Data Matters with Fathom! Dynamic Statistics™ software

Activity 9.3

Now that there is computer software to do all the computational steps of calculating a regression equation, very few people calculate regression equations by hand. You might wonder whether the calculations in Section 9.3 of Data Matters (or your calculations for the exercises) are correct. To double-check them, you can enter the data into Fathom and compare your answers with Fathom’s.

Once your data is entered in a case table, you get a regression model by dragging a new estimate onto the workspace. Select Empty Estimate, Linear Regression. Fathom calls the x-axis attribute the Independent attribute and the y-axis attribute the Dependent attribute. This is because when you create a scatter plot, the tradition is that if you have any suspicions about the direction of causation, you put the cause on the x-axis and the outcome on the y-axis.

Here are the steps for this part of the project.

1. Find the slope of a relationship between two numeric attributes in the population.
2. Take a sample.
3. Calculate the regression equation and standard error.
4. Check and record whether the confidence interval based on the sample’s slope included the population’s slope.

Repeat the steps to check a number of confidence intervals for the population’s slope.

Step 1: Find the slope of a relationship between two numeric attributes in the population.

Open RepUSSampleMarch2001.ftm. Drag a new estimate onto the workspace, select linear regression, and drag the attributes into the Estimate Parameters window. Record the slope in the population.

Step 2: Take a sample.

Click on the March 2001 collection box. Go to Analyze and select Sample Cases. Create a case table for your sample.

Step 3: Calculate the regression equation and standard error.

Drag the attributes from the sample into the Estimate Parameters window.

Step 4: Check and record whether the confidence interval based on the sample’s slope included the population’s slope.

Rather than reporting the standard error of the slope, Fathom reports the confidence interval for the slope.

Repeat these steps to gather more slopes. How often did your confidence interval include the population’s slope?

To get a larger number of samples, click on the Estimate Parameters window, then select Analyze, Collect Measures. Drag a case table onto the workspace to see the statistics from each sample. The margin of error of the slope is labeled SlopeHalfWidth. Inspect the Measures Collection to set the number of measures.

You can add attributes to the Measures Collection by clicking on <new>. You might want to add the top and bottom of the 95% confidence interval. Then you might want to add a marker to indicate whether the population slope was inside the confidence interval. For example, if the population slope were 22, then you could use a formula like (22 > slope – slopeHalfWidth)and(22 < slope + slopeHalfWidth).

That formula will be true only if the population slope is inside the confidence interval.

Explore the slopes and the confidence intervals with estimate parameters.

How does the standard deviation of the samples’ slopes compare with the estimated standard error of the slope that you got in your first sample?