Math 151 Day 10

Math 151 , Sp. '08, Mon., Feb.18, Day 10 .after class. Hit reload..

HW Day10: Ch. 3: Read Density curves pp. 64-9 .// 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. Next: We WILL use table A. Moore doesn't separate out reading the z-table in the following; focus on just the z-table parts on first reading: p. 76-80, Cum. proportion and normal. "Check" 3.22, 3. 23. "Backward" from prop. to z pp. 81-83. We'll revisit and learn to deal with x's.

Some slack on spss:: hand in *at least* 1 spss problem today. rest by Wed.
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
== = = = = = = = = = = = = =
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!)
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DO Reading and questions: due Friday
"None of the above" by Malcolm Gladwell on reserve and outside my door.
Questions: 1) What is the Flynn effect?
2) What is a likely reason for it?

- -postpone rest- -start all, using Applet: Normal Density Curve on your CD or at http://bcs.whfreeman.com/bps4e.
- - Standardize
p. 76, 3.9 mens & women's heights
p. 86, 3.33 ACT/SAT Jacob and Emily (Info above #3.32)

- Using table with "z"'s--standard normal.--
Table use--z: Always sketch a normal curve first, mark the area you are looking for! Do these with the Applet: Normal Density Curve on your CD or at    http://bcs.whfreeman.com/bps4e. , and check with your table answers. (Uncheck the 2-tail box for most uses. Mean 0, s.d. 1)
p.80 3.10 z's to proportions, using Table A.
- -- "Backward"--z :Always sketch a normal curve first, roughly mark the proportion=area you are given.
p.83, 3.13 (backward z) Do with table, check using Applet: Normal Density Curve on your CD or at    http://bcs.whfreeman.com/bps4e.
p. 89, 3.52 Quartiles of normal dist.    Use the Applet and also, use table A to find the quartiles. Your answers may differ in the second decimal place because the Applet only goes by .02's on the z-axis --.64, .66, .68... and Table A goes by .01's.

Start the following, using the APPLET--keeping your paper to complete with the next assignment(s). Do hand computations as we learn how!.= = = = Using table with "x"'s--"raw" values. = = = =
Begin these by drawing and labeling the appropriate normal curve for each question, leaving space for computation. Normal templates-may help. Then use the Applet: Normal Density Curve on your CD or at   http://bcs.whfreeman.com/bps4e. to find the required values. Write these on your paper. Next, calculate the values using Table A. Your answers from each method should be very close (the Table gives a bit more accuracy than the Applet.)
p. 87, 3.37 Jacob's score, and 3.39 top score. Mean, s.d. are before 3.32 on p. 86.
p. 87, 3.46 surprising difference in tails
A. , What proportion of pregnancies last 310 days or more? Find Mean and s.d. in p.74, 3.7 (more next time on this)
p. 80-81 3.11 and 3.12 (locomotive adhesion, 2 dist's)

Read, to discuss

doA. 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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p. 75 3.8 SAT & ACT (Standardize)

Exams not finished.
Questions on SPSS? Day 7 Where have people used it successfully? Files for textbook problems are on your CD! See SPSS Info page for details.
Solutions for SPSS HW problems is posted in Mac 101, 110, linked here.. (Mac 101 is going to be closed a lot.)
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We started this wed. but didn't get very far.
Density curves, BPS4e pp.64-69
GET handout HW sheet: "Density curves" if you didn't
See Day 6 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.

"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.
a 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.

start here wed.
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)

"What proportion"problems: BPS4e pp. 78-80, first pass
Use Applet: Normal Density Curve http://bcs.whfreeman.com/bps4e
Proportion with scores between 100 and 145? below 100? Above 145?
What score is at the 75th percentile?

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ Standard normal table use~ ~ ~ ~ ~ ~ ~ ~ ~ ~
Standard Normal table use. Our tables give area to the left of a z value (Cumulative Proportions)
Using standard normal table: See text p. 76-80. Table A: p.684-5. Table A (Excel)
z | .00 .01 .02 ..... =number in "hundredths place" ...| -2.4 | .0082 .0080 .0078 ....= area to the left of "edge number" ...| 1.4 | .9192 .9207 .9222 ones&tenths

Proportion of z's below -2.40 = P(z < -2.40) = .0082
             = prop. of individuals 2.40 s.d.'s or more below the mean)
P(z < -2.41) = .0080 P(z < -2.42) = .0078 ,     P(z < 1.42) = .922
                                              ?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.)

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
.

Reading table backward: Table A (Excel)
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 value is at 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 has 10% of the observations above it.
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All of these can be checked using the Applet: Normal Density Curve http://bcs.whfreeman.com/bps4e

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