Math 151 , Fall 2001, Monday, Sept. 17, Day 8 Final version

Normal distribution.  Introduction Day 7
   Using standard normal table:  See text p. 58.
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.

        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.
Day 8 lecture ended here

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:  85 is 1 s.d. below the mean.  Computation:  z = (85 – 110)/25 = (–25 raw points)/25 = –1 s.d. from mean.
           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

•     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.

or   P ( 100 < x < 145)  = P ( –.4 < z < 1.4) = P( z < 1.4) – P(z < –.4) = .9192 – .3446 = .5746
     Read "Proportion of x's with 100 <x<145"  for P(100<x<145)

HW  (The first ones repeat the problems at the bottom of Day 7)
Day 8, Feb. 14.  Read  rest of sec. 1.3.  We'll finish next time.

Hand in  p. 64 1.61 eyeball sigma p. 54 1.53&54 Normal, men's hts--68-95-99.7 rule. p. 64 1.63 pregnancies--68etc rule -------------------- table use: Always sketch the distribution first, mark the area you are looking for! p.61 1.57 z's . ------------------- "Backward"Always sketch a normal curve first, roughly mark the proportion=area you are given. p. 62, 1.59 (backward z) -------------------- standardizing:  with day 9

Read, to discuss

Optional (more practice)  1.55 wechsler ais, 68etc rule --------------------   p. 65 1.65 z's -------------------   "Backward"  p. 65, 1.66 (backward z) --------------------

These will be part of Day 9 hw standardizing: p. 56 1.56 SAT/ACT       p. 65 1.64 (cf. batting avgs) table use: Always sketch the distribution first, mark the area you are looking for! p. 64 1.68 a and b.Pregnancies Also, What proportion of pregnancies last 310 days or more? (see below.)  p. 61 1.58 (locomotive adhesion, 2 dist's)   p. 66 1.69 (Stanford-Binet, "superior")

Read, to discuss

Optional (more practice)  1.67  +Learn to use SPSS to find  the proportions--  manual pp. 58-59:"CDF"

[In 1973] the following item appeared in Dear Abby's 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

     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?]