Central Limit Theorem
Day 70 - Lesson 7.3
Explain how the shape of the sampling distribution of a sample mean is affected by the shape of the population distribution and the sample size.
If appropriate, use a Normal distribution to calculate probabilities involving a sample mean.
Activity: Who has Better ACT Scores?
First off, you have to understand that Rockford is our rivalry school — in sports and in academics. So there is a good joke buried in this Activity and students that are sharp will really appreciate it. Consider downloading the Word document and editing in your rivalry school.
Here is the applet that students will be using for this activity (click begin). There are also two activities in Section 7.3 of The Practice of Statistics that use this applet. We have created some context for the applet (ACT scores) which is not really the intention of the applet, but students always learn better when there is a context…so go with it.
Population distribution, distribution of a sample, or a sampling distribution?
While the distinction between these three distributions is a Learning Target from Lesson 7.1, this applet provides a concrete visual representation of the differences. Don’t forget to ask “What does this blue block represent?“
Is 30 a magic number?
Of course not! The sample needed to make the sampling distribution “approximately” Normal depends on two factors. First, how crazy is the population distribution? The crazier the distribution, the larger the sample size needed. Second, what makes a distribution “approximately” Normal? We want to be able to use a Normal distribution to do probability calculations in order to estimate a true probability. The more Normal the sampling distribution, the closer our estimated probability will be to reality. So 30 is not a magic number, but one that we can use to help us in our instruction (and for the AP Exam rubrics!).
In reality, there were a small group of statisticians 300 years ago that met on Tuesday nights at Buffalo Wild Wings. At these meetings they would discuss important statistical concepts. At one of their meetings (over an exceptional batch of buffalo dry rub wings), they had a heated debate about the sample size necessary for Normality. In the end, they settled on 30 as the magic number.