The first few years teaching AP Statistics, I enjoyed teaching students about binomial distributions, but I didn't quite understand how it was connected to the bigger picture. It always felt like an "add on" that we cover so students can answer a few questions that will likely show up on the AP Exam. With more experience, I realized that binomial distributions are a critical step in the progression towards the end goal of the course.
A Path to Inference
Here is the realization that helped me bridge the gap from binomial distributions to inference:
A binomial random variable measures the count of the number of successes in a fixed number of trials. Any inference about categorical variables uses the proportion of successes in a fixed number of trials. In order to change the count of successes into a proportion, simply divide by the sample size. In other words, a sampling distribution of sample proportions is a scaled version (by 1/n) of a binomial distribution.
This means that any of the properties of binomial distributions are also properties of the sampling distribution of sample proportions. Also the formulas for binomial distributions can be easily scaled to become the formulas for the sampling distribution of sample proportions. Here are some ideas that we get from binomial distributions that we will use later with sample proportions:
10% condition - observations can be considered independent as long as the sample size is less than 10% of the population.
Large Counts condition - when the expected number of success and failures are both greater than or equal to 10, the binomial distribution can be approximated using a Normal distribution.
Formulas for the mean and standard deviation.
By the end of teaching this content on binomial distributions, we can even ask students real inference questions:
Mr. Wilcox gets a bag of candy that contains 200 Skittles and he gets only 13 red Skittles. Does this provide convincing evidence that the company's claim that 15% are red is a lie?
Three Lessons to Teach Binomial Distributions
Determine whether the conditions for a binomial setting are met.
Calculate and interpret probabilities involving binomial distributions.
Memphis basketball player Darius Washington is fouled at the end of a game and has a chance to win the game for Memphis if he can make three free throws. Students develop an understanding of the binomial formula by considering the probability that Memphis wins the game.
Determine whether the conditions for a binomial setting are met.
Calculate the mean and standard deviation of a binomial random variable. Interpret these values in context.
This lessons starts with a pop quiz for students. The catch is that there are no questions, and they simply have to guess the answer for every question. Students then use their prior knowledge of random variables to arrive at formulas for the mean and standard deviation for a binomial distribution.
Check the 10% condition to be able to assume independence of observations.
Check the Large Counts condition to us the Normal approximation to the binomial distribution.
Several years ago, Skittles changed the flavor of the green Skittle from lime to sour apple (big mistake!) but this has since been fixed. In this lesson, students arrive at two of the conditions that we will need later for inference: the 10% condition and the Large Counts condition.
And One Lesson to Teach Geometric Distributions
Calculate and interpret probabilities involving geometric random variables.
Calculate the mean and standard deviation of a geometric distribution. Interpret these values.
So everyone knows about the bottle flip challenge. But where did it come from? Answer: It was a viral video of a high school student who successfully performed the challenge at the high school talent show. What you don't know was that he did a lot of his practicing in his AP Stats class. Be sure to show this video at the end of the lesson. Thank you to all-star AP Stats teachers Kelly Pendleton and Alana Braland who designed this lesson at Stats Medic Summer Camp. It is absolute fire. Fun fact: the student in the video that went viral ACTUALLY WAS Kelly's student when the video went viral. This created a ton of buy-in from our students when we told them about our friend Kelly and her student. Follow Kelly and Alana on Instagram @srsly.statistics (best handle ever!).