statistics and their corresponding parameters. These symbols are widely used in statistics and you are expected to know and use these symbols and terms. The symbol for sample mean is read "x bar" and the symbol for sample proportion is read "p hat." PLEASE get out your notebook and write down the following symbols and their meanings:

Use the following scenario for the questions below:

From a large group of people who signed a card saying they intended to quit vaping, 1000 people were selected at random. It turned out that 210 (21%) of these individuals had not vaped over the past 6 months. 

Since we do a lot of small group work in AP Statistics, we’ve decided to sample 2 students from a group of 4 and take the average of their scores to get an estimate for how well students in this group performed on the exam.  **Note: usually people wouldn’t use a sample and population this small, but we’re starting small to help you visualize what we’re doing.**

For this scenario, our population will be 4 students with the following scores: Alexis - 3, Bert - 4, Chris - 5, Devin - 5

Since we do a lot of small group work in AP Statistics, we’ve decided to sample 2 students from a group of 4 and take the average of their scores to get an estimate for how well students in this group performed on the exam. 

For this scenario, our population will be 4 students with the following scores: 3,4,5,5

Use the NetLogo model below to simulate choosing two students (n=2) at a time from this population (N=4).

1) Click "setup"  to start the model. If you are bothered by a person being cutoff on the edge of the screen, click "setup" again.

2) Click "collect sample" to choose your first sample, and record your results in the table below before pressing the button again.

3) Continue pressing "collect sample" and entering data into the table below until all possible samples have been chosen.

*Note: you may have to add additional rows to the data table. 

What you created on the previous page was the sampling distribution of AP scores for that group of 4 students. You took all six possible samples (n=2) from a population of four students (N=4). Sampling distributions are an important concept in Statistics and understanding what they are is the key to everything we do for the rest of the year.

A sampling distribution shows all possible samples of size n taken from a population of size N. In real-life scenarios, we don't create the entire sampling distribution by hand, or even using a computer, because our population is much larger (or it's size is unknown).  For illustrative purposes we will use a NetLogo model to simulate taking samples of AP Statistics exam scores from last year. There were 48 students in our classes last year, and we’ll take samples of 2 students at a time.

Here's the new scenario: 

We ran the model from the previous page until we collected every possible sample of 2 students. Below you can see the ENTIRE sampling distribution for our situation (n = 2, N = 48 students). 

SWBAT:

Understand/use/define Bias vs. Variability

Understand/use/define the relationship between sample size and variability

In the previous lesson we learned about Sampling Distributions. We specifically looked at a sampling distribution of MEANS. In this lesson we're going to use the same model that we used in the previous lesson to create and compare sampling distributions for a few different statistics. (note all sampling distributions created in this lesson are actually approximations, they are not complete, but they are perfectly good for our purposes)

Scroll down to see instructions and questions for this page.

Now we're going to do the same thing, except instead of calculating the mean score for our sample, we're going to calculate the MAXIMUM score of our sample. We (Mr. Mickelson and Mr. Mills) made this model, so we can change it in order to explore a different statistic. We're going to change the model to find the maximum of each sample. 

Note: If you need to start fresh just reload the page and everything will reset.

a) Underneath the model window is a tab called "netLogo Code". Click on that to open the code. Don't freak out, we're just going to make one little change!

b) Scroll down to find line 29. You can see the line numbers on the left side of the workspace. 

c) In lines 30 and 31 you should see the word "Mean." In both places, change the word Mean to the word Max.

d) Press recompile code at the top of the workspace.

e) Click setup and collect samples. This time the model will take the maximum value from each sample.

e) Click "tables," the "data set" again. Click "graph" and this time put the "Max" on the axis of the graph.

f) Add the mean to your graph of sample maximums (click the ruler, then check "Mean").

On the previous pages you probably noticed that the center of the "Mean" sampling distribution lined up pretty closely with the center of the actual population distribution. However, the center of the "Max" and "Min" sampling distribution did not match up with the actual maximum and minimum of the population. The statistics "Max" and "Min" are examples of BIASED estimators. The mean of their sampling distribution is not equal to the actual value of the population parameter. 

Here's the same model, but now we're back to calculating means. Scroll down for instructions!

This is just a page to put together your understanding of bias and variability. A lot of times students say "more biased" when they really mean "less variable", so this is a pretty important thing to nail down!

(Use the figure below for questions 1 and 2) 

The figure shows approximate sampling distributions of 4 different statistics intended to estimate the same parameter.

Students will be able to:

• Calculate and interpret the mean and standard deviation of the sampling distribution of a sample proportion
• Determine if the sampling distribution of is approximately Normal
• Calculate and interpret the mean and standard deviation of the sampling distribution for a difference in sample proportions,
• Determine if the sampling distribution of a difference in proportions is approximately Normal
• Use a Normal distribution to calculate probabilities

Recall from Chapter 2, we had two main calculator functions that were used:

