| Nothing to hand in
Monday. If you kept your HW today, hand it in at the exam time.
Hand in after break Sec. 6.3, and notes Day 34 p. 347 6.58 500 tests for psychic powers 6.62 77 potential schizophrenia markers A. You have a theory that walls painted pale pink will have a mellowing effect on elementary school students and produce better grades. So you receive permission to repaint one classroom from each grade at the local school over Christmas vacation (the others stay as they were). Indeed, the students in the pink classrooms do better on end-of-year tests. What criticism can be made of your experiment, and how could it have been designed to avoid this? = = = = = = = = = = = = = = = = = Ch. 7, Sec. 7.1 "Standard error" & t-distribution family. p. 364 7.1, 7.2, 7.3 p. 373, 7.4 CI p. 386, 7.19 Shrimp ATP A common calculational mistake is to divide the SE by square-root-of-n. But square-root-of-n is already IN SE! Don't divide by it again! (I.e. pay attention to the difference between "standard deviation" and "standard error.") |
Optional
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START HERE Monday
Question last Monday: there are no "numbers" attached to how
big n, how close, how likely, in Law of Large Numbers:
"Take observations at random from a population with population mean
µ. Then as the number of observations n increases, the
sample mean xbar gets closer and closer to µ. "
But Central Limit Theorem gives us a back door
to such numbers: for instance, if n = 100, then 95% of sequences
of 100 randomly chosen numbers (SRS's of size 100) will have their xbars
within about 2(sigma/10) of µ. If we increase
n to 400, then 95% of sequences of 400 numbers will have their xbars within
about 2(sigma/20) of µ. We might be "unlucky"
and get a sequence that stayed away from µ longer, but
eventually (not defined here) we should get close.
The Law of Large Numbers focuses on the behavior of Xbar as we look
at larger and larger samples. The Central Limit Theorem on the behavior
of all possible xbars for a fixed sample size.
Sec 6.3, cont'd: cautions and
limitations:
Hawthorne effect and multiple test warning:
Day
34
= = = = = = = = = = = = = = = = = = = = =
Chapter 7, Inference for Distributions (we'll
do 7.1, 7.2, and the first segment, to p. 414, of 7.3)
Inference for means, using xbar from a SRS to make inference about µ:
|
Sigma known Sigma unknown |
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|
normal
Population is
not normal
|
Xbar is normal;
find z using sigma |
Xbar is normal;
find z using s. |
Xbar is normal;
find z using sigma |
Xbar is normal;
Find t using s |
| Xbar is normal-ish (CLTh);
find z using sigma |
Xbar is normal-ish (CLTh);
find z using s |
Unrealistic. sigma's
only "good" for normal pop's. |
(See p. 381)
If you can't use t, Find a statistician |
|
t-distribution
family: like standard normal only slightly fatter in the tails.
Mean = 0. Symmetrical around 0.
"Degrees of freedom" tell which member of
the t family. t(k) is the t distribution with k degrees of
freedom.
Comparison with normal (Excel
file)
Lower d.f.--fatter tails. Higher d.f.--more
like standard normal.
Table C: upper tail: probability
<--> "critical" t-value.
Start working on green box:
Assume Normal population . Mean µ, s.d. sigma, both unknown.
Take SRS, size n, find xbar, find s (sample standard dev.)
Standard error of the (sample) mean = s/sqrt(n)
Standard deviation of xbar, estimated from the data.
"Standard
error of the mean": s/sqrt(n) SEM, SEXbar,
etc. Just like sigma/sqrt(n), only
s from data replaces sigma.
When you estimate the standard
deviation of a statistic,
the resulting estimate is called the "standard error" of the
statistic.
Standardizing xbar with s instead of sigma results in
t = xbar -µ
s/sqrt(n) the one-sample t statistic
which has the t-distribution with n-1degrees
of freedom.
We'll now repeat all the stuff from Chapter 6, only wherever there was
a z, we'll substitute a t.
Here we go....
"One-sample" t- procedures:
SRS
of
size n. Use Xbar to estimate µ.
Substitute s for sigma in the standardizing
formula. We get t instead of z, with n-1 degrees of freedom.
It's a good idea to check
for at least approximate normality.
Confidence intervals:
xbar + t*
(s/sqrt(n)) Choose t* from table C, using the n-1
row,
and confidence level C.
Special case of common
pattern: estimate + t* SEestimate
Example: bacteria per milliliter in10
specimens of raw milk from one producer. Parameter: actual
mean bacteria/ml.
5370, 4890, 5100, 4500, 5260, 5150, 4900, 4760, 4700, 4870
4|5
n = 10, xbar = 4950,
s = 268.45 SEM = 268.45/sqrt(10)
=268.45/3.162=84.89.
deg. of freedom = 9
4|77
90% CI: from t(9) in table, t*
= 1.833 CI is 4950+1.833x268.45/sqrt(10)
4|889
4950 +1.833x84.89,
or 4950+155.6
bacteria/ml.
5|11
If we had KNOWN Population sigma = 268.45, we'd have used z* = 1.645, gotten
a narrower CI.
5|23
(but we don't know sigma!)
| Activstats t-distribution procedures:
Ch's 18 (CI's) pp 1,2,3 and 20 (tests)pp 1,2. For next, pp 1, 2 of each, or read Moore carefully - p. 18-1 Activities 1 and 2 introduce "Standard Error", review CI's from normal table, large n, s substituted for sigma Activity 3 shows how using s instead of sigma (with n=15) gives CI lengths that vary from sample to sample. - p. 18-2 Activities 1 and 2 introduces t-distribution and a CI with it. Activity 3 shows a t-table, like ours (See note--ours is easier. Activity 4 stresses assumptions.) - p. 20-1 Activities 1 and 2 introduce t-test, analyzing the data (same data as Moore p. 371, Eg. 7.2) (Activity 3, SPSS--we'll come back to that) Activity 4 is self-test. - p. 20-2: Activity 1 using t-tables, repeating sweetness data (Moore p. 371) Activity 2, repeats conditions for t test, same as for CI p. 18-2 activity 4. Activity 3, choosing a test. Cf. my webpage with the chart. |
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