| Hand in, Sec.6.1 :
6.5 IQ test scores Read pp. 312-13 before doing this one. (already assigned) 6.3 density of x-bar, and confidence intervals. This problem combines the text Figures 6.2 and 6.4 For part d, to draw the confidence interval: just choose any point on the horizontal axis of your graph to be x-bar (Like me at the board, my head being xbar). Measure off the distance m (half the width of the shaded interval) and extend a bar m wide to the left and the right of your point,below the curve (My arm length was m). (Like fig. 6.4, the bars with arrows at the ends. The red dots show what the x-bar is for that confidence interval) Choose another point, and repeat.. If your first x-bar was in the shaded interval, pick your second outside the shaded interval, and vice versa. You should note that if x-bar is in the shaded interval, then the confidence interval bar covers mu (280) and if x-bar isn't, then the bar doesn't. - - - - - - - - - - - - - - - - - - - - - - - - - A. Shoeboxes: Separate sheet: You can get a sample of 4 each outside my door, or in class Wed. For each sample of size 4 from a shoebox, find the mean, (know which box you got them from: which: white #s, green box, yellow, red top) and tell whether you believe the population mean for that box is 20, or something bigger. Bring to class Friday to pool. The shoeboxes are outside my door if you missed doing them in class. - - - - - - - - - - - - - - - - - - - - - - - Cautions; general review and extension p. 314 6.14 internet, response rate p.317, 6.19 newts p. 318, 6.22 men/women CI's p.316, 6.18 consumers/pharmacies - - - - - - - - - - - - - - - - - review p. 315, 6.16 enlighten the unstatistical 6.17hotel mgrs. = = = = = = = = = = = = = = Postpone: Sample size for C.I. p. 3.11, 6.10, 6.11, 6.12 |
Read,
to discuss |
Optional |
Forgot to do: New Shoeboxes: Take 4 from each, write them down (White from green box, yellow from red-top box) For HW, find means (for each box separately.) Know which box! Does that box have pop. mean 20, or some number >20?
Closed
book quiz Wednesday:
1) Give a definition of a level C Confidence Interval for a parameter.
2) a) Write down the formula for a level C confidence
interval for the unknown mean of a normal population.
(Assume
the standard deviation of the population is known.)
b) Tell, or show with a picture,
how "C" connects with your formula.
# # # # # # # # # # # # # # # # # # # # # # #
# # # # # # # # # # # # # # # # # # # # # #
Confidence
interval estimate of a(n unknown) population parameter:
(Table
A, or Table C, t dist. bottom row)
Why does the formula work?
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 can be achieved only by
» accepting lower
confidence level (smaller C),
» or by increasing
sample size (bigger n).
Visual example: Author website, whfreeman.com/scc,
Applet: Confidence Intervals.
You can change the C, with the same
xbars, see the m change. (n and sigma are fixed.)
(50
intervals display. More will overwrite the drawings of old ones.
But they accumulate numerically in the right panel till you
reset.)
Sigma: We can't change it, it comes
with the population. But bigger sigma (more population variability)
will give bigger m (wider CI), i.e. less 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.
Start here Wednesday??:
Planning ahead: Choose sample size big enough to
satisfy desired: margin of error, confidence level.
Given C and m (and sigma), find n.
Method: Use C to find z*. Plug in to formula
for m, and solve for
n. Or memorize formula for n and plug in to it.
,
n = (z* sigma / m)2
Note: z* sigma still on top. m
and n change places, and whole thing is squared!
Round
up! If you get n = 5.06, you need a sample of size 6 to get your
margin of error at least as short as you want.
~~~~~~~~~~~~~~~~
Sec 6.2: "Significance tests use an elaborate vocabulary,
but the basic idea is simple:
an outcome that would "rarely" happen
if a claim were true--is good evidence that the claim is NOT true."
(p.314)
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