p. 338, 5.74 common names (What are your chances if you're taking a statistics course? Lower...)
.Postpone the rest
Sec. 6.1, confidence intervals, p.
using formula: ( ConfidenceInterval.xls Excel spreadsheet automates it-to check answers.)
6.11, 6.12 changes of m with n and C
Cautions (pp. 354-5)
.Postpone the rest..
6.27 Job satisfaction, Gallup, margin of error
6.17, 18 biomarkers
Quiz (Probably Wednesday (not before)): Knowing and using:
Mean and st. dev. for X-bar from SRS of size n (Summary p. 308)
Normal? Central limit theorem: says yes for all parent distributions, approximately, for n large.
If population(s) normal to start with, linear combinations stay normal (including X-bar), mean and s.d. follow algebra rules (as last quiz.) Binomial distribution formula (p. 329 bottom)
mean and st. dev. for Binomial: X (count), p-hat (proportion) (Summary p.332)
Homework questions? Day 29
General issue from Central Limit Th. (don't remember saying this...) What if the population is not 10 to 20 times the sample size? The real s.d. of the x-bars will be narrower than the sigma over square root of n formula. You may not "get to" normal as a shape. Sample of 4 grades from a population of 10 All possible samples. ...
Binomial: Tell me mean & s.d.
of X, p-hat.
Some sample p-hats; (n = 25, p = .6, .2) Compare with your graphs of distributions.
Growth of binomial X as n increases: Mean grows
like n, but spread grows like sqrt(n).
Sample Proportion as n increases: Mean stays at p, but spread shrinks like 1/sqrt(n).
Applet "Normal approximation to Binomial", Excel graph of Binomial
Did a bunch of HW Friday, only new material was Binomial formula.
Normal approximation Day 29
Ch. 6: Introduction
to Statistical Inference:
Requires: Random sample or Randomized experiment. (Our theory: Simple Random Sample usually)
First example: Use sample mean xbar to "estimate" (unknown) population mean µ
Mean of 4 grades estimates
population mean of all 10 ("known"= 69.4) Population dist, Dist. of all sample means.
E.g. 69.75, 64.25, 73.5 (Each is a "point estimate")
Interval estimate: xbar + margin of error (fudge factor) estimates population mean µ (69.4)
69.75 + 1: "µ is
between 65.75 and 73.75" True
69.75 + 4: "µ is between 65.75 and 73.75" True
73.5 + 4: "µ is between 69.5 and 77.5" False
73.5 + 5: "µ is between 68.5 and 78.5" True
64.25 + 4: "µ is between 60.25 and 68.25" False
64.25 + 5: "µ is between 59.25 and 69.25" False
A level C Confidence interval estimate of a(n unknown) population parameter: (p. 347)
Birkenstock Shoebox: You're
Confidence Interval of the form estimate + margin-of-error for the mean µ with Confidence level C: (p.346) Does yours capture the real shoebox mean?
Applet: Confidence Interval. Many sample means: shows it's not the individual interval that C describes, but the Method.
Got to here Monday
The Birkenstock box contains numbers from a normally distributed population, with population standard deviation 2.
You each constructed a 60% confidence interval for the unknown mean: (proof:)
n = 4.
Standard deviation of sample mean = 2/sqrt(4) = 2/2 = 1
z* for C = 60% is .841, so margin of error m is .841 times 1= .841.
To get the z* for C = 60% from the normal table, note that this is the middle 60%, which leaves 40% to be split between the 2 tails. So 20% above z*, and 80% below. Go into the body of table A, find 80%= .8000 is between values .7995 and .8023, closer to .7995. The z value with .7995 below it is .84. Table D gives it more precisely as .841.
How many people captured the true mean?
Previous classes,11/20 = 55% , 22/29= 76%. 9/18 = 50% , 11/20 = 55%, 15/22= 68%, 16/24 = 67%, 16/18 = 89%, 7/13 = 54%, 8/16 = 50%, 7/14 = 50%, 5/10 = 50%, 11/14=79%,. 8/17 = 47%, 10/16 = 63%, 12/19 = 63%. Combined 168/270 = 62.2%. This year's classes, (now) 24/31 = 77%, Combined 192/301 = 63.8%
Quite variable for small samples, but settling down?)
Graphed results, (151 combined with 251): Fall '07 CI's Sp. '08 CI's. Fall'08 CI's. Sp. '09 CI's. Sp. '10 CI's.Compare with Applet: Confidence intervals. This year's shows more uniformity (some who were "out" didn't graph!)
Extension: If n is large, we can
use the formula even if population is not normal.
(Because only the distribution of Xbar is used, and Xbar is normal! Central Limit Theorem)
Cautions read pp. 354-5