If a random sample of size n is selected from a population with mean mu and standard deviation sigma, then the shape of the sampling distribution will be approximately normal if the population is approximately normal; for other populations, the sampling distribution becomes more normal as n increases.

1. Start Fathom
2. Drag a new Collection Icon to the Fathom window, as shown in Figure 1.
   
3. Double click the Collection Icon to open its Inspector.
   
4. Create an attribute named x, as shown in Figure 1.
5.  Double-click the formula cell of x to open the Formula Editor, also shown in Figure 1.
   
6. In the Formula Editor, enter the formula randomUniform(-5,5), as shown in Figure 1.

Our intent is that our population will be uniformly distributed on the interval (-5,5). That is why we enter randomUniform(-5,5) in the Formula Inspector as we have in Figure 1. Now, let's get some cases.

1. Close the Formula Editor by clicking OK. C
2. Close the Inspector as well.
   
3. Right click the Collection Icon and select New Case from the popup menu.
4. Enter 1000 in the New Cases Edit Box, as shown in Figure 2.
5. Select OK to close the New Cases Edit box.
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We now should have 1000 random numbers that are uniformly distributed between -5 and 5.

As you've learned in class, the uniform distribution is continuous, and any real number between -5 and 5 has an equally likely chance of being selected. You can see this for yourself.

1. Highlight the Collection Icon, then dragging a Case Table into the Fathom window, as shown in Figure 3.
   
2. Use the scroll bar in the Case Table to verify that there are 1000 random numbers uniformly distributed between -5 and 5.

Of course, a much nicer way of exhibiting the uniform distribution is with a histogram.

1. Delete the Case Table (we have no further need of it --- but don't delete the Collection.
   
2. Drag a Graph Object into the Fathom window.
   
3. Double-click the Collection Icon to open its inspector, as shown in Figure 4. You may have to rearrange objects in the Fathom window as we have so that one object does not cover another.
4. Select the Cases tab in the inspector (if it is not already selected) and drag the attribute x to the horizontal axis of the Graph Object.
5. Select Histogram from the drop down menu in the upper right corner of the Graph Object.

Select Rerandomize from the Collection Menu (Ctrl+Y) several times and note the uniform nature of our population. Note the horizontal axis of the graph. Note that the histogram runs from about -5 to 5, as it should.

We could change the vertical scale on the graph to a Density scale, then superimpose the uniform density function as we have on past activities, but that is not so important here. However, so that we work with an unchanging population,  so

• double-click the formula cell in the Inspector and then delete the formula as shown in Figure 5.
  

You're uniform distribution may look different from that in Figure 5. Again, this is not important, so don't worry about it and continue with the activity. It is not important that we have exactly the same numbers in our population, only that it is uniform on (-5,5).

We're gonna get a little fancy in this activity and caption each data with its value. To do this,

• select the Display tab in the Inspector for Collection 1 as shown in Figure 5a.

Note where it says "a case" for the value of caption. We'd like to change this caption to be the numerical value of the data selected from the population. This is easy to do.

1. Double-click the Formula cell for the caption. This will open the Formula Editor as shown in Figure 5b.
2. Now, enter x for the formula, as shown in Figure 5b.
3. Close the Formula Editor.

Note that the caption is now the actual value of x, as shown in Figure 5c.

Now,

1. Close the Inspector.
2. Right click the Collection Icon and select Sample Cases from the popup menu.
   
3. Expand Sample of Collection 1 with your mouse as we have in Figure 6. Note that each data in the sample is captioned with its value, due to the fact that we changed the caption in the population Inspector.
   
4. Double-click the Sample of Collection 1 with your mouse to open its Inspector, also shown in Figure 6.
   
5. Select the Sample tab in the Inspector, as shown in Figure 6, then enter 5 for the number of cases, as shown in Figure 6.
   
