  Assignment 11

 "The problems of the world cannot possibly be solved by skeptics or cynics whose horizons are limited by obvious realities. We need men and women who can dream of things that never were." -- (U.S. President John F. Kennedy) Age of Reason (600 B.C. - 300 B.C.)The Greeks, beginning with Pythagoras, come to realize that mathematics is a useful tool in describing the universe, as they know it. Golden Age of Geometry (300 B.C. - 300 A.D.)Beginning with Euclid, the Greeks develop geometry to help them exist on the surface of the earth, and to explain the the movement of heavenly bodies in the sky above them. Decline of Mathematical Knowledge (300 - 500): Ecclesiastical absolutism of the Christian Church crippled discussion of intellectual matters. Many Greek mathematical works were destroyed because they were thought to be based in paganism. When Herkimer ran a nudist colony, what did he say to the impatient bill collector? Answer: "Please bare with us." Things Herky would like to know: Should vegetarians eat animal crackers? How do they get a deer to cross at the yellow road sign? Reading: Section 2.5, pages 100-103. Exercise: Page 103-104/8, 9, 10, 11, 12, 13, 14. In problems 11-14, use your TI-83 and produce the equation of the least-squares regression line. Mathematical word analysis:DIAGONAL: From the Greek roots dia (to pass through or join) and gonus (angle). In a polygon (a quadrilateral, for instance), a diagonal joins the vertex of one angle to the vertex on a non-adjacent angle. If the quadrilateral is convex, the diagonal "passes through" each angle.

OK, group! This section really shows the power of mathematics. What I am going to write below simply represents how you can use your calculator to produce a prediction line (the least-squares regression line) for a data set {(x,y)}. Here is a set of x and y data values:

 x 1 4 5 9 11 15 y 4 14 15 26 35 45

Check the Y= option. Turn OFF any functions that are ON.

Go STAT --> EDIT, put the x values in list L1, and the y values in List L2.

Go STAT --> CALC --> LinReg(ax+b).

LinReg(ax+b) will appear on your screen. Add L1, L2, Y1 as indicated.

LinReg(ax+b) L1, L2, Y1
(Careful here. L1 is above the 1 key, L2 is above the 2 key.
To get Y1, you need VARS -->Y-VARS --> FUNCTION -->Y1.)

You will see the following on the screen:

LinREG
y = ax+b
a = 2.939163498
b = 1.1222940431
r2 = .994009064
r = .9970000321

Your prediction line is y = 2.939x + 1.122. The slope is 2.939, and the y-intercept is 1.122.

If you now hit ZOOM 9 (this is ZoomStat) you will see the plotted points and the line that has been "fit" to the data.

Without providing an explanation here, it will simply be noted that it the r value displayed is close to 1 or to -1, then the line "fits" the data reasonably well.

This is MATH POWER at its best.

 Problem: Use your calculator to find the equation of the line containing (3,13) and (5,17). Solution (with communication): Placing the x values in list L1 and the y values in list L2, and following the steps outlined above, our calculator output yields y = ax + ba = 2b = 7r2 = 1r = 1 Noting that r = 1, a statistician would realize that all data points are on a line. This is not surprising here since there are only two points, and two distinct points determine a line. The equation of the line is y = 2x + 7 Note: Given just two points, it would not be efficient to use the calculator in this manner to find the equation of the line. This purpose here is simply to demonstrate the line of best fit, and with just two points this would be the line containing the two points.