### MATH 251, P & S I, Fall 2011, Fri. Sept. 9,Day 7..After class.  Hit reload!

Day 7, Friday.
Reading:  IPS7e 1.3 Density curves 50-54,  Normal distribution, pp.54-64.
For D: [In 1973] the following item appeared in Dear Abby's "advice" column:
Dear Abby: You wrote in your column that a woman is pregnant for 266 days. Who said so? I carried my baby for ten months and five days, and there is no doubt about it because I know the exact date my baby was conceived. My husband is in the Navy and it couldn't have possibly been conceived any other time because I saw him only once for an hour, and I didn't see him again until the day before the baby was born. I don't drink or run around, and there is no way this baby isn't his, so please print a retraction about that 266-day carrying time because otherwise I am in a lot of trouble.
Abby's answer was consoling and gracious but not very statistical:
Dear Reader: The average gestation period is 266 days. Some babies come early. Others come late. Yours was late.
The question here is not whether the baby was late. That fact is already known. At issue is the credibility of the length of the delay. Ten months and five days is approximately 310 days, which means that the pregnancy exceeded the norm by 44 days. [How unusual is that?]

Day 4: B, and 1.76 ab, discussed Day 5 Solutions  B Will be on in-class quiz after a while.  (Only need it for n=3, but work on getting used to "Sigma" computations.
Density handout solutions
Other HW questions?    How is SPSS going?  Morganstore (messy) but has Macintosh instructions

Quiz today at 9:55.  Stop me!  Closed book, notes; I can lend a calculator. Please bring finished quiz to my office, Mac 102 (under the door if I'm not there. ) I'll probably be in Xerox/print room or 101 if you have questions.

Densities--abstraction from histogram.  "Model"  Day 5
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.
--Greek labels "mu" for mean and "sigma" for std. dev. of a Density.

NEW today:  Details Day 5  (high points here)
"Normal" Density :
("Gaussian", "Bell-shaped")  Normal Density  Applet  http://www.whfreeman.com/ips5e/
• Knowing parameters mean (µ "mu") and standard deviation ("sigma") tells you everything. "N(mean, s.d.)"
• "Standard normal": mean = 0, s.d. = 1  Standardized : how many "s.d.'s from the mean".
• 68-95-99.7 rule:  68% within 1 s.d. of mean, 95% within 2 s.d. of mean, 99.7% within 3 (roughly).
Standardizing: " z-score" A "raw value" x is standardized by telling how many standard deviations above the mean it is.
Got this far Friday.

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.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ First standard normal table use, then with "real" values~ ~ ~ ~ ~ ~ ~ ~ ~ ~

OPTIONAL aids:  Normal curve template (you can count squares), Normal practice handout
Standard Normal N(0, 1).  Our tables give area to the left of a z value.  Table A, front flyleaf
Using standard normal table:  p. 77
z |  .00     .01     .02 .....
...|
1.4 | .9192   .9207   .9222 ....
P(Z < 1.40) = .9192,   P(Z < 1.41) = .9207  P(Z < 1.42) = .9222.
...

Find desired area above or between by rewriting as a subtraction involving area(s) to the left of the endpoint(s).

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 is 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 is the 90th percentile.
1.28 has 10% of the observations above it.

- - - - - - Real/Raw data:  -- - - - - - - - - - -

"What proportion"problems:
•     Sketch a normal curve. Mark mean, 1, 2 s.d.'s.  Label with raw values, and z-values below.
•     Mark end points for problem, roughly, and shade area desired.
•     Standardize end point(s).  Use standard normal table to find area. (Draw helper sketches if needed)
•     Check picture to see if it's plausible.
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
"Backward problems"  "What raw (x) value has area ___ to the left/right of it?"
Sketch  the curve, labeled with x values and z values, and the Area, 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.
Convert the z to an x: z is the number of standard deviations above the mean.
x= mean + z (s.d.)
Percentiles:  a "W" score of 142 has 90% of the scores at or below it.  142 is the 90th percentile.

 Sievers home Math251-Fall11/Dayq7.htm 10pm 9/9/11
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