Math 151 Day 7

Math 151 , Fall '07, Friday, Sept.7, Day 7After class Hit reload..

HW Day7: Ch. 3: Read Density curves pp. 64-9 .// Read ahead, please: 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;. 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 or link: if you missed class) Solutions
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
== = = = = = = = = = = = = =
Postpone: Normal distribution: Use the Applet: Normal Density Curve http://bcs.whfreeman.com/bps4e
(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

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)

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

First hourly exam Friday Sept 14, . Sample exam will be handed out TODAY; solutions will be linked here, probably Sunday, outside my door & on reserve Monday. Closed book, but bring one sheet of notes (anything you like) and a calculator.
Exam will cover thru what is assigned Monday in Chapter 3. You may be asked to read SPSS output, but not how to produce it.
You may 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!
Wednesday: bring questions.
Questions on HW Day5?   A. cars down the highway?
    on Standard Deviation? Review Std. Dev. Day 5
Questions on SPSS? Day 6   Where have people used it successfully?
Solutions for SPSS HW problems is posted in Mac 101, linked here..
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Density curves, BPS4e pp.64-69
GET handout HW sheet: "Density curves"
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.

Median, mean, percentiles, standard deviation are defined for a density curve in analogy to those for a histogram.

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.

Monday:

"Normal" distributions:("Gaussian", "Bell-shaped") part 1 (pp. 70-74)
Applet: Normal Density Curve http://bcs.whfreeman.com/bps4e

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.

"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).

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

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. Table A)

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