As part of my effort to bring inferential statistics to the beginning of the semester I had students work through their first inferential project. This occurred on the sixth day of instruction.

Simulation for a Population Proportion

On the previous day we were going over different ways to display qualitative data (frequency tables, pie charts, bar charts). We created a pie chart for a sample of 60 students and it showed that two-thirds of those students were female. I asked students if the pie chart was strong evidence that more than half of all students at our college were female. They did not yet have the tools to make that decision, but they all felt it was strong enough to make that conclusion. I pointed out that the same pie chart would be generated from a sample of 3 students with 2 females and they understood that was not sufficient evidence. I told them we would investigate these ideas the following day.

Using a fair coin, P(H) = 0.5, we can simulate a sample of 60 students by flipping a coin 60 times and letting heads represent a female student. Using StatCrunch to simulate this experiment 1000 times, I only observed a sample containing 40 or more females only 3 times. This tells us that getting a sample containing at least 40 female students is very unlikely if the true percentage of female students at the school is actually 50%. That suggests that the true proportion of females at the college is higher than 0.5.

Project

Here is a link to a pdf of the project: Project 1 Qualitative Simulation

Part 1: I worked through the first investigation with the class. They were told that there was a claim that 40% of all college students own an iPhone. A random sample of 1000 students contained 360 that owned an iPhone. We began by building a StatCrunch coin flip simulation applet with P(Heads) = 0.40 (probability that a student owns an iPhone if the claim is true) that would flip 1000 coins (the sample size) and determine whether the number of heads (iPhone owners) was at most 360 (number of successes in the sample). Students then ran the simulation for 1000 trials.

Students observed that it was unusual to obtain a sample that contained at most 360 successes through simulation, and we discussed the fact that this is strong evidence that the population proportion is less than 0.40. I also had students find critical values that separated off the lower and upper 2.5% of randomly generated samples in the simulation. I explained how this approach can be used to weigh whether the population proportion is different than 40% when we don’t have an idea whether the population proportion is higher or lower than the claimed proportion. The critical values are also a great lead in to the concept of a confidence interval.

Part 2: This was similar to Part 1. I had students complete this on their own, conferring with partners as they went. This is another case where the sample would be considered an unusual observation, indicating that the population proportion is above 60%.

Part 3: During the week leading up to this I had students visually survey their classes to find the total number of females and students in their classes. I had students begin with a claim that 50% of the students at our school are female – which is not true. Most students noted that there sample was an unusual outcome in the simulation and concluded that the true population proportion is above 50%.

I then gave students the actual percentage (55%) for this semester, and they rebuilt the simulation applet with P(Heads) = 0.55. Many, but not all, of the students found that their sample was no longer an unusual outcome in the simulation and we discussed what that meant (we cannot conclude that the percentage of females at our school is different than 55%).

This gave us one more chance to discuss the concept of bias in relation to sampling.

Overall, a very effective day when it comes to helping students develop the big picture of inferential statistics. Feel free to use, or alter, my project. If you give it a try I’d love to hear how it went.

George