Math 151 , Spring 2005, Day 8 Wed. Feb. 16 Hit reload ...After Class

HW Day 8 (Fri. Feb. 11): 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.
Technology:http://www.whfreeman.com/scc/ , Statistical Applets, Normal Density.
Uncheck the 2-tail box for most uses. Mean0,s.d.1 for Z's. 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!

Day8 Hand in (All D&V) (from Day7)68-95-99.7 rule:  Ch6 p. 99ff: Sketch normals&mark, do questions. 11 guzzlers 14 Rivets (d is a judgment call--depends on circumstance to some extent...) 13 downhill (for d:  Data is in order already.  Stemplot or histo-by-hand (widths=1) is quicker than going to SPSS.) 18 %white  (for d, your answer can be rough. Noting where Q1 is may help in guesstimating.) + + + + + + + + + + + +  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 - - - - - - - - - - - - - - - - - p. 99,  #9 Professors 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  (Note:  22d finds what numbers from the 5-number summary?) = = = = = = = = = = = Part of Day9 HW:  Table use:  "Real" or "Raw"values.  Start these tonight, finding the answers to all parts using one of the Technology tools.  Do the parts by hand (paper tables) as we learn how.  "Backward" parts  are marked with *. 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  See Day6 on Normal. - - - - - - - -  More table practice: z's: p.101 #19, 21

Homework Questions?  Compare, write question numbers on board.

Normal distribution.  Introduction Day 7 , using 68-95-99.7 rule
Showed: "Quincunx" falling bead model for Normal distribution--small independent influences.

  ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ First standard normal table use, then with "real" values~ ~ ~ ~ ~ ~ ~ ~ ~ ~
Standard Normal N(0, 1).  Our tables give area to the left of a z value.  TableZ, Appendix E, A-50
Using standard normal table:  See D&V p. 88.  (Wrong side of graph is shaded in my text)
       z |  .00     .01     .02 .....
      ...|
     1.4 | .9192   .9207   .9222 ....
   P(z < 1.40) = .9192,   P(z < 1.41) = .9207  P(z < 1.42) = .9222.
                                              ?z has more than 2 dec. places?  Round to 2.

    Sketch the density, mark the area you're looking for.
    Figure out how to get it using areas to the left of one or more z-values.
        Think cutting up paper bell-curves. (Remember whole area is 1.)  Like handout.

Example:  Proportion of observations between 0.5 and 1.4  P(0.5 < z <1.4) =
            Proportion of observations below 1.4  minus Proportion of observations below 0.5
               P (z < 1.4)  -  P(z < 0.5)  = .9192 - .6915 = .2277

.
Example:  Proportion of observations above  0.5,    P( z > 0.5) =
                ONE minus proportion of observations below 0.5,   1 -  P( z < 0.5) = 1-.6915 = .3085
.  Reading table "backward":
What z value has area ..... to the left/right of it?
        Sketch  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.
Example:  "What z value has 10%  of the observations above it?"  This is the same z as the one for:
        "What z value has 90% of the observations below (to the left of) it." (What z is the 90th percentile.)

        Find in the table  .8997 and .9015 --  .9000, our number, is between them.
                    .8997 is a little closer to.9000, so use it.
           For .8997, the z value is 1.28.   1.28 is the 90th percentile.
            1.28 has 10% of the observations above it.

Start here Friday: Real/Raw data:
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)

•     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.