Download 4-3 Sampling distributions

4.3.
Sampling distributions
1.
Law of large numbers:
a.
if we draw observations at random , and
b.
if the population has a finite mean µ
c.
as we continue to draw larger samples and calculate the sample mean for
each (cumulative) sample
d.
the sample mean eventually will get closer and closer to µ
2.
sampling distribution: the probability distribution of a sample statistic,
when the sample size is a fixed value n
a.
an empirical (simulation) approach to creating a specific sampling
distribution for n=10
(1)
take a large number of samples of size 10
(2)
calculate the sample mean (or sample proportion) for each
sample
(3)
make a histogram of the values of the sample mean (or sample
proportion)
(4)
examine the distribution (in the histogram) for characteristics
such as center, spread, shape, outliers
b.
for the sampling distribution of either the sample mean and the sample
proportion, you will notice that
(1)
the mean of the sampling distribution is the same as the mean
of the variable (eg, µ = µ x )
(2)
the standard deviation of the sampling distribution is less
than the standard deviation of the variable (eg, σ = σx /√n)
(3)
and the shape of the sampling distribution is approximately
Normal
c.
comments about each of the last results:
(1)
we say that is an unbiased estimator of µ. That is, in the
long run, the average of all the s from repeated samples would
be equal to µ
(2)
the standard deviation of a sampling distributiion has a special
name: standard error (SE)(in this case the standard deviation
of is the standard error, in another case it could be the standard
deviation of p-hat would be the standard error)
(a)
notice that the standard error is smaller than the standard
deviation of an individual value of x; and
(b)
the standard error gets smaller as n gets larger, so the
results from large samples are less variable than the
results of small samples
(3)
we can use the Normal distribution to compute probabilities of
approximately when µ and σ are known: ~ N(µ x ,σx /√n)
approximately or more briefly, ~ N(µ, σ /√n)
3.
Central Limit Theorem (CLT)
a.
the CLT add the third result above, about the shape being Normal
b.
actually, the CLT says the shape is more and more like a Normal
distribution as the sample size gets bigger; that's why it is a "limit"
theorem
(1)
the required sample size n depends on how non-Normal the
distribution of x is. See page 245, Figure 4.11 for an example
(2)
a widely used rule of them requires n≥30, but this is quite
conservative
c.
d.
now when using the Normal distribution: z=( –µ)/SE
the "shape" results also applies to more varied situations, eg: for the
sampling distribution of ∑x, not just (∑x)/n
e.
so this helps explain why the Normal distribution is so ubiquitous
(From Wonnacott & Wonnacott) A population of men on a large midwestern campus has a mean
height µ=69 inches, and a standard deviation σ=3.22 inches. If a random sample of n=10 men is
drawn, what is the chance the sample mean will be within 2 inches of the population mean µ?
µ=69 inches
SE = σ /√n = 3.22/√10 = 1.02 inches
P(67 ≤ ≤ 71) = ?
(67-69)/1.02 ≤Z≤ (71-69)/1.02
-1.96 ≤Z≤ 1.96
so … ? = 0.95
use z=( –µ)/SE