AP Statistics Section 9.3ASample Means

In section 9.2, we found that the sampling distribution of is

approximately Normal with _____

and ___________ if what 2 conditions are met? _______ and

_________________

p̂

p̂

p

n

pp )1(

nN 1010p)-n(1 and 10 np

p̂

In Section 9.3, we need to look at the sampling distribution of , the

sample mean.x

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 ______________ than individual

observations.

Means of random samples are ______________ than individual

observations.

variableless

Normal more

The Mean and Standard Deviation of Suppose that 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 is ____ and its

standard deviation is ______.You should use the recipe for the standard

deviation of only when the population is at least ____ times as large as the sample.

x

x x

x

n

10

x

x

The behavior of in repeated samples is much like that of the sample proportion .

Since , is an _________estimator of the population mean .

The values of are _____ spread out for larger samples. Their standard deviation decreases at the rate , so you must take a sample ___ times as large to cut the standard deviation of in half.

xp̂

x unbiased

x less

n 4

x

These facts about the mean and standard deviation of are true no matter what the

population distribution looks like.x

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 .

That leaves just the shape left to discuss.x

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

standard deviation __________

n

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.

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17.29

172167

z

.5684 :Calc

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 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.

x