ࡱ> UWT}[@ &bjbj44 RViVi.<<<P8888$\$P=h(YYY?=A=A=A=A=A=A=$(?RzAe=<SYSSe= z=S|<?=S?=2!:^<< paE.V8K;?==0=e;AoFA4<PPA<<XYZ@4',YYYe=e=PP TPPTSome Fathom Tools for Investigating Statistical Concepts Robin H. Lock, Burry Professor of Statistics, St. Lawrence University, Canton, NY 13517  HYPERLINK "mailto:rlock@stlawu.edu" rlock@stlawu.edu Three tools available in Fathom are particularly well-suited for encouraging students to explore statistical ideas. We show some examples of each of the following. Interactive linked graphics: Adjustments made by click-and-drag are reflected in multiple analyses. Sliders: Allow student to control one or more quantities to observe the effects in plots or analyses. Collections from collections: Easily collect summary results from simulations, re-samplings, or randomizations. ****************************************************************************** Airfare Activity: This activity will develop the idea of a least squares line for predicting the cost of air travel from Syracuse to various cities, based on the distance to those cities. Heres a brief summary of the steps for the activity. Whats a least squares line? : Start with a Fathom file with the airfare data. Produce a scatterplot with distances on the horizontal axis and costs on the vertical axis. Put a movable line on the scatterplot and drag it to adjust the slope and intercept to give a reasonable summary of the trend of the data. Select Show Squares to visually display the squared deviations from the data points to your line and compute the sum of the squared errors. Try to adjust the line further to make the sum of squared errors as small as possible. All calculations and plots are updated dynamically as the line is moved. Choose the Least Squares Line option to add the line which minimizes the sum of squared errors. Hopefully, it will be close to your best line. What does the r2 value next to the regression line mean? Suppose that you could not use the Distance variable to help predict airfare. Turn off the least squares line and adjust the movable line to be as close to horizontal as you can get it. This gives the same prediction regardless of the distance. Adjust the line up and down (but dont change the slope) to again minimize the sum of the squared errors. (Note: the best value should be the sample mean of the airfares.) Now put the least squares line back on. Note that the sum of squared errors for the least squares line should be better (smaller) than the horizontal line. How much better? Compute the percentage improvement, by dividing the difference between the two sums of squares (i.e. the amount of improvement) by the larger value. The result (up to roundoff differences) is  EMBED Equation.3 . Activities can use ready-made Fathom documents where students just manipulate one or more parameters as in the two examples below.  Activity: Tank Estimation Comparing Competing Estimators Setting: The opposing army has numbered all of its tanks consecutively from 1 to (, where ( is the total number of tanks. We would like to estimate ( using a sample of ID numbers from captured tanks (assuming that the captured tanks are a random sample of all the tanks). Basic Idea: We'll create four collections in Fathom: Collection #1: The original population of tanks- a single column with ID's from 1 to(. Collection #2: A sample of size n from the original collection of tank ID's. Collection #3: A collection of estimates (measures) computed from repeated samples. Collection #4: A stacked version of collection #3 with all the estimates in a single column. 1. Open the Fathom file Tanks.ftm. The original Tanks collection contains tank ID's numbered consecutively from 1 to (. 2. (a) Create a new collection with a sample from the original Tanks collection. Click on the Tank collection box to be sure it's highlighted, then from the Fathom menus, choose Analyze> Sample Cases. This creates a new collection (Sample of Tanks) that will have a sample of ten tank ID's. Expand the collection box to see which tanks were "captured" in your sample. Double-click on the Sample of Tanks collection box to bring up its Collection Inspector. This allows you to adjust the parameters of the sample and collection. The "Sample" tab in the Inspector should be active, click to toggle OFF the "Animation on" and "With replacement" options (you can't capture the same tank twice!). Adjust the sample size to 5. 2(b) Enter formulas for the estimators you want to compare as Fathom measures. Click on the "Measures" tab of the Sample of Tanks Collection Inspector. Click where it says and enter a name for the first estimator (e.g. Theta1). Double click in the empty box under "Formula" for the first measure. This brings up Fathom's formula editor. Type in the formula for your first estimator. Click OK to close the editor and see the value of your estimate in the Inspector. Repeat the process to name and enter formulas for additional estimators. Check that the estimates are reasonable. Click to select (highlight) the Sample of Tanks collection box and hit Ctrl-Y again to draw a new sample. Watch how the estimates in the Inspector change as you get samples. 3. Create a new collection to save the estimates from repeated simulated samples. With the Sample of Tanks collection selected, choose Analyze>Collect Measures from the Fathom menus. This creates a third collection (Measures from Sample of Tanks) that contains (by default) the estimates for five samples. Double-click on this new collection box to bring up its inspector, click to turn the animation off and change the number of measures from 5 to 500. Click on "Collect More Measures" to simulate and store the results for 500 samples (be patient). If you want to look at some of the results, click the "Cases" tab in the Measures from... inspector and use the arrows at the bottom to scroll through cases. 4. (a) Create a fourth collection to display the comparative results. With the Measures from... collection box selected, choose Analyze>Stack Attributes from the Fathom menus. This creates a new (and final) collection named Stacked Measures from Sample of Tanks. Double click on the new collection box to bring up its inspector. You should see just two attributes, one named Group (containing the names of the estimators) and the other called Value (containing the values of the estimators for all the samples). 4. (b) Create a plot to compare the estimators Drag down an empty plot (scatterplot icon) and drop it in an open area on the screen. Drag the Value attribute from the inspector and drop it on the vertical axis of the plot. This should create a fairly messy dotplot of all the data. Click on the word "Dotplot" in the corner of the graph and change it to a histogram. Drag the group variable from the inspector and drop it on the horizontal axis of the graph. This produces separate histograms for each of the estimators. Look at the original Tank collection to find the actual number of tanks. Use Graph>Plot Value to show this value of the plot. 4. (c) Compute summary statistics to compare the estimators. Drag a blank summary table (icon with an "S") to an empty spot on the screen. Drag the Value attribute from the Stacked collection inspector to the "row" slot in the summary table (under the downward arrow). The mean for all the estimators (combined) should appear. Choose Summary>Add Formula to bring up the formula editor and enter the formula s(). Drag the Group attribute to the "column" slot near the top of the summary table. Drop it there to show the summary statistics (mean and std. dev.) for each estimator separately. You will probably need to stretch the summary table to view the results for all estimators.   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