Math 151 , Day 33, Friday, Nov. 14, 2008 ..hit reload...

HW Day33  Continue Ch 14; read first to p. 354.  Then reread.  Know (memorize if necessary) the "boxes" pp. 346 and 347.  Continue with computational method, how C, z*, n, and margin of error m relate.
Check p. 356; in this order: intro: 14.12, 14.13.  Then calculating:  14.11, 14, 15,  Then relationship 14.18, 19, 20. READ also Ch. 16, pp. 387-391, remembering that all our knowledge about sampling still applies. (ignore "significance test" parts.) Check, pp.406-716.19, 21, 22, 23, 25, 26
Last, p. 355, choosing n for a desired C and m.   Check,  Finally sample size 14.17

Hand in All. Also,Begin reviewing for the exam; Sample exam (You can do everything but #2', after this assignment.)
Chapter 14, Confidence intervals
p. 348 14.2  margin of error, interval
p. 348 14.3 Applet:  , percent of captures of true mean, C = 80%.
p. 361, 14.38 Applet:  , percent of captures of true mean. C = 90, 95, 99%  Also, Notice the comparative lengths of the intervals!
p. 360 14.34..and
 p. 360 14.35  explaining confidence

Use the ConfidenceInterval.xls Excel spreadsheet to check your computations of confidence intervals below; but do them by hand, as you'll need to for exams.
p. 352, 14.5 analyzing pharmaceuticals
p. 353, 14.6 IQ Test scores.  The sample mean is about 105.84, to check your calculator's result.
p. 359, 14.27 wine stinks

p. 354, 14.7 n and margin of error
p. 354, 14.8  C and margin of error
p. 358, 14. 21, 22, & 23  Hotel managers' personalities
p. 360, 14.30 & 32  Study times, outlier
p. 361, 14.36 Crime, Margins of error

p. 389, 16.1 b only (the answer to a is "yes"; checking it is optional)
p. 391, 16.3 environment--phone poll error
p. 392, 16.4 a and c only  holiday spending  (the answer to b is 237+ 10.59, checking it is optional)
p. 406, 16.29 (Hotel managers again)

Read, 
to discuss

p. 361, 14.37 newspaper  poll
 Optional
A few  problems good to review for the exam
p. 419, 17.7 Day care, parameter or statistic
p. 422, 17.27 and 28 means vs. individuals.  In #27, they're taking the "about what range" to be the interval containing the middle 99.7%--almost all. (Answer to last question of #28 is
"no"--histogram of individual values in sample will be distributed (roughly) like the population.)
p. 421, 17.26 WAIS, n = 1, n = 60 (Answers: a) about .3707, b) 100, 1.936, c).0049, d) a could be quite different; b still correct, c approx. right bcs of Central Lim. Th.))

Exam 4 a week from today.  Covers Ch 10 (continuous dist's) & 11, thru what we cover Monday of Ch 14, 16.   Sample Exam . (Handed out.)  Solutions linked here.
Quiz returned. answers 

Know the difference:
  µ, the population mean (a parameter)
  xbar, a  mean from a sample (a statistic)
  Xbar the Random Variable, has the distribution of all xbars from all possible samples (this distribution also has mean µ).

Fuzzy central limit theorem, Day 32

Ch. 14, Confidence intervals Details Day 32
Recap:
Confidence interval estimate of a(n unknown) population parameter: (pp. 346-7)

Confidence level C:  example C = 90%.  A 90% confidence interval is one made by a method that has success rate 90% at capturing the real mean.  For any particular interval, we don't know if it's one of the 90% that contain the real mean or one of the 10% that miss.
Applet:  Confidence intervals.     You each made one 60% confidence interval using your sample from the shoebox.  This class, CI's.

  ______________
Use sample mean xbar  to "estimate" (unknown) population mean µ
Requires: Random sample or Randomized experiment.  (Simple Random Sample usually)

"Simple conditions" to develop concepts.
     -- SRS. Most important, now and forever.  No "difficulties", no bias   (Population is at least 10 to 20 times as big as sample)
    -- Variable X is perfectly Normal, mean  µ, s.d. sigma.  (We'll extend from this later)
   -- µ is unknown, but sigma is known!  (we'll remove the sigma-known condition later)

Details and examples Day 32:
Confidence Interval of the form  estimate + margin-of-error  for the mean with Confidence level C: (pp.349-50)  so CI is xbar +  z* (sigma)/ sqrt(n)
.
 Check your calculations with the ConfidenceInterval.xls Excel spreadsheet

+ + + + + + + + + + +
Now: Relation of m (margin of error, half width), C (confidence level), and n (sample size), (and sigma)
    C and z* get bigger and smaller together (bigger C means bigger z*, and vice versa) (standard normal sketch)
,    m = z* (sigma)/ sqrt(n)
    Want bigger C?  Must accept bigger m.  Trade off confidence vs. accuracy ("sharpness").
    But bigger n will make smaller m. This makes sense: bigger sample size, more info-->more accurate estimate.
            (square root makes it Expensive: have to quadruple n to make m half as  big)
    So smaller m, margin of error,  can be achieved only by
        » accepting lower confidence level (smaller C, smaller z*),
        » or by increasing sample size (bigger n).       
        » Sigma:  We can't change it, it comes with the population.  But smaller sigma (more population variability) will give smaller m (narrower CI), i.e. more accuracy in prediction (for the same C and n).

Science  projects directed by Prof. Wahl:  Experiments on chickens bred to be "identical"--very low variability from one to the other.  Therefore very small samples suffice.

Extending "simple conditions":  Ch 16
SRS:  NEED this or a reasonable facsimile.  Bad sample, bad result!
If n is large,
-- the sample st. dev. s (calculated from the data) will be very close to the population s.d. sigma, so we can use s instead of sigma in the formula and be close to correct. (n > couple hundred is quite safe.)
-- the distribution of the x-bars  is really what has to be normal for the CI formula, so the Central Limit Th. allows us to use the formula even if the population is not very normal (but outliers in the sample or strong skewness can mess it up). (n> 25 if population is more or less mound-shape, not too skewed) 

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