Math 151 , Fall 2005, Day 8 Wed. Feb. 15 Hit reload...After class

GET  handout HW sheet"Tables for simple models (densities)"
Get Sample exam

Models for quantitative variables, Ch.6    (AS6-2 ¶1)   See Day7
Changing units: (D&V 84-85, AS 6-1 ¶paragraphs 1&2)
    ( + shift ) Measures of middle should shift  along with the raw data.  Measures of spread are unaffected by +
    ( x/ rescale) Measures of middle and of spread should stretch or shrink along with raw data (We assume we only multiply/divide  by positive numbers.)
To recalculate:  Do the same thing to measures of middle as you do to raw data.
                        To spreads, just do the multiplying or dividing part.
 Shapes  (skewness, humps, clumps, outliers) are not affected by shifts and rescaling.

&&Alias/alibi:  When you change units of measurement for all your data values, you can think of the result 2 different ways:
    "Alias (other name)":  The data distribution sits still. You have just changed the ruler stick you measure by.
             (in/cm ruler.  Thermometer)    Most useful for us.
    "Alibi (other place)" :  The ruler stick keeps the 0 the same and 1 the same width, and the data distribution with "new" values moves to the new location.  D&V pp. 84-5.
  Optional SPSS handout to create new computed variables.
= = = = = = = = = = = = = = = = = = = = = = = = = = = = = =
GET  handout HW sheet: "Tables for simple models (densities)"
Models for quantitative variables    (AS6-2 ¶1)
(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 to  Uniform shape.

Abstraction, idealized histogram ("Probability Model") =
Density curve. Describes a theoretical distribution of data.
Any such model is a curve
   --always on or above the horizontal axis
   --has area exactly 1 underneath it.

This allows area to represent proportion (relative frequency) of "histogram" between specified values.
(We will assume the proportion of observations precisely equal to a value is 0.  "So proportion less than 2" is the same number as "proportion less than or equal to 2.")

• For the spinner, the density  is "Uniform on 0 to 1".
• If you have two spinners like this, spin both at once and add the results--the corresponding density  is "triangular, symmetric, on 0 to 2"
• A more complicated mechanism  produces data corresponding to the model  I've called "trapezoid, -1 to 2"
• A very important one is the "normal" distribution family--bell-shaped.

Numerical summary: (D&Vp.86)
Statistic   from data:        xbar         s           Q1   Median    Q3
Parameter   for model :     µ          sigma       Q1   Median    Q3

Many models have tables to describe them.  Especially percentiles tables showing area to the left of (below) a given value
= theoretical proportion of observations below the value.  30% below x, x is the 30th percentile).

•  What "mechanism"?  &&Thing measured is the result of many small independent influences.
• Family:  all the same basic shape, except for measurement units.
• Mean gives middle, standard deviation gives spread..
• Knowing parameters mean (µ "mu") and standard deviation ("sigma") tells you everything. "N(mean, s.d.)"
• Symmetric.  Mean at middle. &&Curvature changes at 1 standard deviation on either side of mean.
• "Standard normal": mean = 0, s.d. = 1  Standardized -- "s.d.'s from the mean".
• "Middle s. d." && A center region one standard deviation wide spans about 38% (AS, not in D&V)
• 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 are further than 3 s.d. from the mean?  What percent  are  higher than 2 s.d. from the mean?  Etc.
Back-of-the-envelope:  Sketch the curve, mark mean, + 1, +2, +3 s.d.'s-- in real units.
Example:  Wechsler Adult Intelligence Scale 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:   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: Wechsler Adult Intelligence Scale scores used to be 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

  ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ First standard normal table use, then with "real" values~ ~ ~ ~ ~ ~ ~ ~ ~ ~
Standard Normal N(0, 1).  Our tables give area to the left of a z value.  TableZ, Appendix E, A-50
Using standard normal table:  See D&V p. 88.  (Wrong side of graph is shaded in my text)
       z |  .00     .01     .02 .....
      ...|
     1.4 | .9192   .9207   .9222 ....
   P(z < 1.40) = .9192,   P(z < 1.41) = .9207  P(z < 1.42) = .9222.
                                              ?z has more than 2 dec. places?  Round to 2.

    Sketch the density, 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.)  Like handout.

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":
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 is 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 is the 90th percentile.
            1.28 has 10% of the observations above it.

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* AS6 Normal Density tool: Use AS30-2 "Normal distribution based Confidence Intervals" tool for best setup.CAUTION: Don't hit the Enter key! It closes the tool-box!   To use it from Tool1 from the menu bar in Ch. 6:  Right click for menu. Choose Show Buttons.  Choose Show Flag Values, Mean, StandardDeviation; Real Values.  Now you can type in mean and s.d. and  the mean + 1,2,3 s.d.'s will show on the axis.  CAUTION: Don't hit the Enter key! It closes the tool-box!  To register a typed number, click in a different box.