Day7 Math151

Math 151 , Spring 2005, Day 7 Mon. Feb. 14 Hit reload...After class

HW Day7(Mon. Feb. 14): Reading: D&V Ch6 pp. 82-98. (today 82-89) (Normal Prob. Plots p. 94-95 is Optional, but don't miss What Can Go Wrong, p95 bottom). AS Ch. 6 is very good, in order. (Today thru 6-3) (Normal Density Tool for you : Use AS30-2 "Normal distribution based Confidence Intervals tool for best setup*CAUTION: Don't hit the Enter key! It closes the tool-box!))

Hand in (All D&VCh6 unless otherwise noted)
Review: p.77#33, Phone Calls (use SPSS) (the confusing one from Day6)

Shift & rescale(D&V 84-85) from Day6
A) See Ch.5 p.72 #9and10, above. Pair #9 c is a shift. Check that the mean shifts correctly and that the s.d. stays the same (use the back of the book and your picture.) Pair #9bset1 and #10b set 2 is a shift followed by a rescale. (w=10(x-9)). Check that the mean undergoes the shift and rescale, but the s.d. undergoes just the rescale.
Ch6 p. 99ff: 1 Payroll (hint for d: each employee gets 110% of previous)
3 (SATtoACT)

B) 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"). Keep the handout.
a) The high temperature a few days ago was 50⁰ F. Calculate the temperature in C. (Check your calculation on the handout scale)
b) If the mean high temperature in Ithaca during Feb. is 40^o, and the standard deviation is 10⁰ F, and you want those in Celsius instead, what do you do? Calculate the results. (Check your results on the handout scale.)
- - - - - - - - - - - - -
C).Complete the Handout: Tables for simple models (densities)
= = = = = = = = = = Postpone the rest
68-95-99.7 rule: Ch6 p. 99ff: Sketch normals&mark, do questions.
11 guzzlers
14 Rivets (d is a judgment call--depends on circumstance to some extent...)
13 downhill (for d: Data is in order already. Stemplot or histo-by-hand (widths=1) is quicker than going to SPSS.)
18 %white (for d, your answer can be rough. Noting where Q1 is may help in guesstimating.)
+ + + + + + + + + + + +
standardizing: Ch6 p. 99: Sketch each Normal model and label its axis with both the "real/raw" values and the "z" values. Mark the observations on the pictures, do questions.
5 temperatures
6 placement exams

Read,
to discuss

Optional
Use technology to check on &
picture your Normal models:
Moore website http://www.whfreeman.com/scc/
Uncheck the 2-tail box for most uses. OR
ActivStats Normal Density Tool for you :
for best setup*
Use AS30-2 "Normal distribution based
Confidence Intervals tool"
CAUTION: Don't hit the Enter key! It closes the tool-box!

Normal Prob. Plots (D&Vp. 94-95).
Do AS6-4¶3, print out your graphs.
(Problem: SPSS Data set has Hospital charges (money) as String/Nominal, because the missing values were imported as characters.
Change the Type String to Numerical, & then you
can change Measure to Scale.)

Cluster chairs in 4's. Pool (write down) answers to p. 73#13 Marriage age. Ithaca Journal Jan 22, '05 had quiz answers: "How old is the average bride? 24.5 years.... How old is the average groom? 26.5 years." Give some reasons that could account for the big difference between these numbers and the graphed numbers.
HW questions? p.73#13 Marriage age. p. 79#38Holes

Shifting and Rescaling Day 6
, Optional SPSS handout to create new computed variables.
GET handout HW sheet: "Tables for simple models (densities)" Spinner. Use 248x310 pixels
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.

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.")

Many, many models are possible, modeling many phenomena: (Histograms of data for some models)

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.

Median, mean, percentiles, standard deviation are defined for a density model 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.

Numerical summary:
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).

You will make and use tables for the simple models on the handout. These are similar to the table we will use to describe the normal model. (Table Z, appendix E, p. A-50)

Start here Wednesday:
Symmetric, unimodal, no outliers, (not uniform) candidate for
"Normal" Model:("Gaussian", "Bell-shaped") AS6-1,2,3 are good. 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! ), acts like 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 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).

higher

Back-of-the-envelope: Sketch the curve, mark mean, + 1, +2, +3 s.d.'s-- in real units.

Example:

Standardizing: 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.

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 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

- - - - - - - - - - - - - - - - - - - - - - - - - - - -
* 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.

Sievers home

Math151-Sp05/Days7.htm

3pm

2/14/05

This page belongs to Sally Sievers who is solely responsible for its content. Please see our statement of responsibility.