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AP Statistics
Sampling Distributions.
How does x bar behave?


If the sample is collected correctly (easier said than done) AND the population size is at least 10
times the sample size.
X   and s 

n
More importantly the CENTRAL LIMIT THEOREM SAYS

If you draw a SRS from any population with mean  and standard deviation  if n is

sufficiently large the sampling distribution of x-bars will be approximately normal.
How large is large?
So what does this mean in a practical sense?
EXAMPLE: Suppose female heights are normally distributed with mean 64.5 inches and standard
deviation 2.5 inches.
1) What is the probability of choosing a female at random and her height being more than 66 inches?
2) What is the probability of choosing a sample of 5 females at random and having their average height
greater than 66 inches?
3) What is the probability of choosing a sample of 20 females and having their average height greater
than 66 inches?
EXAMPLE: Incomes in a corporation are not normal. They are skewed right. The mean is $75,000 and
the standard deviation $14,000.
4) What is the probability of choosing a sample of 5 employees and having their mean salary less than
$65,000?
5) What is the probability of choosing a sample of 40 employees and having their mean salary less than
$65,000?
What about p-hat?
If the population is at least 10 times the sample size then

p  hat  p

 p hat 
p(1  p)
n
More importantly as n increases the sampling distribution of p-hat is approximately normal. Make sure
and check np  10 and n(1  p )  10
EXAMPLE
5) You are interested in the proportion of American made cars driven by students. Suppose that you
suspect 60 percent of students drive American made cars. What is the probability that in a sample of 20
students you find the proportion is larger than 65%?
6) Same situation as #5. What is the probability that more than 65% drive American cars in a sample of
40 students?