Building understanding of the Central Limit Theorem

Today I focused on building understanding of the Central Limit Theorem and how it can be used. I used a series of problems to help students work directly with sampling distributions and then moved to using the results of the central limit theorem.

To start things off, we had a data set of 5 basketball players and their heights. We looked at all possible samples of two individuals. Students found the mean height for each of the 10 samples. They calculated out the mean and standard deviation of their 10 samples and we compared that to the mean and standard deviation of the population. We also constructed a histogram of the sample mean distribution. We then compared these results to the theoretical model. In this case, the means aligned and the standard deviation was off slightly but it was close. As I explained to the class, models are useful to help explain behavior but they may not be accurate. We also calculated the probability of sample mean equaling the population mean and the probability of the sample mean being  within 1.0 inches of the sample mean.

Next, we looked at a couple of situations, discussed the population and variable of interest and then compared the sampling distribution models for two different sample sizes. This helped students to get comfortable with specifying the sampling distribution model we were using.

Finally, we looked at two problem situations that assumed specific population parameters and then asked what percent samples of a certain size would fall within given ranges. This basically gets back to finding z-scores and working with a normal model. Students still wanted to use the population standard deviation when determining probabilities, but with some reinforcement, most students seemed to understand why the sampling distribution had a different standard deviation.

Afterward, we went through a review activity that I use often and described in a previous post. The second mid-term is tomorrow, after which we will begin developing the concept of confidence intervals.