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4.5 Sample Means
The Binomial Setting
Note. Two facts that contribute to the popularity of sample means in
statistical inference are
• Averages are less variable than individual observations.
• Averages are more normal than individual observations.
Definition. The mean and standard deviation of a population are
parameters. We use the Greek letters to write these parameters: µ for
the mean and σ for the standard deviation. The mean and standard
deviation calculated from sample data are statistics. We write the
sample mean as x and the sample deviation as s.
The Mean and the Standard Deviation of x
Definition. 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 µ and its standard deviation
√
is σ/ n.
Note. The behavior of x in repeated samples is much like that of the
sample proportion p̂:
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• The sample mean x is an unbiased estimator of the population
mean µ.
• The values of x are less spread out for larger samples.
√
• You should only use the recipe σ/ n for the standard deviation of
x when the population is at least 10 times as large as the sample.
Note. Notice that these facts about the mean and standard
deviation of x are true no matter what the shape of the population distribution is.
Example 4.24. The height of young women varies approximately
according to the N (64.5, 2.5) distribution. This is a population distribution with µ = 64.5 and σ = 2.5. If we choose one young woman at
random, the heights we get in repeated choices follow this distribution.
That is, the distribution of the population is also the distribution of
one observation chosen at random. So we can think of the population
distribution as a distribution of probabilities, just like a sampling distribution. Now measure the height of an SRS of 10 young women. The
sampling distribution of their sample mean height x will have mean
√
√
µ = 64.5 inches and standard deviation σ/ n = 2.5/ 10 = .79 inch.
The heights of individual women very widely about the population
mean, but the average height of a sample of 10 women is less variable.
Figure 4.18 (and TM-74) compares the distributions.
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Note. The fact that averages of several observations are less variable
than individual observations is important in many settings.
The Central Limit Theorem
Note. Draw an SRS of size n from a population that has the normal
distribution with mean µ and standard deviation σ. Then the sam√
ple mean x has the normal distribution N (µ, σ/ n) with mean µ and
√
standard deviation σ/ n.
Theorem (Central Limit Theorem). Draw an SRS of size n from
any population whatsoever with mean µ and finite standard deviation
σ. When n is large, the sampling distribution of the sample mean x is
√
close to the normal distribution N (µ, σ/ n) with mean µ and standard
√
deviation σ/ n.
Example 4.25. Figure 4.19 (and TM-74) shows the central limit theorem in action for a very nonnormal population. Figure 4.19(a) displays
the density curve for the distribution of the population. The distribution is strongly right skewed, and the most probable outcomes are
near 0 at one end of the range of possible values. The mean µ of this
distribution is 1 and its standard deviation σ is also 1. This particular
distribution is called an exponential distribution from the shape of its
density curve. Exponential distributions are used to describe the lifetime in service of electronic components and the time required to serve
a customer or repair a machine. Figures 4.19(b), (c), and (d) are the
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density curves of the sample mean of 2, 10, and 25 observations from
this population. As n increases, the shape becomes more normal. The
mean remains at µ = 1 and the standard deviation decreases, taking
√
the value 1/ n. The density curve for 10 observations is still somewhat skewed to the right but already resembles a normal curve with
√
µ = 1 and σ = 1/ 10 = .32. The density curve for n = 25 is yet more
normal. The contrast between the shape of the population distribution
and the distribution of the mean of 10 or 25 observations is striking.
The Law of Large Numbers
Note. The Law of Large Numbers states: Draw observations at random from any population with finite mean µ. As the number of observations drawn increases, the mean x of the observed values gets closer
and closer to µ.
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