Transcript AP Statistics Section 9.3A Sample Means

AP Statistics Section 9.3A
Sample Means
In section 9.2, we found that the
sampling distribution of pˆ is
p
approximately Normal with  pˆ _____
p (1  p )
n
and  pˆ  ___________
if what 2
conditions are met? _______
N  10n and
_________________
np  10 and n(1 - p)  10
In Section 9.3, we need to look at
the sampling distribution of x , the
sample mean.
A basic principle of investment is that diversification reduces
risk. That is, buying several stocks rather than one reduces the
variability of the return on the investment. The figure on the
left below shows the distribution of returns for all 1,815
stocks listed on the NYSE for the entire year 1987 (a very
volatile year on the market). The mean return for all 1,815
stocks was –3.5%.
The figure on the right below shows the distribution of returns
for all possible portfolios that invested equal amounts in each
of 5 stocks. A portfolio is just a sample of 5 stocks, and its
return is the average return for the 5 stocks chosen. The mean
return, , is still –3.5% but the variation among portfolios is
much less than the variation among individual stocks.
There are two principles that you should
understand at the end of this section:
Means of random samples are
______________
less variable than individual
observations.
Means of random samples are
______________
more Normal than individual
observations.
The Mean and Standard Deviation of x
Suppose that x is the mean of an SRS of size n
drawn from a large population with mean  and
standard deviation  . Then the mean of the
sample distribution of x is  x  ____
 and its

standard deviation is  x  ______.
n
You should use the recipe for the standard
deviation of x only when the population is at
least ____
10 times as large as the sample.
The behavior of x in repeated samples is much
ˆ.
like that of the sample proportion p
Since  x   , x is an _________estimator
of
unbiased
the population mean .
The values of x are _____
less spread out for larger
samples. Their standard deviation decreases at
the rate n , so you must take a sample ___
4
times as large to cut the standard deviation of in
half.
These facts about the mean and standard
deviation of x are true no matter what the
population distribution looks like.
In order to describe the behavior of any
distribution, we must discuss shape, center
and spread. We have already discussed the
mean (center) and standard deviation
(spread) of the sampling distribution of x .
That leaves just the shape left to discuss.
Sampling Distribution of a Sample Mean
from a Normal Population
Draw an SRS of size n from a population that has
a Normal distribution with mean  and
standard deviation  . Then the sample mean
has a Normal distribution with mean ____
 and

n
standard deviation __________
Example: Men have weights that are Normally
distributed with a mean of 172 lbs and a standard
deviation of 29 lbs. Find the probability that one
randomly selected man will weigh more than 167 lbs.
167  172
z
 .17
29
1  .4325  .5675
Calc : .5684
Example: Men have weights that are Normally distributed
with a mean of 172 lbs and a standard deviation of 29 lbs. Find
the probability that 12 randomly selected men will have a
mean weight that is greater than 167 lbs.
.7248
In the previous example, we knew the SRS came
from a population with a Normal distribution,
and we could, therefore, assume that the
distribution of x was Normal. What happens if
the SRS comes from a population where the
shape of the distribution is unknown or is
known to be non-Normal? This question will be
answered in our next section.