6. We want to see a live animation, so make sure the "Animation On" checkbox is checked.
   
7. We want to sample from the population with replacement, so make sure that the "With replacement" checkbox is checked.
   
8. Finally, every time we get a new sample of five cases, we want to replace the old sample with the new, so make sure that the checkbox "Replace existing cases" is also checked ("Empty this collection first" in Fathom 1)..

Your Sample of Collection 1 might contain a different set of numbers than what is showing in Figure 6. This is not a problem, so continue with the activity.

Next, we will create a dotplot of the data in the Sample of Collection 1.

1. Drag a Graph object into the Fathom window, as shown in Figure 7.
   
2. Click the Cases tab of the Inspector for the Sample of Collection 1 and drag the attribute x to the horizontal axis of the new Graph object, as shown in Figure 7.
   
3. Right-click the new Graph object, select Plot Value from the popup menu, then enter mean(x) in the Formula Editor to place the mean of the sample on the plot, also shown in Figure 7.

At this point, it is very important that you get a good grasp of what we've done thus far. In the Sample of Collection 1, click the Sample More Cases in the upper right corner. You should see a new sample of five numbers appear in the Sample of Collection 1. Now, note the mean of this particular sample in the dotplot of Sample of Collection 1. Click Sample More Cases again and note that you get a new sample of five numbers and a new mean.

Here are two very important points to consider.

1. We are sampling from a uniform distribution on the interval (-5,5). The mean of this population is clearly zero.
2. Every time we select a sample of five points from this collection, we get a different mean, usually not zero.

Collecting Sample Means

1. Select the Measures tab in the Inspector for the Sample of Collection 1, as shown in Figure 8.
   
2. Create a new attribute named xbar as shown in Figure 8.
3.  Double-click the formula cell of xbar to open the Formula Editor.
   
4. When the formula editor opens, enter mean(x), then close the Formula Editor.
   

You might try clicking Sample More Cases to see what happens. Each time you get a new sample of five numbers from the uniform population on (-5,5). The dotplot and mean of the sample is updated, as is the matching measure xbar in the Inspector for the Sample of Collection 1. Click Sample More Cases several more times and study the action. Let's continue.

1. Close the Inspector for the Sample of Collection 1.
2. Right-click the Sample of Collection 1 and select Collect Measures from the popup menu.
   
3. Use your mouse to expand the Measures from Sample of Collection 1 as we have in Figure 9. Double-click the Measures from Sample of Collection 1 to open its inspector, also shown in Figure 9.
   
4. We want to check "Animation on," as well as "Replace existing cases" ("Empty this collection first" in Fathom 1).
   
5. We want to collect a single measure, so enter 1 in the Cases edit box, as shown in the Inspector in Figure 9.

1. Click the the Display tab in the Inspector for the Measures from the Sample of Collection 1, as shown in Figure 10.
   
2. Double click the formula cell for the caption and enter the formula xbar, which is the attribute name we've given the measure (statistic) that we are collecting.
3. Close the Formula Editor.

Numbers might not match at the moment, but close the inspector and click Collect More Measures in the Measures from Sample of Collection 1. You'll get one measure (statistic) which is the mean of the sample of five numbers from the uniform distirubtion on (-5,5). See Figure 11 for an example.

• Click Collect More Measures several more times.
  

1. A new sample of five numbers is selected from the uniform distribution on (-5,5). These are shown in the Sample of Collection 1 box.
2. This new sample of five numbers is plotted in the Dot Plot and the mean of the sample is computed. In the case of Figure 11, the mean of the sample is 0.280887.
3. A measure (statistic) is collected from the Sample of Collection 1. In particular, the mean of the Sample of Collection 1 is collected and stored in the Measures from Sample of Collection 1. The caption of this measure is the value of the mean and should match the mean of the sample calculated in the Dot Plot.

It would be nice if we could collect a large number of "sample means" and collect them in the Measures from Sample of Collection 1 box. Then we could graph the collection of measures and make further important observations. This is exactly what we will do now.

