MATH 251, Probability and Statistics I, Fall 2005, Day 7

Day 7, Friday Sept.9
Handout on Assessing Normality (Normal probability plots)
Read rest of section 1.3, including Normal Quantiles p. 80-84, and handout.  Chapter 2 for next time.
Hand in: using the table  (copied from day 5)
Always sketch the curve, mark the area(s) you need.
Non-standard normal, table problems: 
    X is normal with mean 3 and s.d. 2. Find: 
    The proportion with X<1.5.  1.5 < X < 4.5. 
   The x for which 30% of the observations are 
              smaller than it. The 30th percentile.
1.96, 1.97  (Wechsler WAIS)
1.112 (osteoporosis)
1.113 (tails)
1.115 a, b, c (pregnancy)
p. 98, 1.141 d.  78's are Raw scores.  I don't know why the first sentence is here--seems irrelevant to the question.

B.  What percent of pregnancies last 310 days? 
    (cf. 1.87, 1.115, see quote below)

p. 90, 1.116 (compute quartiles, pregnancy.  cf. 1.90)
1.101 (ACT equiv. SAT)
1.106 (SAT goal)
= = = = = = = = = = = = = =
Normal quantiles: Handout
1.121 (distances: granularity)
1.122  (match the quantile plots)
Use SPSS to make histograms or stemplots, Q-Q plots, and Method 2 Normal quantile plots (like IPS's) for the following.  Comment on what you see.
1.125 (logging) To use each group of data separately, Data>Select Cases (SPSS handout p.5 bottom)     
 1.127 To create the data,  Transform>Compute: RV.Uniform(0,1) (SPSS handout p. 8 bottom)		 Read, discuss 
 
 
 
 



= = = =
Normal quantiles: 
1.119, 1.120
 Optional 

X is normal with mean 3 and s.d. 2. [Find the x for which 80% of the observations are smaller than it.  The 80th percentile.] 

[(Following 1.116, do 1,117, 1.118, IQR and outliers in Normal]
 


Find answers to normal table problems with SPSS (Handout p.9) 

[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?]
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HW questions from Day 4?  SPSS questions?

Normal distribution, continued:  Table use
  Find proportions from raw x values Day 5
"Backward problems"  "What raw (x) value has area ___ to the left/right of it?"
        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: "W" test:  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 "W" score x= mean + z (s.d.) =  110 + 1.28 (25)=  110 + 32  = 142
Percentiles:  a "W" score of 142 has 90% of the scores at or below it.  142 is the 90th percentile.
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How do you know if it's safe to treat a data set as if it comes from a Normal Density model?
 Assessing Normality
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