Math 151 , Fall 2005, Day 10 Fri. Sept. 16 Hit reload...

--Exam 1  Friday Sept 23, Day 13, in class, closed book.   Bring a simple calculator. I will give you copies of the Normal table.  You may stay late into the lunch hour if you like. If you will need extra time, and can't stay into the lunch hour, make arrangements with me by next Wednesday, please.
Covers Part I:through p.112  but NOT: "backward" normal problems with raw data like p.90..Percentiles II, p. 92-3 questions 2,3.  You will need to read SPSS output, but not tell how to produce any. You will need to calculate "by hand" a standard deviation for four numbers. (As well as medians, quartiles, etc.)   Problems like HW + some true-false or multiple choice types.
 Sample exam  will be available Monday; solutions outside my door.
Day 9(Fri. Feb. 18): Reading: Finish D&V Ch6 pp. 86-98.  (Questions 2&3 p.92-3 are optional. Normal Prob. Plots p. 94-95 is  Optional, but don't miss What Can Go Wrong, p95 bottom).  AS Ch. 6, in order. Note 6-4 ¶1, Normal table does UPPER tail.  Almost all normal tables do LOWER tail.
Then: Quick Review p.103.  Ahead, D&V Ch7 Scatterplots, first thru 117 (AS7-1&2), then Correlation, the rest. (AS7-3&4)
Technology:http://www.whfreeman.com/scc/ , Statistical Applets, Normal Density.  Uncheck the 2-tail box for most uses.  OR ActivStats Normal Density Tool : for best setup Use AS30-2 "Normal distribution based Confidence Intervals tool" CAUTION: Don't hit the Enter key! It closes the tool-box!
Hand in (All D&V)
standardizing: Ch6 p. 99:  Sketch each Normal model and label its axis with both the "real/raw" values and the "z" values.  Mark the observations on the pictures, do questions.
  5 temperatures
  6 placement exams
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Table use: Always sketch the model first, mark the area you are looking for!  Find the answers using Table Z, Appendix p. A-30. Check your answers with one of the Technology Normal tools (see above)
p.101 #20, 22 z's (Note:  22d finds what numbers from the 5-number summary?)

Raw data problems. Do unstarred parts with table Z.  "Backward" parts  are marked with *--do them with a technology tool (see above)
p.102,  25 Cholesterol  a, b, c,  d*, e* 
26 Tires  a, b, c,  d*, e* 
28 Body Temperatures: a, b, c*  Also:  I have a theory as to where the "wrong" number 98.6F came from. Early work on temperatures all took place in Europe. Convert 98.6F and 98.2F to Celsius (subtract 32, and divide by 1.8).  What's my theory? 		 Read, 
 to discuss 		 Optional  (more practice)

More table practice: z's:
p.101 #19, 21

p.102, 27 Kindergarten a, b*, c* 

p.109, 25 BeQuick a,b,c,d,e*,f * 
 
Review exercises p. 104-112  These are good to look at and decide how to do, as exam prep.  Here's a list of problems NOT to try: 16a, 33, 34e,f(vague), 37,38.
  Homework questions?  Day 9
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Standardizing and Standard normal table use    Day 9
Standardizing: (review) A "raw value" x is standardized by telling how many standard deviations above the mean it is.
    Find z:  Subtract the mean from x.  This tells how far "above" the mean x is, in "raw" units. (Below the mean gives negative.)  Find how far this is in "standard deviations" by dividing by the standard deviation.

Examples: (Wechsler 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:  (D&V p.90-2:  "....I", "More..." question 1)  http://www.whfreeman.com/scc/

Example:  Proportion with scores between 100 and 145?

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?" (D&V p.90-2:  "....II")
        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)=  110 + 32  = 142
Percentiles:  a Wechsler score of 142 has 90% of the scores at or below it.  142 is the 90th percentile.

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