The Central Limit Theorem Preview CT-STEM

Reflection

The Central Limit Theorem states that sample means from any population accumulate in a distribution that approaches a normal curve, as long as the sample size is "large enough". Our textbooks define "large enough" as \(n \ge 30\). This means that in order to produce a sampling distribution that is approximately Normal, we must sample at least 30 individuals from the population (if the population distribution shape is unknown or non-Normal). If the population distribution is Normal, the sampling distribution of \(\bar x\) will also be Normal, no matter what the sample size \(n\) is.

Questions

Please answer the questions below.

Question 3.1

Mr. Mills takes a sample of only 10 people and records their score on a particular IQ test. He is confident that he can make inferences about this sample using a Normal approximation. Why can he do this, even though his sample size was less than 30?

Question 3.2

In real life, we usually don't know what the population distribution looks like. Why can we make inferences about the population mean based on a large sample size?

Question 3.3

Explain how this physical model (see link below, called a Galton Board) can be used to describe the Central Limit Theorem. Click on link below to view a GIF of the Galton Board in action.

Galton Board

Question 3.4

Are there any other mathematical topics that you can think of when looking at the Galton Board?

Notes

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