Math 151 Day 8

Math 151 , Fall 2002, Monday, Sept. 16 , Day 8 After Class

Helpers page has been updated with hours.
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HW Day 8
Read for this, pp. 46-55, 57-8. Read ahead, rest of sec. 1.3

Hand in
Handout on Densities
p. 51, 1.50, 51, 52 general densities, mean &median

Postpone the following till after next class:
p. 64 1.61 eyeball sigma
p. 54 1.53&54 Normal, men's hts--68-95-99.7 rule.
p. 64 1.63 pregnancies--68etc rule

Do but don't hand in on Wed. Day 8. Keep and hand in Friday.
table use: Always sketch the distribution first, mark the area you are looking for!
p.61 1.57 z's . Do these with the Statistical Applet "Normal Curve Density Calculator" at http://www.whfreeman.com/scc/
(Uncheck the 2-tail box for most uses. Mean 0, s.d. 1)
Next class we'll learn how to do these with the book's tables and the areas methods.

Read, to discuss

Optional (more practice)

Postpone
1.55 wechsler ais, 68etc rule

p. 65 1.65 z's

(Activstats: 4-4 standardizing, 5-3 Normal distribution)
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>Homework questions. Clarification for problem A, SPSS. You need to Reset the Scatterplot box for it to recognize that you've changed the Measure on Pulse.

Spinner. Use 248x310 pixels

You should have handout HW sheet: "Density curves"

Density curves, pp.46-51
(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.

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.
This allows area to represent proportion 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.")

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.

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

Start here Wednesday
"Normal" distributions:("Gaussian", "Bell-shaped") part 1 (pp. 51-5, 57-8) http://www.whfreeman.com/scc/

What "mechanism"? Thing measured is the result of many small independent influences.
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.
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).

These are back-of-the-envelope tools. Sketch the curve, mark mean, + 1, +2, +3 s.d.'s-- in real units.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ First standard normal table use, then standardizing~ ~ ~ ~ ~ ~ ~ ~ ~ ~
Standard Normal table use. Our tables give area to the left of a z value.
    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.)

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

. bell curves. Use 202x515 pixels to print.

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
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