AP
Statistics
Section
9.3 “Sample Means”
Focus for this section is the mean because …
(i)
Mean is a commonly used statistic
(ii)
Means are less variable than individual
observations
(iii)
Means are more normal than individual
observations
is an estimate of p.
is
an estimate of 
.
If we take the mean of many
repeated samples and examine the sampling distribution of 
(with mean 
and standard deviation 
),
then
we’ll notice that its characteristics are very similar to those of the
distribution of 
:
(1)
is
an unbiased estimator of 
because the mean of
its sampling distribution IS .
(2)
The larger the sample, the less spread there
will be. The standard deviation
decreases at a rate of 
, so to cut the spread in half, you would need to take a
sample 4 times as large.
(3)
In order to calculate the standard deviation,
the population should be at least 10 times as large as the sample.
(4)
One fact re: sampling
distributions that holds true:
If a population has a normal
distribution N(
), then the sample mean 
will also have a
normal distribution N(
).
What our penny activity has
shown is that a population distribution doesn’t have to be normal to obtain the
normal sampling distribution of .
This fact is stated as THE
CENTRAL LIMIT THEOREM:
“Draw an SRS of size n from ANY
population with mean 
and finite standard deviation 
. When n is large
(rule of thumb: 
), the sampling distribution of the sample mean 
is close to the normal
distribution N(
).
*The further the shape of
the population distribution is from normal, the larger
the sample size n will need to be to see the normal sampling distribution.
(Ex)
of CLT in action
Ex.
9.12 p. 521-522
(Ex)
of how CLT can be used
Ex.
9.13 p. 522-523
When we worked with
probability, we said that it described the likelihood of an event over many,
many trials.
(ex)
Flipping a coin a few times, P(heads) may be 66%, but
over thousands of trials, it will even out to be approximately 50%.
This idea is referred to as the
LAW OF LARGE NUMBERS and can be applied to the mean:
With more and more
observations, the mean 
gets closer and closer
to the actual value of .
ASSIGNMENT:
P. 518-519 (9.31 – 9.34)
P. 524-525 (9.35 – 9.38)
Additional CLT ditto