Day 11 Hand in FRIDAY
= = Using table with
"x"'s--"raw" values. = =
Optional (more practice)
p. 88, 3.29 (backward z)
p. 90-91 3.45 Osteoporosis (2 dist's)
Normal Distribution: Using Standard Normal Table "backward" (Day7 for details), Normal curve templates --> you can use these and actually "count squares" to (approximately) check your work if you like.
Using Standardizing and the Standard Normal Table together to do more general problems.
Normal Practice Handout
Review Standardizing: 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.
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", mean 110, s.d. 25)
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: BPS5e pp. 79-83
or P ( 100 < x <
145) = P ( –.4 < z < 1.4) = P( z < 1.4) – P(z
= .9192 – .3446 = .5746
Read "Proportion of x's with 100 <x<145" for P(100<x<145)
Proportion with scores above 145? 1– proportion with scores below.
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"Backward problems" "What raw (x) value has area ___ to the left/right of it?" BPS5e pp. 83-86.
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" it.)
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. Table A (Excel)
The "Classic IQ test"score x= mean + z (s.d.) = 110 + 1.28 (25)= 110 + 32 = 142
Percentiles: a "Classic IQ
test" score of 142 has 90% of the
at or below it. 142 is the 90th percentile.
Did it all. Next time, exams back, questions on the above work, start Ch 4.
**[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.Abby's answer was consoling and gracious but not very statistical:
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?]