1. Double click the Measures from Sample of Collection 1 box  to open its inspector, as shown in Figure 12.
   
2. Uncheck "Animation on" as it takes too long to collect a large number of measurses when the animation is "on."
   
3. Check "Replace existing cases" ("Empty this collection first" in Fathom 1).
4. Enter 100 in the Measures edit box, as shown in Figure 12.

Now,

1. Click Collect More Measures in the Measures from Sample of Collection 1 and 100 sample means will be collected.
   

1. Move the Inspector aside, as shown in Figure 13.
2. Drag a Graph Object into the vacant space. Click the Cases tab in the inspector, as shown in Figure 13.
   
3. Drag the attribute xbar onto the horizontal axis of the Graph Object.
   
4. Select Histogram from the upper right corner of the Graph Object.
   

Go ahead and

1. Close the Inspector.
2. Select the Histogram of the sample means (xbar).
3. With the histogram selected, select Scale from the Graph menu, then select Density to create a density scale on the vertical axis. This density scale is shown in Figure 14.
4. Next, right click the histogram of xbar and select Plot Function from the popup menu.
5. When the Formula Editor opens up, enter the formula normalDensity(x,mean(xbar),stdDev(xbar)) and close the Formula Editor. This superimposes the normal curve on the histogram as shown in Figure 14.

Click Collect More Measures several more times and study what happens. For example, note that the last sample of five appears in the Sample of Collection 1 box, and its mean appears in the Dot Plot, but where is this number in the Measures from Sample of Collection 1 box? That's right, it's at the end. Use the scroll bar to scroll down and note that the last measure matches the mean in the Dot Plot.
 

Increasing the Sample Size

1. As the sample size increases, the histogram of the sample means will become more and more normal in shape.
2. The histogram seems to be approximately centered at zero, which is the mean of the uniform population on (-5,5).
3. As the sample size increases, the histogram of the sample means will begin to narrow. Currently, the range of the histogram in Figure 14 is about 8 (from -4 to 4). Keep an eye on that range as you increase the sample size.

1. Double-click the Sample of Collection 1 to open its inspector.
2. Click the Sample tab, as shown in Figure 15, and increase the sample size to 10, also shown in Figure 15.

Now, to update your configuration,

• click Collect More Measures in the Measures from Sample of Collection 1 box.
  

What is important to notice?
 

1. The sample size is now 10. This can be seen in the Dot Plot at the bottom left of Figure 16. Note that there are now 10 dots. If you wish, you can also verity this by expanding the Sample of Collection 1 box, which will reveal 10 balls, reflective of the fact that we are now drawing 10 numbers from the uniform distribution on (-5,5).
2. The histogram of xbar (of sample means) seems to be "even more normal" in shape.
3. The histogram seems to be approximately centered at zero, which is the mean of the uniform population on (-5,5).
4. The range of the histogram has decreased. It now runs from about -3 to 3, so the range is about 6. Compare this with the range in Figure 14.

1. Double-click the Sample of Collection 1 box to open its inspector.
2. Increase the sample size to 25, as shown in Figure 17.

Next,

1. Close the Inspector.
2. Click Collect More Measures in the Measures of Sample from Collection 1 box.

You will get an updated Fathom window similar to that shown in Figure 18.

Again, it is important that you note the following:
 

1. The sample size is now 25. This is reflected by the Dot Plot of the sample at the lower left corner of the Fathom window. You can expand the Sample of Collection 1 box to verify that it now has 25 blue balls, signifying a sample of 25 has been selected from the uniform distribution on (-5,5). You'll note that all of the numbers in the Sample of Collection 1 box are between -5 and 5. This is also reflected in the Dot Plot.
2. The histogram of xbar appears to be "more normal in shape."
3. The histogram seems to be approximately centered at zero, which is the mean of the uniform population on (-5,5).
4. The range of the histogram has again decreased. Note that it runs now from about -2 to 2, so the range is about 4. Compare this with the earlier histograms in Figures 14 and 16.

Homework