Math 151 , Day 34, Monday, Nov. 12, 2007  after class. hit reload...

HW Day34 .  Ch 14; read first to p. 354.  Then reread.  Know (memorize if necessary) the "boxes" pp. 346 and 347.  Continue with computational method  pp 349-50, 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 (Test to here.)

Last, p. 355, choosing n for a desired C and m.   Check,  Finally sample size 14.17
Moore Ch. 14, Day 34  Hand in Wednesday  .Yes, all.
p. 348 14.2  margin of error, interval
p. 348 14.3 Applet: Confidence Interval  , percent of captures of true mean, C = 80%.
p. 361, 14.38 Applet: Confidence Interval , percent of captures of true mean. C = 90, 95, 99%  Also, Notice the comparative lengths of the intervals!
p. 360 14.34 and 14.35  explaining confidence

 Use the ConfidenceInterval.xls Excel spreadsheet to check your computations of confidence intervals; but do them by hand, as you'll need to for exams.
p. 352, 14.5 analyzing pharmaceuticals (find the sample mean by hand)
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")
p. 391, 16.3 phone poll error
p. 392, 16.4 a and c only  holiday spending
p. 406, 16.29 (Hotel managers again)
Read, 
to discuss

p. 361, 14.37 newspaper  poll
 Optional
A few review problems:
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.
p. 421, 17.26 WAIS, n = 1, n = 60
Exam 4 this Friday.  Covers  Ch. 10 p. 257 on (Continuous models and R.V.'s), plus of course being able to use the rules p. 253 for areas, in the continuous case. Ch. 11 up to p. 286 only, Ch. 14 to p.354, Ch. 16 to p. 391. (i.e. thru tonight's HW).   Sample Exam  (Handed out Fri. Outside my door..)  Solutions .
   Sign up Wed. for early start Friday:  Confirm with me any other time to take it.

Sampling distribution of the (sample) mean, Central Limit Theorem. 

"Fuzzy Central Limit Theorem:"
Data whose variation is due to  many   small    independent   random influences will have an approximately normal distribution.
  Balls and pins, heights of women, etc.  (p. 281, after the yellow box)
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  See Day  33 for details.  Quick summary:
Introduction to Inference: Chapter 14, Confidence intervals
Statistical Inference: drawing conclusions about a population from sample data.
    Requires: Random sample or Randomized experiment.  (Simple Random Sample usually)

"Simple conditions":  to develop concepts.
    --SRS.  No "difficulties", no bias   (Population is at least 10 to 20 times as big as sample)
    --Variable X  (population distribution) 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)

Use sample mean xbar  to "estimate" (unknown) population mean µ
   (xbar is a "point estimate" of µ)

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 made one from the shoebox.

Confidence Interval CI of the form  estimate + margin-of-error  for the mean with Confidence level C: (pp.349-50) ( Table A. , or  Table C., t dist., z* row (Moore, back flyleaf.) Example:  Sample of size 9 from a Normal population with unknown mean and pop. s.d. sigma = 6,  xbar = 12.
  Find a 90% CI estimate for the unknown mean µ: 
              z* = 1.645               (sigma)/ sqrt(n) = 6/3=2, so m = 3.290;
                       CI is 12 + 3.290, or  8.710 to 15.290.
    Check your calculations with the ConfidenceInterval.xls Excel spreadsheet

.New:.
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.
    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.
This part in gray is the only one we didn't touch on in class Monday.

Extending "simple conditions":  Ch 16
SRS:  NEED this or a reasonable facsimile.
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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