Data Matters with Fathom! Dynamic Statistics™ software

Activity 6.3

Section 6.3 makes some very precise claims about the means of random samples. It says that if you gathered a bunch of random samples, you would find that they were roughly normally distributed and that the standard deviation of the means would be equal to the standard deviation of the observations in the population divided by the square root of the sample size. This standard deviation of sample means is called the standard error.

Your task in this project is to test those claims.

These are the steps for this task.

1. Pick a variable to work with and find its mean and standard deviation in the population.
2. Take random samples, saving their means.
3. Find the mean and standard deviation of the means and get a histogram of the means to see whether these claims are correct.

Step 1: Pick a variable to work with and find its mean and standard deviation in the population.

Open Rep US Sample. Select Analyze, Estimate Parameters, Empty Estimate, Estimate Mean. Drag the variable you want to work with onto Attribute (continuous): <unassigned>.

Step 2: Take random samples, saving their means.

Select Analyze, Sample Cases. Inspect the sample case and set the sample size and Animation as you like. The sample size has to be greater than 1. Click on Measures and add a measure, sampleMean, with the formula mean([attribute]). (Replace [attribute] with the name of the attribute you are working with.)

Select Analyze, Collect Measures. Drag a case table onto the workspace so you can see the first sample mean. Drag a new Graph onto the workspace and drag sampleMean onto Drop an attribute here.

Inspect the Measures Collection to set the number of samples and Animation as you like.

Step 3: Find the mean and standard deviation of the means and get a histogram of the means to see whether these claims are correct.

Select Analyze, Estimate Parameters, Empty Estimate, Estimate Mean. Drag sampleMean onto Attribute (continuous): <unassigned>.

Are the sample means normally distributed? How does the mean of the sample means compare with the population mean? How does the standard deviation of the sample means compare with the standard deviation of the population observations?

Sort the sample means and check that 5% fall at least two standard errors from the population mean.

Try other sample sizes. Does sample size affect how well the equation for the standard error works? Is sample size related to the shape of sample distributions?

Try attributes that start with other distributions. Does the shape of the underlying distribution matter when there are small sample sizes? Does the shape of the population distribution matter when there are large sample sizes?