Math 151 , Fall 2002, Wednesday Day 15, October 2Hit reload to get most current version After class

Exams still not finished.  Friday for sure (I hope!)
HW assignment Day 15, due Friday day 16
Reading:  (Re)read Moore 2.3 thru p. 114.  (Read ahead,Moore, p. 108,Moore, p. 116-119,Moore, p. 119-122. Then sec. 2.4. We'll skip Moore 2.5, start Ch. 3  next.)
Hand in Friday:  
If you didn't already do these for today, copied from Day 14 page, do:
Review of straight lines: 
p. 124, 2.39, 2.40. Most people did fine on lines on the pretest. If these are a problem, ask someone NOW! Any MathClinic assistant can help with these.  Also Just the Basics on reserve covers it.

B.Practice fitting "least squares best fit" lines:  Author's website,  http://www.whfreeman.com/scc,  (ClickNetscape toolbars to minimize them, if needed.)
  Choose "Statistical Applets",  Correlation/Regression.  Check in the "Show least-squares line" box and put in some data points.   Check in the "Show Mean X &Mean Y lines" box; see if #3 below holds.  Repeat for a few data sets.
--Try fitting the line yourself:  Put in some data points.  Now click Draw Line.  Click and drag in the picture and you'll get a line with 3 blobs. Drag the center and it will go up and down, Drag an end and the slope will change. Put the line in the best place for predicting y's from x's.  If you do well by the "least squares" criterion, the green bar up top will shrink close to 0.   Check in the "Show Mean X &Mean Y lines" box; adjust your line.  Check in the "Show least-squares line" box and see how you did. Try to fit at least 4 different data sets. Write down on your paper what you discovered (were your judgment errors consistent in any ways--did you have any surprises?)

Moore p. 111, 2.31 acid rain No data, therefore no SPSS (draw the line by hand)
C. Use the SPSS handout and graph  the regression line for govsal on avgpay (as shown), also the lines for the 4 separate groups (either on one graph or on panels.)

Moorep. 111 2.32 (Manatees) Import the dataset into SPSS (Class Materials\Math151) In SPSS,  Print the plain graph, and one with the regression line. Draw the regression line BY HAND as best you can on the plain graph. Check with the other one. For part b, pencil in the new points on the graph with the printed line. Find the mean by hand(calculator)...
 p. 126, 2.44 p. 129, 2.48 Sarah grows.... Use SPSS for parts a and b, calculator for the rest.

D. For the data of Moore, p103, 2.22 (metabolism), Print out a graph with the regression line for all the
 people, and another with 2 separate lines (M and F). Use the equations  to calculate the predicted
 metabolic rate for 
     a) a person of mass 45 kg. 
     b) a female of mass 45 kg. 
     c) a male of mass 45 kg. 
 Now use the "up and over" method of Fig. 2.10 p. 107, with a pencil and straightedge to mark the
 predicted values on the y-scale. Write down your computed answers next to them.  Make sure the two
 methods give consistent answers.
+ + + + + + + + + + + + + + + + + + + + + + +
A. Open the Excel file RegressionSlope (or in the folder RegressionDemos in ClassMaterial\Math151).  Change x-y values in the yellow boxes and watch the line change.  Change x-values in col. F and watch the "run" (red line) change. Notice the slope = the coefficient of x = the rise/run = increase in y per unit increase in x.  Fix it so the increase in x (the "run") is exactly 1.  Print the page to hand in.
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With the 4 "facts": 
p. 114, 2.33  prof. swims--two lines x->y, y->x Also,Make both graphs in SPSS, each with its regression line.  Use SPSS to find the means for time and pulse, and draw (by hand is ok) the xbar, ybar lines on each graph.  Note the Regression lines won't coincide if you flip one graph.

p. 111, 2.30 heating degree days,  checking formulas on p. 104. Import the dataset
 into SPSS. Use SPSS to get the formula in part a (again), and the mean, s.d., and correl. coeff. in part b.  Then use your calculator to calculate the slope and intercept.  Compare with SPSS's. 

