Math 151 , Spring 2002, Friday, Day 9  After class

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

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"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.

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)
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Start here Monday:
"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.
            Multiply z by the size of 1 standard deviation.  Now you have distance above the mean, measured in raw units.
            Add the mean.  Now you have the "raw" value x. (You have "unstandardized")
Example:  What x value has 10%  of the observations above it?  This is the same x as the one for:
        What x value has 90% of the observations below (to the left of) it.

The table gives z = 1.28, approximately.  The Wechsler score x= mean + z (s.d.) =  110 + 1.28 (25) = 142

Percentiles:  a Wechsler score of 142 has 90% of the scores at or below it.  142 is the 90th percentile.

HW Day 9, Feb. 14.
Moore: Read  rest of sec. 1.3.  Ahead, Ch. 2.

Hand in (All from Moore.  Use the Activstats tool to check, if you like.) >>"Backward z"Always sketch a normal curve first, roughly mark the proportion=area you are given. p. 62, 1.59 (backward z) >>standardizing: I recommend drawing the two "rulers"--raw and z-score. 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 (with 1.68), 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") These backward normals will be part of Day 10 HW   >>"Backward Normal"Always sketch a normal curve first,   roughly mark the proportion=area you are given.   p. 76 1.89 (soldiers' heads)   p.64 1.68 c (pregnancy) p. 66 1.70 (z-scores of quartiles)  This one you can do, day 9

Read, to discuss

Optional (more practice)  "Backward"  p. 65, 1.66 (backward z) -------------------- 1.67                "Backward"  p. 62 1.60 (WAIS)

[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?]