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The second part of the Central Limit Theorem is:

The sampling distribution of the mean is normally distributed, with standard deviation equal to the population standard deviation (often denoted as the greek letter, sigma) divided by the square root of the sample size (often denoted as n):

σn\frac{\sigma}{\sqrt{n}}

The standard deviation of a sampling distribution is also known as the standard error of the estimate of the mean. In many instances, we cannot know the population standard deviation, so we estimate the standard error using the sample standard deviation:

standard deviation of our samplesample size\frac{standard\ deviation\ of\ our\ sample}{\sqrt{\text{sample size}}}

Two important things to note about this formula is that:

  • As sample size increases, the standard error will decrease.
  • As the population standard deviation increases, so will the standard error.

Instructions

1.

In the workspace, you can see a population distribution and a sampling distribution (scroll down to see the sampling distribution). Right now, the sample size is set to 10.

Increase the sample size to 50 and note the change in the shape of the sampling distribution.

A smaller standard error means that the distribution will be taller & skinnier. Is that the case?

Remember to scroll down to see the 2nd plot after you hit run.

2.

Now increase the standard deviation of the population to 30.

This means that the population distribution will have more variation (and will therefore appear wider and flatter). The sampling distribution will also become wider and flatter because the standard error will increase (due to the larger numerator).

Does what you see line up with what you expect?

3.

Play around with the two variables samp_size and population_std_dev some more.

Keep in mind that:

  • As sample size increases, the standard error will decrease.
  • As the population standard deviation increases, so will the standard error.

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