| Hand in
(All D&V) Sampling distributions: means 17, 18 Sampling distribution 19 GPAs 20 home values Sampling dist. of means, cont. p. 350 B): "Normal" body temperature 98.6 deg. on average. (Assume this is true.) Assume normal distribution, & s.d.among many people is 0.6. What is the-- Probability that one (random) healthy individual's normal temperature is above 98.8? Probability that the mean of a sample of 4 is above 98.8? Probability that the mean of a sample of 36 is above 98.8? Probability that the mean of a sample of 100 is above 98.8? Note as n grows, SD shrinks, but only by square root of n! 21 c, d Pregnancy 23 Pregnancy skewed 22 Rainfall 24 At work 28 Potato chip bags Do the following on a separate page, and hand in with Day 30's assignment C) SD vs. SE: (p. 347, SE) a) You know that in the simulation of flipping a fair coin 30 times, p=.5. Calculate the SD(p-hat). (Check with the bottom of Day 28's covered material. This is the same no matter what actual proportion you got, since it is based on the population proportion p=.5. Now suppose you actually got 21 heads, p-hat = 0.7. And suppose you DON'T know p. Your best estimate of the SD(p-hat) is found by putting 0.7 in to the formula for SD. Do it. This is SE(p-hat), the "standard error of the sample proportion" (It's different depending on your sample). Compare with SD(p-hat), the "standard deviation of the sample proportion." b) Suppose you measure the heights of (a sample of n =) 25 daffodils and find they have a mean of 30 cm and a standard deviation of 2 cm. The seller of the bulbs says that the population of this variety of daffodils has standard deviation 1.7 cm. Find SD(sample mean) and SE(sample mean). A)Having gotten your 30 slips of paper from the green shoebox and counted how many 1's you have--Calculate the proportion of 1's, p-hat, for your sample (you did this already). Also plug in p-hat (instead of p) to the formula for SD(p-hat): so if you got 12/30, p-hat = .4. "q-hat" = 1 - p-hat = 1-.4 = .6. SD formula: square root of (p ·q/n)= square root of (.4 ·.6/30) = square root of .008 = .089 (This is SE(p-hat), p. 347). Bring these estimates of p and SD(p-hat) to class next time. (Shoebox is outside my door if you still need to do it.) |
Read,
to discuss |
Optional |
Homework questions? Day 28
Sampling DistributionsCh.
18 Details: Day 27
and
Day 28
: recap:
Take a Sample from a population. SRS!.
Imagine (simulate) what would happen if you
took "all possible" SRS's. For each sample, calculate
a
statistic.
Take n independently sampled values from a population with
population
proportion p:
Sampling distribution of sample proportion p-hat
(read: p-hats from all possible samples) is well modeled by N(p, 
)
if np & nq are >10 (at least 10 successes
and
10 failures expected), and n < 10% of population.
Sampling distribution of the
mean,
y-bar: Details: Day 28
Distribution of all means from all possible random samples of size
n from a population.
Need Random Sample, Independence (in particular,
for sampling without replacement, n < 10% of population.)
Population has mean µ and
standard deviation 
.
Whatever
the shape of the population distribution that we draw the sample
from, 
IF the population is Normal,the
sampling distribution of the y-bars is Normal.
The Central
Limit Theorem (CLT) In
any case, for "large" n, the sampling distribution of the y-bars is
Approximately Normal.
Average of
Spinners, Day 28
Start here Friday:
How large is "large"? How approximate is
"approximate"?
If the population was close
to normal, n doesn't need to be very large.
Even if the population is
pretty weird, n=25 gives a pretty good approximation to
normal.
But if we have really Big outliers or really BAD skewness, many need much
more.
Pictures on overhead.
Moore applet, "Central Limit theorem"
Next job: (Ch. 19) We usually DON'T KNOW the
population
parameter; use the statistic from our sample to ESTIMATE it.
YOU don't know the real proportion of 1's in the green shoebox.
Each of you has an estimate. In "real life" you won't have a
bunch
of classmates with other samples; you'll only have your own. (Also, in
this case, I know the real proportion. Not so in "real life") How
"good" is your estimate of the real p?
You know how the sampling distributions of sample proportions (and
sample
means) behave; we'll use that. But we want to know how much they
are spread, and for that we need the parameter p (and q) for
proportions,
(and the parameter 
for means)
And we don't know those! So we use the sample
statistics
p-hat and s in place of them.
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