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Chapter 18, Sampling distributions: proportions 1, 3 Coin tosses 5 More coin 9 Loans 13 Apples 14 Genetic defect Sampling distributions: means
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Homework questions? Day 27
Sampling DistributionsCh.
18
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. We'll look at 2 important ones:
p-hat, the proportion of some characteristic
Day 27
y-bar, the mean of some variable (e.g. sample
height)
Take n independently sampled values from a population with population
proportion p:
Sampling distribution of p-hat (read: distribution of p-hats
from all possible samples) is well modeled by
N(p, 
)
(if n < 10% of population,& np and nq >10)
Example: Flip "fair" coin 25 times.
Probability that my experiment will produce 15 or more heads?
15/25 = .60. P( p-hat > .60)=
P(Z >
1) = 16%, using the 68-95-99.5 rule.
Look at your flips. Some issues with "granularity"
-- only certain values are possible. But general shape looks right.
Sampling distribution of the mean,
y-bar:
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 µand
standard deviation 
.
Whatever the shape of the population
distribution that we draw the sample from, 
The mean of the y-bars = the
mean of the population
The standard deviation of the
y-bars =
the s. d. of the population divided by the square root of n.
* y-bar "hits" the population mean on average
(doesn't systematically go too high or too low.)
* Averages are less variable than individual
observations. Averages from large samples are less variable than averages
from smaller samples (because of dividing by the square root of n)
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.
Start here Wednesday
Example: "Normal" body temperature
98.6 deg. on average. (Assume this is true.)
Assume normal distribution, & s.d.among many
people is 0.6.
Probability that one individual's
normal temperature is below 98.0 degrees?
Take SRS
of 9 people. Sampling distribution of the mean? Probability
that the mean is below 98.0?
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!
SPSS simulation: average of spinners
which
can land on any number between 0 and 1.
Population--one spinner. distribution flat between
0 and 1, mean .49 s.d. = .29
n = 2, Average of 2 spinners is Xbar. Distributionof
x-bars is triangular between 0 and 1, mean .50, s.d. .21.
.29/sqrt(2) =.205
n = 4, Average of 4 spinners is Xbar. Distributionof
x-bars is normalish between 0 and 1, mean .50, s.d. .15.
.29/sqrt(4) =.145
n = 15, Average of 15 spinners is Xbar. Distributionof
x-bars is normal between 0 and 1, mean .50, s.d. .09.
.29/sqrt(15) =.076
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