Have you ever wondered how many Starburst you could pick up with one hand? If you had a bigger hand, might you be able to pick up even more candy? Have you ever envied the bigger kids at Halloween? In fact, did you ever think you might be able to predict how much candy a person could pick up?
Our goal with this project is to investigate the relationship between the size of a person’s hand and how many Starburst that person can pick up. If our model is good enough, we can predict the number of Starburst someone can pick up based on his or her hand span.
Graphing Number_of_Starburst VersusHand_Span (cm)
Now let’s look at the relationship between the two attributes, Number of Starburst and Hand span (cm).
- Drag the graph icon from the shelf to an empty area in the document.
- Drag the attributeHand_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 attributeNumber_of_Starburst 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 to the right and write a sentence to
describe your scatterplot (make sure to note the direction, form and strength).
- Switcherroo. Would the description of your scatterplot change if you plotted Number_of _Starburst on the x-axis and Hand_Spanon the y-axis?
Place your cursor on the label Number_of _Starburston 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_Spanback on the horizontal axis and Number_of _Starburston the vertical axis.
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 Starburst on the y-axis and Hand spanon the x-axis, and that you have saved all of your data.
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 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.
Fitting a Movable Line
Recall that our goal is to be able to predict the number of Starburstcollected 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) 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 VA in the Tootsie Pop Grabbing Event at an upcoming competition. With the graph selected, Choose RemoveMovable 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 Starburst.
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 Starburst ± error).
Influential Points
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 theGraph menu) and choose Plot Function. Type in the equation of your least-squares regression line (in Fathomyou 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 affects 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.
- 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.