Math 151 , Fall 2006, Friday, Sept. 8, Day 7Hit reload.. After class

HW Day7:  Ch. 3: Read Density curves pp. 64-9 //  Normal & 68-95-99.7% rule pp.70-74. Use Normal Density Applet curve to check concepts and computation. "Check" problems p. 84: 3.15, 16, 17, 18;Got this far in class. 19, 20. //Standardizing to standard normal pp.74-76, "Check" 3.21.  Ahead, rest of chapter.  We WILL learn to use table A.   
Hand in 
A. Complete the Handout on Densities (get from outside my door if you missed class)
p. 66, 3.1 Sketch density curves
p. 69, 3.2 & 3.3Uniform distribution This is the same density as A on the Handout on Densities.
p. 69 3.4 means and medians
= = = = = = = = = = = = = = = = = = = = = = = = = = 
Normal distribution:  Use the Applet: Normal Density Curve   http://www.whfreeman.com/bps/
(or on your book's CD) to check your answers.
- - - - -Shape related to mean and s.d., 68-95-99.7 rule.  
p. 74 3.5 Women's hts, sketch
postpone the rest:
p. 74 3.6  Normal, women's hts--68-95-99.7 rule.
p. 74 3.7 pregnancies--68etc rule (This distribution may not apply to planned births, of which we now know there are a lot!)
- - - - - Standardize
p. 76, 3.9 mens & women's heights
p. 86, 3.33 ACT/SAT Jacob and Emily (Info above #3.32)
 		Read, to discuss
A. Look at table A, pp. 685-6 and compare with the Handout on Densities tables (table A has more numbers; just look at the left 2 columns for now...)  See if you can read from the table that
the area for z less than 0 is .5000,
the area for z less than 1 is .8413,
 the area for z less than -1 is .1587.
 Optional (more practice) 
 
 

= = = = = = = = 
- - - - - - - - - - 
Postpone: p. 75 3.8  SAT & ACT  (Standardize)



 
First hourly exam Day 10 (Sept. 15), a week from today . Sample exam available Monday, solutions will be outside my door & on reserve. 
Exam will cover thru what is assigned Monday, probably the material below on this page, maybe a little more.
Questions on Standard Deviation? A. cars down the highway?   Day 5
Questions on SPSS? Day 6  Output for HW problems will be posted in Mac 101.
-  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -  -Spinner. Use 248x310 pixels

Density curves, BPS4e pp.64-69
GET  handout HW sheet: "Density curves"

    (When values can take on any of a continuous interval of numbers)
Example:  Spinner:  Label edge with continuous values from 0 to 1. Spinning should produce 1/10 of all spins in each colored sector.  Simulations of 500, 3000 spins show roughly true. More spins would get closer.
(Histograms of simulations)

Abstraction, idealized histogram ("Mathematical model") = Density curve. Describes a theoretical distribution of data.

Any density curve:  is a curve
   --always on or above the horizontal axis
   --has area exactly 1 underneath it.
Many, many density curves are possible, modeling many phenomena.
  • For the spinner, the density curve is "Uniform on 0 to 1".
  • If you have two spinners like this, spin both at once and add the results--the corresponding density curve is "triangular, symmetric, on 0 to 2"
  • A more complicated mechanism will produce data corresponding to the density curve I have called "trapezoid, -1 to 2"
  • A very important one is the "normal" distribution family.
    • Median, mean, percentiles, standard deviation are defined for a density curve in analogy to those for a histogram.
    -- median has half of area below and half above.
    -- mean is balance point.  On the long-tail side of median if distribution is skewed. Same as median if symmetric.
    --First quartile has 1/4 of area below, 3/4 above. Etc. for others.

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

  • You will make and use  "Cumulative Proportion"tables for the simple distributions on the handout.  These are similar to the table we will use to describe the normal distribution.

    • "Normal" distributions:("Gaussian", "Bell-shaped") part 1 (pp. 70-74) Applet: Normal Density Curve   http://www.whfreeman.com/bps/ Example:  "Classic IQ test" scores are approximately N(110, 25).  mean=110, mean +1s.d. = 135, mean + 2s.d.'s = 160,  mean -1s.d. = 95, etc.  See picture below.

    Standardizing: (p. 74-5) 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.
    Values in any normal distribution, after standardizing, become values in a N(0,1) "standard normal" ("Z") distribution.

    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.)

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