Next, we looked at the idea of sampling distributions and the Central Limit Theorem (CLT). This took about 45 minutes to go through and hopefully established some foundation of the concept and form of sampling distribution models. Next class we'll work through some problems and then review for the upcoming mid-term.
Below is the outline of the lesson with italicized comments enclosed in square brackets [like this].
—Sample
Distribution of Sample Mean
·
Take random samples of 3 students and calculate
mean height [Took 7 random samples and compared means to the class mean.]
o
What happens? [Did any sample have a mean the same as the class? Were the means all the same?]
o
Sampling error – each sample yields a different
mean [Each sample differs, this is sample variability.]
·
What would you expect to happen if took larger
samples? [Students thought spread would decrease; continue to increase sample until sample entire population then how much variation-none!]
o
Mean stays the same but the spread decreases
·
What will the mean of the means be? [Mean will be close to population mean, more samples of larger size provides even closer results.]
o
Population mean
·
What will the standard deviation be?
o
Population standard deviation / sqrt(n) [Students realize variability decreases, just now how.]
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What is distribution of the sample means?
o
Create a histogram [Created histogram of the seven sample means, unimodal, slightly skewed but symmetric enough-call this nearly normal. A normal model can be useful in helping to understand behavior.]
·
CLT
o
Discuss mean and standard deviation of sampling
distribution
·
A look at sampling distributions [Showed different sample populations, graphs of large number of samples for various sizes; all look normally distributed but the standard deviation gets smaller.]
·
Calculate for siblings
o
Use sample size of 5 then 10 then 20 to
demonstrate law of diminishing returns [Started by having students calculate population (the class) mean and standard deviation. Showed that the sampling distribution model would be of the form N( mean, std dev / sqrt(sample size) ). Next asked students to calculate the sampling distribution standard deviation for a sample size of 5, then 10, then 20. Showed that doubling size each time has less and less effect on decreasing the size of the sampling model standard deviation. This is the law of diminishing returns. Students wondered why the standard deviation of the sampling model was so much smaller than the standard deviation of the population. It was a good opportunity to remind students we are looking at the distribution of means calculated from drawn samples. It turns out that the standard deviation of this sampling model is connected to the standard deviation of the population but as the samples we select increase in size the variation we see in the means of the samples decrease until we sample the entire population. At this point every sample will be the same and there will be no sample variation in the sample means.]
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