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. 
Read ahead:  Normal quantile plots, 65-67; then Ch. 2

Hand in: 
Normal density p. 84ff. Always sketch the curve, mark the area(s) you need.
p. 57, 1.101&2, test scores, Rule
1.120 (sketch) 1.122 (pregnancies, rule)
1.165 compare 2 curves (graph, p. 76)
1.112 (Women talk more?)  Use the Rule to give the limits containing 68, 95, and 99.7% of the speakers, M & F; and use that to answer (b).  (I.e. are these data Normal?)
1.24 Use  Normal Density Curve  Applet to find Quartiles in Normal
p. 681.103 test score z-scores
1.132&1.33 (SAT/ACT, compare)  Find the z-scores.  Also sketch both normal curves, labeling  the axes  in points, at + 1 s.d.  and mark all the students' scores on them.


Postpone all the rest:  But you can, if you want, use the applet to get (approximate) answers, sketch the curves; leave space for the computations.
1.126 (forward), 1.128 (backward) (standard normal table)
- - - - - - - - - - - -
The rest of these on separate paper as part of Day 8: Do all of these using the  Normal Density  Applet  http://www.whfreeman.com/ips7e/ to get answers. Leave space to do the calculations to get the answers from Table A. Always sketch the curve, mark the area(s) you need.
B. Simple warmup: Non-standard normal, table problems: 
    X is normal with mean 3 and s.d. 2. Find: 
    The proportion with X<1.5.  1.5 < X < 4.5. 

p. 63, 1.105 & 106 test scores, proportions

1.144 (osteoporosis)  Also sketch both curves on the same axis.
1.145 a, b (pregnancy)  (c will be assigned later) 
p.77 1.176d    (75's are Raw scores.  You could transform each to the N(100,20) distribution, if you remembered your formulas from parts a and b, or (shorter) you can find the percents asked for directly from the separate grades' distributions.  In each case you've found the percentile for  a grade of 75.)

C.  A small difference in means may give a surprisingly large proportional  difference in tails: 
a) Return to the data of 1.165, p. 75&6, and calculate what proportion of Females score below 450 and what proportion of Males score below 450.  Answer this:  The proportion of Males scoring below 450 is ____ times  the proportion of Females scoring below 450.
b)  Some of the difference in  (a) may be because the Male s.d. is larger, not just the difference in means.  Recalculate the proportion of Males who would score before 450 if their mean were still 565 but their s.d. 49 (= F s.d.)  The proportion is now ______.
This assumes, of course, that the Normal model holds up into the tails;  always questionable.

D. What proportion of pregnancies last 310 days or longer?  (see 1.145).

 Read, discuss 
  p. 69, 1.109&10: Sketch Normal;  shift and stretch



Optional 


Normal density
1.121 Use Normal Density Curve  Applet to check on the Rule
1.23 (horse preg's , rule)

Postpone the rest:

1.127 (forward), 1.129 (backward) (standard normal table, more practice)
1.130 & 131  (Wechsler WAIS, more practice)

Use   Normal Density Curve Applet  http://www.whfreeman.com/ips7e/  (or theapplet for ips6e, etc.) to check on all your Normal calculations!  (The Applet goes by .02's, and the text by .01's so the answers may differ slightly)


   
 



 
 

 
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.
San Diego Reader
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/ 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).                                 

.  Reading table "backward":
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:   - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
"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.
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