Data Matters with Fathom! Dynamic Statistics™ software

Activity 2.2

Section 2.2 claims that if you create random samples from some population and create a histogram of the proportions in the samples, the histograms will show shapes that are roughly bell-shaped. The bell-shaped histograms will have centers at roughly the probabilities or population proportions involved in generating the samples’ proportions. In this project, you will see for yourself whether that is true.

The project in Section 2.2 requires these three steps.

1. Set the probability where you would like it, set the sample size where you would like it, and use the random number generator to simulate taking samples.
2. Record the proportions that appear in the samples and make a histogram of those proportions. (Check whether the histograms are roughly bell-shaped with centers at the probability that you chose.)
3. Repeat the process with larger samples. You’ll notice that your histograms are narrower with larger samples. That’s the law of large numbers.

Here’s how to do each step.

Step 1: Set the probability where you would like it, set the sample size where you would like it, and use the random number generator to simulate taking samples.

In Fathom, you’re going to set the probability by creating different kinds of populations. You will be using random sampling with replacement. To get a 50% sample, you could use a population of two people and call them Yes and No. Because you are using With replacement, you are drawing from the same pair of people on each draw, and the chance of drawing each person is equal. The chance of Yes is 50%, and the chance of No is 50%.

If you would like to work with a 10% chance, make a population of 10 people: 1 Yes and 9 No. That will give you a 10% chance of Yes.

To create your population, drag a case table onto the workspace. Click on <new> and enter a name for your attribute: Person . Click on the white rectangle below the name and start typing in Yes and No to set the proportion you would like to work with.

As you were setting the name of the attribute, a collection appeared. That is the collection of the case table you created. Click on that collection and select Analyze then Sample Cases. A new collection appears called something like Sample of Collection1. It is a sample collection.

The sample collection will hold each sample that you draw during this simulation. To see the sample’s first sample, click on the sample collection to select it, then drag a case table onto the workspace. The case table will show you what is in the sample collection. It starts with a default sample size of 10.

To set the sample size, click on the sample collection, then select Edit, Inspect Collection.

In the box that pops up, there is a white rectangle next to cases. That will hold the sample size you want to work with. Type your sample size into that box, then click on Sample More Cases to take a sample. You will see the contents of the sample collection changing.

We want to calculate the proportions in the sample. To get started, click on the Measures tab at the top of the Inspect Collection box. (The Measures tab may be partially hidden, showing only Mea….)

When you can see the Measures sheet of the Inspect Collection box, click on <new> and enter a name for the proportions: Proportion .

Right-click the name you typed in and select Edit Formula. In the formula box, type in Proportion(person=”Yes”) and click OK. The Inspect Collection box will now show the proportion of the current sample that is Yes.

You can go back to the Sample tab and click on Sample More Cases, then go back to the Measures tab to see that the proportion has changed.

Step 2: Record the proportions that appear in the samples and make a histogram of those proportions. (Check whether the histograms are roughly bell-shaped with centers at the probability that you chose.)

To record the proportions, you will use a new collection and create a Measures Collection. To create the Measures Collection, click on the Sample Collection, then select Analyze, Collect Measures. Creating the Measures Collection will prompt the Sample Collection to sample cases five times.

The default for a Measures Collection is to gather the measures from five random samples. To see the measures that the Measures Collection has pulled, select the Measures Collection and drag a case table onto the workspace. To see how those numbers are distributed, pull a graph onto the workspace and drag the measure name (Proportion) from the Measure Collection’s case table to Drop an attribute here in the graph. Click on Dot Plot and select Histogram.

There’s the histogram. At this point, you are welcome to say, “Well, that’s not a bell shape.” Yours might be a little bell-shaped, but I doubt it. To see the bell shape, you will have to use more than five samples.

To change the number of samples, click on the Measures Collection, then select Edit, Inspect Collection. Type in some number—at least 50—then click on Collect More Measures.

Now you can see a bell-shaped histogram, I bet. Jot down for yourself how wide the histogram is. What proportion is at the bottom (left) of the histogram? What proportion is at the top?

Step 3: Repeat the process with larger samples. You’ll notice that your histograms are narrower with larger samples. That’s the law of large numbers.

To change the sample size, you have to go back to the Inspect Collection box for the Sample Collection. If that Inspect Collection box is no longer showing, click on the Sample Collection, then select Edit, Inspect Collection. Type a larger number in the white rectangle. Then go back to the Measures Collection’s Inspect Collection box, rearrange the boxes so you can see the histogram, and click on Collect More Measures.

Now how wide is the histogram? You used a larger sample size. Was the histogram narrower?