Transcript 5.3 The Central Limit Theorem

5.3 The Central Limit Theorem
LEARNING GOAL
Understand the basic idea behind the Central Limit
Theorem and its important role in statistics.
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5.3-1
Suppose we roll one die 1,000 times and record the outcome
of each roll, which can be the number 1, 2, 3, 4, 5, or 6.
Figure 5.23 shows a histogram of
outcomes. All six outcomes have
roughly the same relative
frequency, because the die is
equally likely to land in each of the
six possible ways. That is, the
histogram shows a (nearly) uniform
distribution (see Section 4.2).
It turns out that the distribution in
Figure 5.23 has a mean of 3.41 and
a standard deviation of 1.73.
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Figure 5.23 Frequency and
relative frequency distribution
of outcomes from rolling one
die 1,000 times.
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5.3- 2
Now suppose we roll two dice 1,000 times and record the mean
of the two numbers that appear on each roll. To find the mean
for a single roll, we add the two numbers and divide by 2.
Figure 5.25a shows a typical result.
The most common values in this
distribution are the central values
3.0, 3.5, and 4.0. These values are
common because they can occur in
several ways.
The mean and standard deviation
for this distribution are 3.43 and
1.21, respectively.
Figure 5.25a Frequency and relative
frequency distribution of sample means
from rolling two dice 1,000 times.
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5.3-3
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5.3- 3
Suppose we roll five dice 1,000
times and record the mean of the
five numbers on each roll. A
histogram for this experiment is
shown in Figure 5.25b.
Once again we see that the central
values around 3.5 occur most
frequently, but the spread of the
distribution is narrower than in the
two previous cases.
The mean and standard deviation
are 3.46 and 0.74, respectively.
Figure 5.25b Frequency and
relative frequency distribution
of sample means from rolling
five dice 1,000 times.
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5.3- 4
If we further increase the
number of dice to ten on each
of 1,000 rolls, we find the
histogram in Figure 5.25c,
which is even narrower.
In this case, the mean is 3.49
and standard deviation is 0.56.
Figure 5.25c Frequency and
relative frequency distribution of
sample means from rolling ten
dice 1,000 times.
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Table 5.2 shows that as the sample size increases, the mean
of the distribution of means approaches the value 3.5 and the
standard deviation becomes smaller (making the distribution
narrower).
More important, the distribution looks more and more like a
normal distribution as the sample size increases.
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The Central Limit Theorem
Suppose we take many random samples of size n for a
variable with any distribution (not necessarily a normal
distribution) and record the distribution of the means of
each sample. Then,
1. The distribution of means will be approximately a
normal distribution for large sample sizes.
2. The mean of the distribution of means approaches the
population mean, m, for large sample sizes.
3. The standard deviation of the distribution of means
approaches σ/ n for large sample sizes, where s is
the standard deviation of the population.
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5.3-7
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Be sure to note the very important adjustment, described
by item 3 above, that must be made when working with
samples or groups instead of individuals:
The standard deviation of the distribution of sample
means is not the standard deviation of the population, s,
but rather s/ n , where n is the size of the samples.
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TECHNICAL NOTE
(1) For practical purposes, the distribution of means
will be nearly normal if the sample size is larger than
30.
(2) If the original population is normally distributed,
then the sample means will be normally distributed
for any sample size n.
(3) In the ideal case, where the distribution of
means is formed from all possible samples, the
mean of the distribution of means equals μ and the
standard deviation of the distribution of means
equals σ/ n.
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5.3-9
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5.3- 9
Figure 5.26 As the sample size increases (n = 5, 10, 30), the distribution of sample
means approaches a normal distribution, regardless of the shape of the original
distribution. The larger the sample size, the smaller is the standard deviation of the
distribution of sample means.
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5.3-10
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EXAMPLE 1 Predicting Test Scores
You are a middle school principal and your 100 eighth-graders are
about to take a national standardized test. The test is designed so
that the mean score is m = 400 with a standard deviation of s = 70.
Assume the scores are normally distributed.
a. What is the likelihood that one of your eighth-graders, selected
at random, will score below 375 on the exam?
b. Your performance as a principal depends on how well your
entire group of eighth-graders scores on the exam. What is the
likelihood that your group of 100 eighth-graders will have a
mean score below 375?
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The Value of the Central Limit Theorem
The Central Limit Theorem allows us to say something about
the mean of a group if we know the mean, m, and the standard
deviation, s, of the entire population. This can be useful, but it
turns out that the opposite application is far more important.
Two major activities of statistics are making estimates of
population means and testing claims about population means. Is
it possible to make a good estimate of the population mean
knowing only the mean of a much smaller sample?
As you can probably guess, being able to answer this type of
question lies at the heart of statistical sampling, especially in
polls and surveys. The Central Limit Theorem provides the key
to answering such questions.
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5.3-12
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