Download Section 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.
Copyright © 2009 Pearson Education, Inc.
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.
Copyright © 2009 Pearson Education, Inc.
Figure 5.23 Frequency and
relative frequency distribution
of outcomes from rolling one
die 1,000 times.
Slide 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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Slide 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.
Copyright © 2009 Pearson Education, Inc.
Slide 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.
Copyright © 2009 Pearson Education, Inc.
Slide 5.3- 5
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.
Copyright © 2009 Pearson Education, Inc.
Slide 5.3- 6
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.
Copyright © 2009 Pearson Education, Inc.
Slide 5.3- 7
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.
Copyright © 2009 Pearson Education, Inc.
Slide 5.3- 8
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.
Copyright © 2009 Pearson Education, Inc.
Slide 5.3- 9
TIME OUT TO THINK
Confirm that the standard deviations of the distributions
of means given in Table 5.2 (slide 6) for n = 2, 5, 10
agree with the prediction of the Central Limit Theorem,
given that σ = 1.73 (the population standard deviation
found in Figure 5.23). For example, with n = 2, σ/ n =
1.22 ≈ 1.21.
Copyright © 2009 Pearson Education, Inc.
Slide 5.3- 10
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.
Copyright © 2009 Pearson Education, Inc.
Slide 5.3- 11
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?
Solution:
a. In dealing with an individual score, we use the method of
standard scores discussed in Section 5.2. Given the mean of
400 and standard deviation of 70, a score of 375 has a standard
score of
data value – mean
z = standard deviation = 375 – 400 = -0.36
70
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Slide 5.3- 12
EXAMPLE 1 Predicting Test Scores
Solution: (cont.)
According to Table 5.1, a standard score of -0.36 corresponds to
about the 36th percentile— that is, 36% of all students can be
expected to score below 375. Thus, there is about a 0.36 chance
that a randomly selected student will score below 375.
Notice that we need to know that the scores have a normal
distribution in order to make this calculation, because the table of
standard scores applies only to normal distributions.
Copyright © 2009 Pearson Education, Inc.
Slide 5.3- 13
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.
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?
Solution:
b. The question about the mean of a group of students must be
handled with the Central Limit Theorem. According to this
theorem, if we take random samples of size n = 100 students
and compute the mean test score of each group, the distribution
of means is approximately normal.
Copyright © 2009 Pearson Education, Inc.
Slide 5.3- 14
EXAMPLE 1 Predicting Test Scores
Solution: (cont.)
Moreover, the mean of this distribution is m = 400 and its standard
deviation is s / n = 70/ 100 = 7. With these values for the mean
and standard deviation, the standard score for a mean test score of
375 is
data value – mean
z = standard deviation = 375 – 400 = -0.357
7
Table 5.1 shows that a standard score of -3.5 corresponds to the
0.02th percentile, and the standard score in this case is even lower.
In other words, fewer than 0.02% of all random samples of 100
students will have a mean score of less than 375.
Copyright © 2009 Pearson Education, Inc.
Slide 5.3- 15
EXAMPLE 1 Predicting Test Scores
Solution: (cont.)
Therefore, the chance that a randomly selected group of 100
students will have a mean score below 375 is less than 0.0002,
or about 1 in 5,000.
Notice that this calculation regarding the group mean did not
depend on the individual scores’ having a normal distribution.
This example has an important lesson. The likelihood of an
individual scoring below 375 is more than 1 in 3 (36%), but
the likelihood of a group of 100 students having a mean score
below 375 is less than 1 in 5,000 (0.02%).
In other words, there is much more variation in the scores of
individuals than in the means of groups of individuals.
Copyright © 2009 Pearson Education, Inc.
Slide 5.3- 16
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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Slide 5.3- 17
The End
Copyright © 2009 Pearson Education, Inc.
Slide 5.3- 18