Math 151 , Fall '08, Mon., Sept 15, Day 8 .After class. Hit reload..

First hourly exam, this  Friday Sept. 19, Day 10 .
Sample exam handed out today.   solutions are linked here,  Closed book, but bring one sheet of notes (anything you like) and a calculator.
Exam will cover thru what is assigned today (No more than the work outlined on Friday's page ), Plus reading SPSS output.  You may be asked to read SPSS output (as we see it on the sample exam), but not how to produce it.  Sample exam may go further than we cover.  We'll know at end of class Mon.
You may  start early and/or stay late, if you don't have another class. Let me know ahead on clipboard. Not before 9(?) You don't have to work in the classroom;  you just have to sign in and say where you'll go (in the building!), on the clipboard.  If you want more than an hour, and have obligations before and after--or other problems-- see or email me to make a plan before Wednesday! 
Questions on last HW? Day 7
Questions on SPSS? Day 6   See also  SPSS Info page for details--I'll try to keep it updated on "issues".
       SPSS is now also on machines in Mac 304 (Except machine Reed didn't have Class Material, as of Fri.)
  Solutions for SPSS HW problems are posted in Mac 101, 110, linked here.
Class EMAIL:  Math151@wells.edu

"Percentile:" 
"The 38th percentile is 25 pounds"= "25 pounds is at the 38th percentile" = 38% of the observations are at or below 25 pounds.  
25th percentile of salaries = 1st quartile:  25% of the salaries are at or below the $ value of Q₁.
    Note, the "somethingth percentile" is a number in the x-units; in the units of the variable you're looking at.  What Percentile is x at?  If you draw the density or histogram, it's the Area  to the left of x. (Cumulative Proportion)

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"What proportion"problems:  BPS4e pp. 78-80, first pass
Use  Applet: Normal Density Curve   http://bcs.whfreeman.com/bps4e

  table.  Written for Z--N(0,1); learn to read first, then to use for a different mean and s.d.
Standard Normal table use.  Our tables give area to the left of a z value (Cumulative Proportions)
Written for Z--N(0,1); learn to read first, then to use for a different mean and s.d. See text p. 76-80. Table A: p.684-5. Table A (Excel)
    Sketch the density, label axis, mark the area you're looking for.
    Figure out how to get it using areas to the left of one or more z-values.
        Think cutting up paper bell-curves. (Remember whole area is 1.)

Reading table backward:  Table A (Excel)
What z value has area ..... to the left/right of it?
        Sketch  roughly.
        Restate (if needed) as "What z value has area A to the LEFT of it."
        Look in body of table for the value closest to A.
        Go to edge(s) of table to find what z that goes with.

Exam 1 will go no further than this.
Got to here in class Monday.  = = = = = = = = = = = = = = = = = = = =
Next:  Using Standardizing and the Standard Normal Table to do more general problems.
Review Standardizing:  A "raw value" x is standardized by telling how many standard deviations above the mean it is.
    Find z:  Subtract the mean from x.  Now you know how far "above" the mean x is, in "raw" units. (If it's below the mean, the number will be negative.)  Find how far this is in "standard deviations" by dividing by the standard deviation.
That's the z-score.

Standardizing:   A way of comparing an individual against its pack.
                                Comparing individuals from different packs, each relative to its own.
                        Removes "units of measurement" from the discussion.
                        Enables use of the standard normal table.

Examples: ("Classic IQ test", mean 110, s.d. 25)
  85 is 1 s.d. below the mean.  Computation:  z = (85 – 110)/25 = (–25 raw points)/25 = –1 s.d. from mean.
 145 is how many s.d.'s above the mean?
                Computation: z = (145 – 110)/ 25=  (35 raw points above mean)/25 = 1 2/5 = 1.4 s.d. above mean

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"What proportion"problems:  BPS4e pp. 78-80

•     Sketch a normal curve. Mark mean, 1, 2 s.d.'s.  Label with raw values, and z-values below.
•     Mark end points for problem, roughly, and shade area desired.
•     Standardize end point(s).  Use standard normal table to find area. (Draw helper sketches if needed)
•     Check picture to see if it's plausible.

or   P ( 100 < x < 145)  = P ( –.4 < z < 1.4) = P( z < 1.4) – P(z < –.4) = .9192 – .3446 = .5746
     Read "Proportion of x's with 100 <x<145"  for P(100<x<145)
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"Backward problems"  "What raw (x) value has area ___ to the left/right of it?"   BPS4e pp. 81-83.
        Sketch  the curve, labeled with x values and z values, and the Area, roughly.
        Restate (if needed) as "What z value has area A to the LEFT of it."
        Look in body of table for the value closest to A.
        Go to edge(s) of table to find what z that goes with.
        Convert the z to an x: z is the number of standard deviations above the mean.
            Multiply z by the size of 1 standard deviation.  Now you have distance above the mean, measured in raw units.
            Add the mean.  Now you have the "raw" value x. (You have "unstandardized" it.)
Example:  What x value has 10%  of the observations above it?  This is the same x as the one for:
        What x value has 90% of the observations below (to the left of) it.  

The table gives z = 1.28, approximately.  Table A (Excel)
The "Classic IQ test"score x= mean + z (s.d.) =  110 + 1.28 (25)=  110 + 32  = 142

Percentiles:  a "Classic IQ test" score of 142 has 90% of the scores at or below it.  142 is the 90th percentile.