| Hand in, Sec.6.1 :
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.
|
Read,
to discuss |
Optional |
New Shoeboxes: (Neglected to pass around) 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: Answers
are just below here
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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