Illinois State University Mathematics Department


MAT 312: Probability and Statistics for Middle School Teachers

Dr. Roger Day (day@ilstu.edu)



Assignment #3: Possible Solutions


Assignment made on Wednesday, 6/23/04
Assignment due on Thursday, 6/24/04

Use the Couples' Ages data set, handed out in class, to complete the following tasks.

1. Create the equation for the median-median line of this data set, using husband's ages as the first element (x) of each ordered pair and wife's ages as the second element (y) in each ordered pair. Show all steps in the process as we illustrated in class and express your equation in slope-intercept form. You can check the equation of your median-median line using your calculator.

Here is the scatter plot of the data. Below it, I show each of the three clusters of data points from which we generate the summary points.

Summary Point (x1,y1) = (25,24)
Summary Point (x2,y2) = (31,28)
Summary Point (x3,y3) = (54,46)

Using the outer summary points, the equation of the line containing them is

y=(22/29)x + 146/29

or, approximately,

y=0.7586x + 5.0345.

When x = 31, this equation gives us y = 828/29, or approximately 28.5517.

Our second summary point, however, has a y-value of 28, so we need to move our line one-third of the distance between y = 28 and y = 828/29. This distance is one third of 16/29, or 16/87, which is approximately 0.1839.

So we decrease the y-intercept of our first line by 16/87, giving us the equation

y=(22/29)x + 422/87

or, approximately,

y=0.7586x + 4.8506.

Your calculator can confirm for you the ordered pairs comprising the summary points and the equation of this median-median line. Here is the scatter plot of the data with the median-median line on it.

2. Use your calculator to generate the least-squares linear regression equation for this data set, using husband's ages as the first element (x) of each ordered pair and wife's ages as the second element (y) in each ordered pair. Copy all the information your calculator shows you when you generate the least-squares equation. Express your equation in slope-intercept form.

After entering the data into a TI-83 calculator and selecting the command for Linear Regression, the calculator shows the following:

LinReg
y=ax+b
a=.8789973259
b=2.445803822
r^2=.8909477545
r=.9439002884

So the least-squares linear regression equation is

y=0.8789973259x+2.445803822

Here is a scatter plot of the data with both regression lines on it. The red line is the median-median line and the green line is the least-squares line.

 

3. Calculate the sum of the squared error terms (SSE) for each of the models you just created (i.e., for the median-median line and for the least-squares line). You are encouraged to use your calculator's "lists" features to carry out your calculations. Express each sum rounded to the nearest hundredth of a unit.

The SSE for the median-median line is 617.95 and the SSE for the least-squares line is 461.22. Look at the last plot above and convince yourself, graphically, why the SSE is smaller for the least-squares line.




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