Tootsie Pops and Hand Span Activity

Tootsie Pops and Hand Span Activity

Making a Collection and Entering Data

Enter the data you collected into Fathom.

Open Fathom by double clicking on the shortcut or alias to Fathom on the desktop.

Click on the middle icon in the upper right hand corner of your screen to maximize your work area.

Making a Collection and Entering Data

We have data from our activity in the form (hand span in centimeters, number of tootsie pops). Given just these measurements, how well can we predict the number of tootsie pops a person could pick up for other hand spans?

Across the top of the Fathom window is a shelf with tools for making objects.

Drag a case table from the shelf into the document.

When you click, your mouse pointer becomes a closed fist. As you move the mouse over the document, you see a frame that shows you where the case table will be placed when you release the mouse.

You should get an empty table similar to the one shown at right.

Click once on <new>.

Type Hand_Span for the first attribute and press Enter.

When you press enter, a box representing an empty collection appears next to the table.

Make the table wider by dragging on its right edge. This will give us room to see all of our attributes at once.

Make another attribute Number_of_Tootsie_Pops for the number of tootsie pops the person picked up.

Double-click on the label Collection 1 in the title of either the table or the collection. This should bring up a dialog for renaming the collection.

Type Tootsie Pop Activity and click OK.

Entering Data

Now we are ready to enter the values for hand spans and number of tootsie pops from our class activity.

Click in the empty cell below Hand_Span.

Enter your first hand span and then press Tab to enter the number of tootsie pops. Hit Tab to enter the next hand span.

Enter the rest of your data. Use Tab to move to the next cell. At the end of each case, Tab will take you to the first cell in the next case.

Create a folder with your name on it in My Documents folder. Save your document to your folder with the name tootsie pops.

Undo and Redo

Fathom records all changes that you make to your document and allows you to undo and redo them.

Click the Edit menu.

The first item in the Edit menu shows you what you can undo. If the last change you made was to type in a value, the undo command is Undo Value Change.

Choose Undo from the Edit menu several times.

Each time you choose Undo, Fathom takes you one step further back. You can continue this until you get to the state of the document when you opened (or created) it. The shortcut key is Ctrl+z.

Choose Redo from the Edit (ctrl r) menu.

Redo is the inverse of Undo. You can tell from the label in the Edit menu what it is you are about to redo.

Make sure you redo all of your data back before continuing.

Graphing Number_of_Tootsie_Pops Versus Hand_Span (cm)

Now let’s look at the relationship between the two attributes, Number of tootsie pops and Hand span (cm).

Drag the graph icon from the shelf to an empty area in the document.

Drag the attribute Hand_Span (from your case table) to the horizontal axis of the graph over the spot labeled Drop an attribute here. As you move the mouse over the x-axis, a black border appears, showing that you can release there.

Drag the attribute Number_of_Tootsie_Pops and drop it on the vertical axis of the graph.

We normally describe a scatterplot by noting its

direction (positive/negative),
form (linear/curved) and
strength (degree of the linear association between the variables).

Draw a rough sketch of your scatterplot below and write a sentence to describe your scatterplot (make sure to note the direction, form and strength).

Switcherro. Would the description of your scatterplot change if you plotted Number_of _Tootsie_Pops on the x-axis and Hand_Span on the y-axis?

Place your cursor on the label Number_of _Tootsie_Pops on the y-axis and drag it to the horizontal axis. The attribute Hand_Span will automatically appear on the y-axis.

Describe any changes in the form, direction or strength of the relationship.

Put hand span back on the horizontal axis and number of tootsie pops on the vertical axis.

Changing Units. How would your description of your scatterplot change if you measured the hand span in inches instead of cm? Write your prediction below.

Create a new attribute called inches by clicking in the box labeled <new>. Type in the name and hit enter.
Click on the attribute inches to select the column.

To create a formula to change centimeters to inches, place your mouse over the attribute inches and right click to bring up the formula editor (alternately, with the inches column selected, you could also go to the Edit menu and choose Edit Formula).

The formula editor will open on your screen. We want to type the formula hand span * 2.54. You can click on the + sign next to Attributes to see the attributes you have defined. Double click on Hand_Span to paste it in the formula editor. Click on the multiplication symbol on the calculator or on your keyboard and type in the value 2.54. Click OK to close the formula editor.

Create a new scatterplot of number of tootsie pops v. inches and describe any changes to the form, direction or strength of the association between the variables.

Note: The correlation coefficient (r) is a value that quantifies the strength and direction of a linear association of the variables in a set of bivariate data. The value of the correlation coefficient is found using the formula .

Examine the formula for the correlation coefficient r. Why doesn’t a change in units (for example, cm to inches) affect the form, direction or strength of the relationship?

Why was the form, direction and strength of the relationship not affected whether we plotted Hand_Span v. Number_of_Tootsie_Pops or Number_of_Tootsie_Pops v. Hand_Span?

Rescaling Axes and the Help System

It is easy to rescale the axes in Fathom. Suppose I need to look at higher values on my x-axis then what is presently shown on the graph. Position your mouse on the upper values on the x-axis. Your mouse should turn into a white hand shape. Drag the axis to the left to see higher values on the x-axis (if you want lower values, position your mouse on the left-hand side of the axis and drag to the right). You can do the same thing on the y-axis. Try it! Don’t forget to Edit Undo (Ctrl Z) before you continue.

Dragging Data

You can drag data in a graph to change it, making it easy to observe the effect of changes in the data on graphs and analyses.

Before you start, make sure you have the attribute number of tootsie pops on the y-axis and Hand span on the x-axis, and that you have saved all of your data[RC1].

