Math 151 , Fall '08, Fri., Sept 12, Day 7 .After class. Hit reload..

First hourly exam, next  Friday Sept. 19, Day 10 .
Sample exam handed out today.   solutions are linked here,  paper copy to read is outside my door.  Closed book, but bring one sheet of notes (anything you like) and a calculator.
Exam will cover thru what is assigned this coming Monday (No more than the work outlined on today'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.  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 HW Day4?   (5 number summary, details) Canadian/U.S. weekend births?p.60,2.35 What's deceptive about that graph?  Day 4
Questions on HW Day5?   (standard deviation)   Review Std. Dev. Day 5
Questions on SPSS? Day 6   See also  SPSS Info page for details--I'll try to keep it updated on "issues".
 Solutions for SPSS HW problems are posted in Mac 101, 110, linked here.
-  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -

Density curves, BPS4e pp.64-69
GET  handout HW sheet: "Density curves" if you didn't Monday .
See Day 5 for notes & handout link.  Outline:

Any density curve:  is a curve --always on or above the horizontal axis    --has area exactly 1 underneath it.
 This allows area to represent proportion of "histogram" between specified values.

Many densities have tables to describe them.  Especially tables showing area to the left of (below) a given value ("Cumulative Proportion").

• What "mechanism"?  Thing measured is the result of many small independent influences.
• a Family:  all the same basic shape, except for scale. Mean gives middle, standard deviation gives spread.
• Knowing mean ("mu") and standard deviation ("sigma") tells you everything.  
• "N(mu, sigma)"= "Normal with mean mu, standard deviation sigma"
  
• Symmetric.  Mean at middle. Curvature changes at 1 standard deviation on either side of mean. 
  
• "Standard normal": mean = 0, s.d. = 1
• "Middle s. d." A center region one standard deviation wide spans about 38% (not in text)
• 68-95-99.7 rule:  68% within 1 s.d. of mean, 95% within 2 s.d. of mean, 99.7% within 3 (roughly).

What percent is further than 3 s.d. from the mean?  What percent  is  higher than 1 s.d. from the mean? Etc.
These are back-of-the-envelope tools.  Sketch the curve, mark mean, + 1, +2, +3 s.d.'s-- in real units.

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" scores are approximately N(110, 25)
   A score of   85 is 1 s.d. below the mean.  Computation:  z = (85 – 110)/25 = (–25 raw points)/25 = –1 s.d. from mean.
           (About the 16th percentile--16% get scores < 85)
   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
           (What percentile is this?  What percent get scores < 145?  Need a table for between the "whole" s.d.'s.  Next.  Table A)

"What proportion"problems:  BPS4e pp. 78-80, first pass
Use  Applet: Normal Density Curve   http://bcs.whfreeman.com/bps4e
Proportion with scores between 100 and 145?   below 100?  Above 145?
What score is at the 75th percentile?

 If you don't have a handy "Applet" or a user friendly calculator?  Must use a 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~ ~ ~ ~ ~ ~ ~ ~ ~ ~
Standard Normal table use.  Our tables give area to the left of a z value (Cumulative Proportions)
Using standard normal table:  See text p. 76-80. Table A: p.684-5. Table A (Excel)
       z |  .00     .01     .02 ..... =number in "hundredths place"
      ...|
    -2.4 | .0082   .0080   .0078 ....= area to the left of "edge number"
       ...|
     1.4 | .9192   .9207   .9222
 ones&tenths

  Proportion of z's below -2.40 = P(z < -2.40) = .0082
             = prop. of individuals 2.40 s.d.'s or more below the mean)
  P(z < -2.41) = .0080  P(z < -2.42) = .0078 ,     P(z < 1.42) = .922
                                          ?z has more than 2 decimal places?  Round to 2.

    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.)

Example:  Proportion of observations between 0.5 and 1.4  P(0.5 < z <1.4) =
            Proportion of observations below 1.4  minus Proportion of observations below 0.5 
               P (z < 1.4)  -  P(z < 0.5)  = .9192 - .6915 = .2277

.

Example:  Proportion of observations above  0.5,    P( z > 0.5) =
                ONE minus proportion of observations below 0.5,   1 -  P( z < 0.5) = 1-.6915 = .3085
. 
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.
Example:  "What z value has 10%  of the observations above it?"  This is the same z as the one for:
        "What z value has 90% of the observations below (to the left of) it?"   "What z value is at the 90th percentile?"

        Find in the table  .8997 and .9015 --  .9000, our number, is between them.
                    .8997 is a little closer to.9000, so use it.
           For .8997, the z value is 1.28.   1.28 has 10% of the observations above it.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
All of these can be checked using the Applet: Normal Density Curve   http://bcs.whfreeman.com
Exam 1 will go no further than this.