Math 151 , Fall 2002, Friday Day 16, Oct 4 Hit reload to get most current versionAfter class

HW assignment Day 16, due Monday
Reading:  Finish 2.3, read 2.4.   Skip 2.5. Ahead in Ch. 3.

Hand in Monday A.  Use ResidualsRSquared from the website or the lab to graph these data sets, along with a graph of the residuals.  Print the results, and describe the shape of the residuals (it may help to connect the dots with pencil, to see the pattern.)  a)  x 1 2 8 4 6 9      y 1 3 6 6 7 5  b) x 1 2 7 4 6 9    y 7 6 2 4 2 1 Moore p. 122, 2.36 speed&gas again a, b, c, d.   There is a data file for problem 2.36, and its third column is the residuals (check them against the book). B. Use Author's website, http://www.whfreeman.com/scc, ...Correlation/regression.   Make a cloud of data (about 15 points), put in the regression line.  Play with an outlier: drag a point to the far left (right) and drag it up and down.  Try it if it's in the middle range of x's.  Write answer: Where is it most influential? Now add a bunch more points (50 is max.)  Play with an outlier  again.  Does the outlier have more or less influence with a larger data set? Moore p. 123, 2.38 Gesell first word-point in middle of x range. Get the data into SPSS, delete child 19, graph and get the regression line and r2.  Use the formula on p.117 and graph the line for the full data set by hand on your printout.   r2  for the full data set is on p. 122.  Moore p. 122, 2.37 Calories (You saved these, I think--or, from Moore's files, in  TA02-04) Graph and get lines in SPSS with and without the outliers.  Graph the line for "without outliers" by hand on the printout for "with outliers" so you can compare them better.  Print one more graph (with outliers) and keep it for problem C below. = = = = = = = = = = = = = = = =  Postpone  Sec. 2.4  p. 131, 2.53 farm population (SPSS)    Also connect the dots, or plot the residuals--is there any curve to the relationship? p. 132  2.54 Dow average/stocks p. 138 2.63 math&verbal r, states/individuals C.  Look again at p. 122, 2.37.   These values are averaged values, over a bunch of people's guesses.  What would the graph look like if all the individuals' separate guesses had been graphed?  Add points to your graph .

Read,

Optional  SPSS will make residuals:  Do Analyze>Regression>Linear (a new menu for us)  Click your variables into Independent (X) and Dependent(Y).  Hit the Button "Save...": Checkbox Residuals: Unstandardized. Continue, Ok out of the menus.  You'll get output; ignore it.  You'll get a new variable, the residuals.  Try it with the data file for problem 2.36, with speed and gas.  You'll get a fourth variable that should be the same as the residuals variable.

Regression-- Review comments
ANY Straight line y = a + bx  (or bx + a):  b, the coefficient of x, is the slope of the line.  If x changes one unit, y changes b units, so b is the rate of change of y with respect to x.  (If y is weight in pounds, and x is height in inches, b is the number of pounds  we expect to see weight go up by, per inch that height goes up by.

"Regression line of weight on height":  height = horizontal (x) axis, weight = vertical (y) axis.

LEAST SQUARES PROPERTY
"Residual at x" = y - yhat  = distance between observed y and  predicted y (what's left over after predicting)
    ( Positive if observed is bigger than predicted, negative if observed is smaller than predicted)
Least squares principle:  Find the line that minimizes the sums of the squared residuals.(Here, or in Mac 101, ClassMaterials\Math151\ RegressionDemos\RegressionLine.xls, Squares tab)
       This method of finding a "best fit" straight line for predicting y's from x's was derived mathematically to work well with "joint normal" data--elliptical clouds.  For data of this sort, the line does  give the mean of the y's for each given x (at least in the abstract.)

Drawback if the data is not the "elliptical cloud" type:
     Outliers get their residual distance squared:  May be very influential  in determining where line sits.
             Especially if at lowest or highest x-values, may change slope of line a lot.
                 Author's website,http://www.whfreeman.com/scc, ...Correlation/regression.   Play with an outlier.
 (Outliers toward the middle x's may not change the slope, but may affect r and r².)

Plotting residuals:  This amounts to making the regression line into a new x-axis--If you plot the residuals themselves vs. the original x values, without the distraction of the slanted line, outliers and patterns other than the linear (if any) can emerge.
(Here or  ClassMaterials\Math151\RegressionDemos\ResidualsRSquared.xls , Graph of Residuals tab.(doesn't have tiny unlined graph)
SPSS can make a new variable of residuals, which you then can use to make a scatterplot. Optional HW.
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Will start here Monday, do all of 2.4  Cautions  Sec. 2.4
Plot the data: Summary formulas and numbers don't tell the whole story.  (Anscombe's quartet, p.127, 2.46-7, also in ACT HW ch.9)

Extrapolation-- extra (outside) polation (putting a point): Using the line to predict outside the range of x's you have data for.  Unavoidable if x is time; but inevitably dangerous--nothing says the mechanism you see will persist in a wider range.

Averaged data will produce a stronger relationship (higher correlation, R²) than the merged raw data from individuals (the averaging hides much variability)  You did a problem on height vs. age--they were averaged values.

Lurking variables and association/causation next time.