Data Matters with Fathom! Dynamic Statistics™ software

Activity 7.1

If you were thinking of doing a z-test of a sample mean, you might subtract that mean from the population mean you were interested in and divide by the standard error that you estimate from the sample’s standard deviation. Section 7.1 claims that what you would get is something that is not normally distributed, but distributed more like a narrow volcano.

In this project, you will test this claim. To start, you will take samples with two observations each. You’ll be taking the samples from a normal distribution with a mean of 0 and standard deviation of 1. For each of several hundred samples, you will calculate a statistic that is the sample mean divided by the standard error estimated from the sample standard deviation.

Drag a case table onto the workspace. Add an attribute, z. It will be drawn from what is called a standard normal distribution, which is a normal distribution with a mean of 0 and a standard deviation of 1. Draws from standard normal distributions are z scores.

Right-click on z and edit its formula to randomNormal(0,1). Right-click on z again and add two cases.

Press “Ctrl-I” to get the Collection Editor, then select Measures. Add a measure, t. The measure’s equation is mean(z)/stdError(z). StdError() is the standard deviation divided by the square root of the sample size.

If you would like, you could subtract the null hypothesis mean from the mean z, but the null hypothesis is that these observations are coming from a distribution with a mean at 0, so you would be subtracting 0. (Note that in this simulation, the null hypothesis is true.)

Add a second measure, Mean, that is the mean of z (formula: mean(z) ). The means will be normally distributed, and you will be able to compare t’s distribution to a normal distribution.

Select Analyze, Collect Measures. Drag a case table onto the workspace so you can see the first measures. Drag two graphs onto the workspace. Drag t onto one of the graphs and Mean onto the other.

Select the Measures Collection and press “Ctrl-I” to get the Collection Inspector. Set the sample size to something above 300 and collect more samples.

What do you think about Gosset’s claim discussed on page 392 in the text? Does it look as if you could do a regular z-test with those statistics?

Sort the statistics and check the value that is 2.5% up the list. With two observations, there is one degree of freedom, and Gosset claimed that 95% of the t-values would fall between –12.7 and 12.7. Did that work for your data? Try 4,000 samples. Do –12.7 and 12.7 work with 4,000 samples? Why does how many samples you take make a difference?

If you used 2 and –2 as your cutoffs for significance, how often would you have rejected the true null hypothesis?

Try other population standard deviations and other sample sizes. Do they make a difference?