• normalcdf(lower bound, upper bound, mean, st dev): used to find the area under the Normal distribution between two bounds
  • Example: IQ scores are normally distributed with mean 100 and a standard deviation of 15. Calculate the proportion of individuals that have IQ scores above 135.
  • Solution: normalcdf(135, ∞, 100, 15) ≈ 0.0098 --> approx 1%
• invNorm(area under the Normal distribution to the LEFT of a z-score ("percentile"), mean, st dev): used to find the z-score or observation in a Normal distribution that is located at the nth percentile. 
  • Example: IQ scores are normally distributed with mean 100 and a standard deviation of 15. Mr Mills claims he scored in the 98th percentile on an IQ test. What is his IQ score?
  • Solution: invNorm(0.98, 100, 15) ≈ 130.81

Answer the following questions below using your knowledge from Chapter 2

While normal distributions provide good models for some distributions of real data, the distributions of some the common variables are usually skewed and therefore distinctly non-normal. Examples include economic variables such as personal income and total sales of business firms, the survival times of cancer patients after treatment, and the lifetime of electronic devices. It is risky to assume that a distribution is normal without actually inspecting the data, so it is important to check a distribution for normality. You can assess the normality of a distribution by plotting the data using a dotplot, stemplot, or histogram or checking whether the data follow the 68–95–99.7 rule. However, just because a plot of the data looks normal, we can't say that the distribution is normal. For a better assessment of whether a data set follows a normal distribution, we can use a normal probability plot.

To assess the normality of a distribution from its normal probability plot, look at the plotted points, and see how well they fit the normal line. If they fit well, you can safely assume that your process data is normally distributed. If your plotted points don't fit the line well, but curve away from it in places, you may have a non normal distribution.

Follow the steps below to create a Normal Probability Plot! Answer the questions as you go through each step.  

Let's switch gears and look at sample proportions. The main question will investigate on this page: 

When can we approximate the sampling distribution for proportions as Normal? 

The model below allows you to select a sample size of your choice and set the population proportion to anything you want. We can use this model to evaluate claims about sample proportions in any context.  You may want to drag the slider at the top to generate your samples more quickly. Scroll down to see instructions and questions.

IMPORTANT - PLEASE READ

As you saw on the previous slide, the sampling distribution for \(\hat p\) depends on \(n\) and \(p\) . When \(p\) is closer to 0.5, and \(n\) is larger, the sampling distribution looks more Normal.

Here's a summary: Choose an SRS of size \(n\) from a population of size \(N\) with proportion \(p\) of successes.  Let \(\hat p\) be the sample proportion of successes. Then, the following is true for the sampling distribution of \(\hat p\) (as long as the conditions are met):

We will check these 3 conditions EVERY TIME we do a sampling problem about proportions.

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. 

By the end of this lesson, you should be able to:

• Explain how the shape of the sampling distribution of  is affected by the shape of the population distribution and the sample size (aka, The Central Limit Theorem)
• Explain why the Central Limit Theorem is one of the fundamental theorems in statistics
• Calculate the mean and standard deviation of the sampling distribution of a sample mean \(\bar x\) and interpret the standard deviation
• Calculate the mean and standard deviation of the sampling distribution of a difference in sample means \(\bar x_1- \bar x_2\) and interpret the standard deviation
• Determine if the sampling distribution of \(\bar x_1- \bar x_2\) is approximately Normal
• If appropriate, use a Normal distribution to calculate probabilities involving \(\bar x\) and  \(\bar x_1- \bar x_2\)

Use the NetLogo model below to answer the questions.

The Central Limit Theorem states that sample means from any population accumulate in a distribution that approaches a normal curve, as long as the sample size is "large enough". Our textbooks define "large enough" as \(n \ge 30\). This means that in order to produce a sampling distribution that is approximately Normal, we must sample at least 30 individuals from the population (if the population distribution shape is unknown or non-Normal). If the population distribution is Normal, the sampling distribution of \(\bar x\) will also be Normal, no matter what the sample size \(n\) is. 

From our formula sheet:

\(\mu_\bar X = \mu\)                     \(\sigma_\bar X = \frac {\sigma}{\sqrt n}\)

Trains carry iron ore from a mine in Brazil to an aluminum processing plant in Peru in hopper cars.  Filling equipment is used to lode ore into the hopper cars.  When functioning properly, the actual weights of ore loaded into each car by the filling equipment at the mine are approximately normally distributed with a mean of 70 tons and a standard deviation of 0.9 tons.  If the mean is greater than 70 tons, the loading mechanism is overfilling.

Dr. Lopez gives students 90 minutes to complete the final exam for her course. Most students use almost all the time allowed, and relatively few students finish early, so the distribution of times that it takes students to finish the exam is strongly skewed to the left. The mean and standard deviation of the finishing times are 85 and 10 minutes, respectively. 

Suppose we took random samples of 40 students and calculated \(\bar x\) as the sample mean finishing time. We can assume that the students in each sample are independent.

From our formula sheet:

ACT scores at Ardrey Kell High School are Normally distributed with mean 26 and standard deviation 3.  ACT scores at Providence HS are skewed to the right with mean 25 and standard deviation 5.

The heights of young men follow a Normal distribution with mean \(\mu_M\)= 69.3 inches and standard deviation \(\sigma_M \)= 2.8 inches. The heights of young women follow a Normal distribution with mean \(\mu_W\) = 64.5 inches and standard deviation \(\sigma_W\) = 2.5 inches. Suppose we select independent SRSs of 16 young men and 9 young women and calculate the sample mean heights \(\bar x_M\) and \(\bar x_W\).

Please try to do your best to answer the following truthfully