p. 116, 2.35  beavers (prop. explained.) Do parts a and b on SPSS, c is just to answer.

p. 128, 2.47  Julie's grade (Not SPSS, just calculator) 
p. 129, 2.51 "regression"  (Not SPSS, just calculator)   Hint below*

B (another B).  Use the Excel RSquared page. ( R-Squared (or R-squared tab in ResidualsRSquared.xls: ClassMaterial\Math151\RegressionDemos)). Shift points around and get an r2 close to .8 (80%) (Between .75 and .85 is good enough.).  Note that if r = +.9, then  r2 = .81.   Now shift the points so that r is negative and r2 is close to .8.  Print the resulting page to hand in. (Data and graph)

 Read Optional 
 
 
 

 
*Hint:  ybar = 46.6 + .41xbar [why?].  Let c be the amount Octavio's final is predicted to exceed the mean.
Then (ybar +c) = 46.6 + .41(xbar + 10) [why?].  Use the two equations and solve for c.  If your algebra skills are not strong enough, don't get upset; this calculation is not central to the course.  Read the answer!
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(Activstats: Next time I'll discuss  the least squares criterion ( ACTp. 9-3, 1st 2 activities)Residuals (ACTp. 9-3, 1st activity), finish this chapter)
= = = = = = = = = = = = = = = = = = = = = =
HW questions?
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Regression line: Moore 2.3 Predicts or estimates a y (vertical) value for a given x (horizontal) value.  "Regressing y on x" .
    Formula yhat= a + b x.    a is y-intercept. b  is slope:  If x increases one unit, yhat increases b units.
(In a straight-line relationship, the amount that y increases for one unit increase in x is the same no matter what value of x you start with)          RegressionSlope.xls or in ClassMaterial\Math151\RegressionDemos

Facts (Moore pp. 112-14)

  1. The Regression line is trying to predict the "average y" for a given x (with the added requirement that it is a straight line).

    2. Unless the data lies perfectly on a straight line, the line for predicting weight from height -- "regressing weight on height" --(for example) will NOT be the same line as that for predicting height from weight--"regressing height on weight".  (In-class demonstration) (The picture on p.113 is about this. )
     
  2. A change of one standard deviation in x corresponds to a change of r standard deviations in y, along the regression line.

    3.  The slope b expresses change in y-units per x-unit. (Suppose x is inches, y is pounds. Then b is in pounds per inch.) You can find b by multiplying r by the standard deviation of the y's (that's in pounds)  and dividing by the standard deviation of the x's (that's in inches)
    In "algebra", b = r times (s.d. of y)/(s.d. of x)  (Equation p. 104)
           If we standardize both the x-values and the y-values, the slope will just = r !
     
  3. The regression line goes through the point given by the two means, (xbar, ybar). http://www.whfreeman.com/scc

    4. --If you know this, you know ybar = a + b (xbar).  You can solve this for a, a = ybar - b (xbar). (OtherEquation p. 104)
    --So knowing 2 and 3 give you the equation of the line from the means, s.d.'s, and r.
    --And if you draw the two lines, y on x and x on y, they will intersect at (xbar, ybar)
     
  4. r2 ("Coefficient of Determination") = Proportion of variability in y-values explained/predicted by knowing x and using the least squares regression line.  (Exactly what that means mathematically is hard.  Just get used to it as a measurement.) More:R-Squared (or R-squared tab in ResidualsRSquared.xls: ClassMaterial\Math151\RegressionDemos)

    5. r2 is the square of the correlation coefficient r!  (-, + Sign gets lost.)
    If r = .7, about half (.49) of the variability  in the y's is explained by using the regression line relationship to predict y from x. (If weight and height have a correlation of .7, then half of the variability in weight can be explained by knowing height.)
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