Math 151 ,  Sp. '08, Wed, Feb. 20, Day 11 .After class. Hit reload.

HW  Day 11  Ch. 3:  Use Normal Density Applet curve to check concepts and computation. //Standardizing to standard normal pp.74-76, "Check" 3.21. We WILL 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. 76-80, Cum. proportion and  normal.  "Check" 3.22, 3. 23. "Backward" from prop. to z pp. 81-83.  We'll revisit and learn to deal with x's. 

Reading and questions:  due Friday
"None of the above" by Malcolm Gladwell  on reserve and outside my door. 
Questions: 1) What is the Flynn effect?
2) What is a likely reason for it?

Normal distribution:
(Normal templates-optional-you can count squares--may help!)
DONE DAY 10:
p. 74 3.5 Women's hts, sketch
p. 74 3.6  Normal, women's hts--68-95-99.7 rule.
p. 74 3.7 pregnancies--68etc rule (This distribution may not apply to planned births, of which we now know there are a lot!)
p. 88, 3.51 check 68-95-99.7 rule , using applet: Normal Density Curve on your CD or at   http://bcs.whfreeman.com/bps4e.

Now- - - - Standardize: Draw and label the normal density curve, the "raw" axis and the "z" axis together, mark your value(s), as well as calculating.
p. 76, 3.9 mens & women's heights
p. 86, 3.33 ACT/SAT Jacob and Emily (Info is above #3.32)

- Using table with "z"'s--standard normal.---------
Table use--z: Always sketch a normal curve first, mark the area you are looking for!  Do these with the Applet: Normal Density Curve on your CD or at    http://bcs.whfreeman.com/bps4e. , and check with your table answers.  (Uncheck the 2-tail box for most uses. Mean 0, s.d. 1) 
  p.80 3.10 z's to proportions, using Table A.
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"Backward"--z :Always sketch a normal curve first, roughly mark the proportion=area you are given. 
p.83, 3.13 (backward z) Do with table, check using Applet: Normal Density Curve on your CD or at    http://bcs.whfreeman.com/bps4e.
p. 89, 3.52  Quartiles of normal dist.    Use the Applet and also, use table A to find the quartiles.  Your answers may differ in the second decimal place because the Applet only goes by .02's on the z-axis --.64, .66, .68... and Table A goes by .01's.

= = continue, keep your paper for the rest.
Standardize (find z-scores) for all the "x's"= =
= =  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/bps4e. 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. 87, 3.37 Jacob's score, and 3.39 top score. Mean, s.d. are before 3.32 on p. 86.
p. 87, 3.46 surprising difference in tails
 A. , What proportion of pregnancies last 310 days or more? Find Mean and s.d. in p.74, 3.7 (more next time on this) 
 p. 80-81 3.11 and 3.12 (locomotive adhesion, 2 dist's) 

..--- postpone---- "Backward Normal"-----------
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/bps4e. 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. 83, 3.14 IQ test
p. 87, 3.41 Abigail, top 20%.  Mean, s.d. are before 3.32 on p. 86.
p. 87, 3.42  quartiles  Mean, s.d. are before 3.32 on p. 86.

p. 179, 7.27 breaking bolts, (a, b +).  For (a), think carefully about which side of 90 you want: Does a bolt that breaks at 95 ksi qualify?  Does a bolt that breaks at 85 ksi qualify?   ALSO:  If they test every bolt and  throw away all bolts that break at 70 ksi or below, what proportion do they throw away?

 Read, to discuss Optional (more practice)

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p. 75 3.8  SAT & ACT  (Standardize)
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p. 86 3.30 z's to proportions
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"Backward" 
p. 86, 3.31 (backward z)
p. 89, 3.53
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postpone
"raw", "backward"
p. 87, 3.43, quintiles Mean, s.d. are before 3.32 on p. 86.  Quintiles are used by the government to report much economic census data.

new email: math-151@wells.edu

Clinic Schedule (current now) (Hit reload on the Clinic Schedule too)  Helpers

Questions on HW Day10?       Day 7   For 3.2 check your handout drawing and count squares or calculate areas.  3.3 is in the back of the book. 
3.1:  a) Graph (c) of 3.4 would do.  b) Graph (b) of 3.4 would do.
p.69, 3.4:  a) C is the mean, pulled to the right of the median (whether the median is A or C) by the long tail.  B is the median because it cuts the area in half:  A is clearly too far to the left to be the median (it's the "mode")
b) B is both mean and median; since the distribution is symmetric, the balance point (mean) and half-area point (median) are together at the center.  A and C mark the two "modes". 
c)  A is mean, B is median, C is "mode"--same reasoning as for (a) 
      Density handout   and Solutions!

"Percentile:" 
"The 38th percentile is 25 pounds"= "25 pounds is at the 38th percentile" = 38% of the observations are at or below 25 pounds.  
25th percentile of salaries = 1st quartile:  25% of the salaries are at or below the $ value of Q1.
    Note, the "somethingth percentile" is a number in the x-units; in the units of the variable you're looking at.  What Percentile is x at?  If you draw the density or histogram, it's the Cumulative Proportion to the left of x.

Show? "Quincunx" falling bead model for Normal distribution--small independent influences.
Normal distribution.   Day 10 + Today are notes for "all"of Ch. 3.  See how far we get!
 Handout:   Normal Templates: you can count squares for approximate answers using these density curves.
Applet: Normal Density Curve   http://bcs.whfreeman.com/bps4e
  Introduction Day 10: shape, mu & sigma,  using 68-95-99.7 rule, last time.  New:
      Standardizing: Day 10:
      Standard Normal table use.  Day 10:   and backward Day 10:

Start here Friday
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:  BPS4e pp. 78-80

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

x = 145 gives z = 1.4  (done above.)      Area to left of z = 1.4 is .9192
x = 100 gives z =  –.4                           Area to left of z = –.4 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)
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"Backward problems"  "What raw (x) value has area ___ to the left/right of it?"   BPS4e pp. 81-83.
        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.  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.

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