Math 151 , Sp. '12, Wed., Feb. 15, Day 11 .After class. Hit reload..

HW :  Ch. 3:  Use Normal Density Applet curve to check concepts and computation. .Rest of chapter. //Standardizing to standard normal pp.77-79 "Check" 3.21.  We WILL learn to use table A. Moore doesn't separate out reading the z-table in the following; focus on just the z-table parts on first reading:  p. 79-83, Cum. proportion and  normal.  "Check" 3.22, 3. 23. "Backward" from prop. to z pp. 83-86.  Then we'll revisit and learn to deal with x's.  "Check" 3.24.
Read ahead:  Read Ch. 4 (Scatterplotts and correlation) to p. 104 Check p.112  4.14, 15, 16,   and NEXT  pp. 104-112 (correlation) Check 4.16 thru 4.22.  You do not have to be able to calculate r by hand.  You should be able to guess roughly at an r for a swarm of data; as p.108-9, and know and  be able to use facts 1-4, p. 107, and Cautions 1-4 pp. 108,110.

Day 11 Hand in  FRIDAY MONDAY:  
- Using table with "z"'s--standard normal.--
= = = = = = =
"Backward"--z :Always sketch a normal curve first, roughly mark the proportion=area you are given. 
p.84, 3.13 (backward z) Do with table, check using Applet: Normal Density Curve on your CD or at    http://bcs.whfreeman.com/bps5e.
p. 93, 3.53  Quartiles of normal dist.    Use the Applet and also, use table A to find the quartiles. 

= =  Using table with "x"'s--"raw" values.  = =
 Begin these by drawing and labeling the appropriate normal curve for each question, leaving space for computation.  Normal templates-may help. Then use the Applet: Normal Density Curve on your CD or at   http://bcs.whfreeman.com/bps5e. to find the required values.  Write these on your paper.  Next, standardize the endpoints for each area (proportion) you are asked for. Next calculate the values using Table A.  Your answers from each method should be very close (the Table gives a bit more accuracy than the Applet.)
p. 83, 3.11 Monsoon rains
p. 90, 3.39 Jacob's score
p. 90, 3.43 surprising difference in tails
A. , What proportion of pregnancies last 310 days or more? (Assuming distribution is Normal)  Find Mean and s.d. in p.87, 3.19 (Why do I care? See Dear Abby** very bottom of this page)
p. 89, 3.34 making tablets (adjust mean)

= = = = Backward Normal - - x problems .= = = =
Begin these by drawing and labeling the appropriate normal curve for each question, leaving space for computation.  Then use the Applet: Normal Density Curve on your CD or at  http://bcs.whfreeman.com/bps5e. to find the required values.  Write these on your paper.  Next, calculate the values using Table A.  Your answers from each method should be very close (the Table gives a bit more accuracy than the Applet.)
p. 89-90, 3.36, 3.37, 3.38 Gas Mileage (10th percentile, quartiles, quintiles.  Mean & s.d. are above #3.35)
p. 86, 3.14  quartiles, fruit flies 

=====Postpone the rest:  you can do the computations now but this spreads the "Normal" work out a bit. "Approximately" Normal ======
p. 91, 3.47  ACT scores (whole numbers)
p. 92, 3.50 (Use SPSS )  Monsoon rains  If you use Graphs to make the histogram, you can have it put a Normal curve over your histogram.

Read,
to discuss

Optional (more practice) 
 


"Backward" 
p. 88, 3.29 (backward z)


...

Normal Practice Handout

p. 90-91 3.45 Osteoporosis (2 dist's)  x-->Prop.

 
p. 86, 3.14 fruit flies (backward x, quartiles)
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EXAM 1 still not finished.  Sorry!

Questions on last HW? Day 10
  Normal:  68-95-99.7% rule.  Standardization.  Table A, z to proportion.Day7
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New material:
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

Example:  Proportion with scores between 100 and 145?  Table A (Excel)

x = 145 gives z = 1.4  (done before.)      Area to left of z = 1.4 is .9192  Proportion with scores below 145 is .9192
x = 100 gives z =  –.4                           Area to left of z = –.4 is  .3446  Proportion with scores below 100 is .3446
                                                Desired area = Difference=  .5746;  about 57%.  Looks about right from picture.

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)

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 scores at or below it.  142 is the 90th percentile.
Did it all.  Next time, exams back, questions on the above work, start Ch 4.

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**[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
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?]