Note: You can delete objects you no longer need by clicking on the object to select it and pressing the delete key. You can hide an object by selecting it and going to the Object menu and choosing the Hide command.

We’re ready to observe the effect of changing data values. Move the mouse near the point you want to move so that the tip of the arrow changes from a white northwest pointer to a black west pointer . This is your clue that you are positioned to drag.

Notice also that the status bar at the bottom left of the Fathom window shows the coordinates of the case.

Drag the data point. As you drag, you should see the values for the two attributes changing in the case table.

Choose Undo Drag from the Edit menu. The point returns to its original position.

You can set data from being accidentally changed by selecting the collection (box of gold balls) and going to the Collection menu and choosing Prevent Changing Values in Graphs. Make sure this command is unchecked before continuing with the activity.

Fitting a Movable Line

Recall that our goal is to be able to predict the number of tootsie pops collected from the person’s hand span (cm). Do the points in your scatterplot appear to follow a linear pattern?

Select the graph by clicking on it once. This activates the Graph menu (If the Graph menu does not appear, make sure that your graph is selected). Choose Movable Line from the Graph menu.

You can also right click (Windows) Control-click (Mac) on the graph to bring up a menu with commands that apply to the graph.

The line that appears in the graph is not a fitted line. You can change its slope and intercept by dragging it. Dragging on the end of the line causes it to rotate around the other end. Dragging the middle of the line moves it parallel to itself. The cursor changes shape to suggest what will happen when you drag. Notice that the equation of the line shown below the graph updates as you drag.

Experiment with dragging the line. Position it so that it appears to give a good fit to the data.

While eyeballing the fit through the data points is sometimes sufficient, we often need some criterion for best fit. A commonly used criterion is least-squares fit. You can see how least squares works.

With the graph selected, choose Show Squares from the Graph menu.

The graph now shows a square constructed from each point to the movable line; a square whose length is equal to the difference between the actual and predicted value for the point is called the residual.

Residual = Actual Value – Predicted Value

Experiment with dragging data points. Notice that the squares change as you drag, but the line does not move. (Be sure to use Undo to put the points back where they started from before proceeding.)

Experiment with dragging the line. Notice that the squares change and that the sum of the squares reported below the graph changes.

Adjust the line so that the sum of the squares of the residuals is approximately at a minimum.

The line that satisfies this criterion is called the least-squares regression line. It is the line that makes the sum of the squared residuals as small as possible. Fathom can compute this line.

With the graph selected choose Least Squares Line from the Graph menu.

How closely did you manage to adjust the movable line to match the least-squares regression line? Write your regression equation below. Explain what the slope and the y-intercept mean in the context of this activity.

Making a Prediction – Looking at Residuals

Suppose that you have been selected to represent GA in the Tootsie Pop Grabbing Event at an upcoming competition. With the graph selected, Choose Remove Movable Line from the Graph menu. The movable line disappears.

Move the mouse pointer along the least-squares line, noting that the coordinates of the tip of the arrow are reported in the status bar in the lower left corner of the window. When the x-coordinate of the mouse pointer is at your hand span, you can read off the predicted number of tootsie pops.

This prediction was probably a bit off. Let’s see how much off we might expect it to be.

With the graph selected, choose Make Residual Plot from the Graph menu.

A plot of the residuals appears below the main scatterplot. Corresponding to each point in the original graph is another point below whose y-value is the difference between the predicted value and the actual value (its vertical distance from the line).

Drag one of the points in the top graph. Notice how its residual changes in the residual plot. Notice also that, since the least-squares line is changing in response to dragging the point, the other residual points are changing as well.

Choose Undo (ctrl Z) to return the data to their original values.

Notice the scale of the residual plot. It tells you how far from the line your data fall. Write your prediction, giving a “margin of error” (number of tootsie pops ± error).

The Influence of an Outlier

Remove the residual plot by going to the graph menu and selecting it so that it is no longer checked.

Let’s play around a bit with the idea that one data point may, or may not, have a big effect on the regression line. First, we need to fix our regression line so that we can use it as a point of reference.

To do this, right-click on the graph (or, with the graph selected, go to the Graph menu) and choose Plot Function. Type in the equation of your least-squares regression line (in Fathom you only type in the values to the right of the = sign (see graphic below). You may have to reposition the formula editor so that you can see your equation. When you are done, click OK to close the formula editor.

What you’ve done is created an independent function with the same equation as your least-squares regression line, but one that is not defined by your data (you’ll be able to see it once you move a data point). When you drag a data point, your least-squares regression line will change, but your plotted function will remain fixed, giving you a reference for how much an individual data point effects your least-squares regression line.

We need a little room to create an outlier. Manipulate the axes by dragging the upper and lower ends of each axis towards the middle. You should have something similar to the graph shown at right. (The bounds of the axes have changed.)

Drag the rightmost data point far from the line. Notice how wildly the slope and intercept of the regression line can change in response to changes in just one of the data points. A least-squares regression line is quite sensitive to outliers, making it especially important that you look at your data before reporting a least-squares slope or intercept.

Return the data point to its original value. You can rescale the graph by reselecting scatterplot (click on the triangle in the upper-right hand corner of the graph).

Record the value from your least-squares regression line. This value is called the coefficient of determination. It tells us the percent of variation in our response variable (y) that can be explained or attributed to our explanatory variable (x). Write a sentence, in context (always;-), stating the correlation coefficient for your data and its interpretation.

Ideas for Further Exploration

The Graph menu has a command to show a median-median line. Investigate this line’s behavior when data is dragged and compare it to the behavior of the least-squares regression line.

To find out more about the median-median line, go to the Fathom Help Menu. Click on the Search tab and type in median-median line

[RC1]1Amen!