Data Matters with Fathom! Dynamic Statistics™ software

Activity 4.3

Section 4.3 makes a set of claims about p-values of chi-square statistics. They are summarized in Table 4.3.3 from Data Matters.

Your task in this project is to check those critical values. You will do that by setting up a workspace in which the null hypothesis is true, then collecting measures from random samples, saving the chi-square values. You can then sort the chi-square values and make sure that only 5% are above each critical value.

One Degree of Freedom

Here is how to check the critical value for df of 1.

To have one degree of freedom, we have to have two rows and two columns.

Drag a case table onto the workspace and add two attributes, Row and Column. Right-click on each attribute to edit its formula, setting their formulas to floor(2random()).

Here’s how that formula works: random() produces random numbers from 0 to just under 1. When those numbers are multiplied by 2, you get numbers ranging from 0 to 1.999999 that floor then rounds up to 1 or down to 0.

With this equation, the 0’s and 1’s are equally likely. With floor(1.5random()), 0’s are twice as likely as 1’s. If you would like, you could play with the relative probabilities to see if it made a difference.

Right-click on an attribute and add cases. You may use as many cases as you like, as long as all of your expected counts are 5 or more.

Take measures off a Test Hypothesis table. Select Analyze, Test Hypothesis. A Test Hypothesis table appears. Select Empty Test, Test of Independence. Drag Column onto First attribute: <unassigned> and drag Row onto Second attribute: <unassigned>.

Now select Analyze, Collect Measures. Drag a case table onto the workspace to see what measures get collected. The first column is our chi-square statistic!

To collect more samples, click on the Measures Collection and press “Ctrl-I” to get the Collection Inspector. Set the number of samples to something reasonably high. I like a number above 400, but you can try out different numbers of samples and think about which seems to work well (and why!).

To check the cutoffs in Table 4.3.1, sort the chi-square values by right-clicking on them and selecting Sort Descending. You can then find the chi-square values that were higher than 95% of the chi-square values.

How did the table do? Is 3.84 a good cutoff?

For Degrees of Freedom Over One

To work with cross-tabs with more rows and/or more columns, go back to the formulas for Row and Column. Floor(3random()) will produce three categories, Floor(4random()) will produce four categories, and so on.

Check all five rows of Table 4.3.3. Find the critical cutoff for six degrees of freedom.

Try different cross-tab designs that have the same degrees of freedom, like a three-column-by-three-row design and a two-row-by-five-column design. Do the cutoffs stay in the same place? Do the chi-square values really have the same distribution when the null hypothesis is true?

Check whether sample size matters.

Check whether it is really true that the expected counts must be at least 5. You can do this by using a two-column-by-two-row design with 19 observations in each sample.

In the previous section, you found the chi-square values for tests of correlations in the U.S. population. You were working with the representative U.S. sample that is available from the Data Matters Web site at www.keycollege.com/dm. Now get the p-values and report those tests in two ways: once in the style of the popular press and once in the complete style of the sciences.