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AP Statistics Notes
9.3: Sample Means
Name_______________________________
Date__________________________
Sample Proportions vs. Sample Means
 ________________________ (Section 9.2) are most often used when we are
interested in __________________________.
 For example, “yes” or “no” questions.
 ____________________are most often used when we are interested in
numerical or _____________________________.
 For example, the average rate of return on stocks in the stock market.
The distribution of returns for 1,815 New York Stock Exchange common
stocks in 1987
The distribution of returns for all possible portfolios that invested equal
amounts in each of five stocks in 1987
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Life Lessons Learned from the Stock Market
1) Averages are less variable than individual observations.
2) Averages are more normal than individual observations.
 These two facts contribute to the popularity of sample means in statistical
inference.
Mean and Standard Deviation of a Sample Mean
 Suppose that x is the mean of an SRS of size n drawn from a large population
with mean μ and standard deviation σ.
 Then the mean of the sampling distribution of x is
 The standard deviation is
Sample Mean Basics
 The sample mean x-bar is an unbiased estimator of the population mean μ.
 The values of x-bar are less spread out for larger samples.
 You should only use the recipe for standard deviation of x-bar when the
population is at least 10 times as large as the sample.
 These facts are true no matter what the shape of the population distribution.
Ex 1: Young Women’s Heights
 The height of young women varies approximately according to the N(64.5,
2.5) distribution.
 We could safely say the if we repeatedly select one woman at random, the
heights we get will also follow this distribution.
 But, what will happen if we begin choosing samples of 10 women at random?
 What will be the sample mean height x-bar of the sampling distribution?
 What about the standard deviation?
Sampling Distribution of a Sample Mean from a Normal Population
 Draw an SRS of size n from a population that has the normal distribution with
mean μ and standard deviation σ. Then the sample mean x-bar has the normal
distribution
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Ex 2: More on Young Women’s Heights
 What is the probability that a randomly selected young woman is taller than
66.5 inches?
 What about the probability that the mean height of an SRS of 10 young women
is greater than 66.5 inches?
The sampling distribution of the mean height x-bar of 10 young women
compared with the distribution of the height of a single woman chosen at
random
What have we learned?
Question…
 Does x-bar still have a normal distribution even when the population
distribution is not normal?
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Central Limit Theorem
 Draw an SRS of size n from any population whatsoever with mean μ and
standard deviation σ. When n is large, the sampling distribution of the sample
mean x-bar is very close to the Normal distribution
How large is large enough?
 How large a sample size n is needed for x-bar to be close to Normal depends
on the population distribution.
 More observations are required if the shape of the population distribution is far
from Normal.
Three Scenarios to Consider…
1) The population has a Normal distribution – shape of sampling distribution:
2) Any population shape, small n – shape of sampling distribution:
3) Any population shape, large n – shape of sampling distribution:
The sampling distribution is normal if the population distribution is
normal. It will be approximately normal for large samples regardless of
the shape of the population distribution.
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