Math 151 , Spring 2002, M Feb. 11, Day 7  Hit reload to get most current version Corrected HW.

Get handout HW sheet: "Density curves"
    Correction Bc) " .92 BELOW it"

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.")

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.

Many densities have tables to describe them.  Especially tables showing area to the left of (below) a given value. Or to the right of (above) a given value.

•  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%
• 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 2 s.d. from the mean? Etc.

HW assignment Day 7, Mon. Feb 11
ACT: From Activstats Homework
Moore:  From David S. Moore, The Basic Practice of Statistics
Reading:  HW Day 7,
Read for this, pp. 46-55.   Read ahead, table use pp.57-8,  rest of sec. 1.3

Hand in *(Temperature handout and Handout on Densities are in the folder outside my door, if you missed class.) A) The U.S. is almost the only country left that uses Fahrenheit to measure temperatures. To change F to C (Celsius), you subtract 32, and divide by 1.8.   HANDOUT with both scales ("Alias").  a)  The temperature as I write this is 500 F.  Calculate the temperature in C, and mark the temperature on the handout. (Check your calculation on the scale) b)  If the mean high temperature in Ithaca during  Feb. is 40o, and the standard deviation is 100 F., and you want those in Celsius instead, what do you do? Calculate  the results.  Check your results with the handout scales.    >Handout on Densities (fill in everything) Moore: p. 51, 1.50, 51, 52 general densities, mean &median Do with  Day 8: Moore 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

Read, to discuss

Optional (more practice)                          1.55 wechsler ais, 68etc rule