Download Central Limit Theorem (as an “experimental fact”)

11.4 CENTRAL LIMIT THEOREM
A theorem is a statement of some mathematical property or principle that
can be proven by logical arguments. The central limit theorem is a very
important theorem in statistics. We will not prove it in this book, but we will
look at examples that demonstrate what it tells us and its great usefulness.
Central Limit Theorem (as an “experimental fact”)
Suppose we take a very large number of random samples, each of size n, from a population with mean ␮ and standard deviation ␴ .
Then if the sample size n is relatively large
(n ⱖ 20), the shape of the relative frequency
histogram of the sample means X will ap-
proximate the shape of a normal density
whose mean is ␮ and whose standard deviation is ␴ /冪n. Equivalently, the relative
frequency histogram of (X ⫺ ␮ )/(␴ / 冪n) will
approximate the standard normal density
tabulated in Appendix E.
The fact that the relative frequency histogram of a very large number
of Xs must be close to an appropriately scaled normal density provided n,
the sample size used to form each X, is relatively large (n ⱖ 20) amounts to
asserting that we can use the normal distribution to calculate probabilities
involving X as if X were distributed exactly normally with mean ␮ and
standard deviation ␴ / 冪n. Thus the experimental fact claimed in the above
statement of the central limit theorem translates into the practical result
that we can compute probabilities involving an X based on a sample of at
least 20 using the normal distribution regardless of the shape of the original
population the Xs were sampled from. Thus to compute p(X ⱕ x) we let
z ⳱ (x ⫺ ␮ )/(␴ / 冪n) and enter Table E using this value of z. Here ␮ and
␴ are the population mean and standard deviation and n is the sample size
used to obtain X.
Example 11.8
Using a box model of size 200, we constructed a nonnormal population having
a population mean ␮ ⳱ 63.5 and a population standard deviation ␴ ⳱ 12.0. We
then randomly drew 100 samples of size 4, 16, and 36 with replacement from this
population, and we calculated the sample mean for each of the 100 samples at each
sample size. The resulting sample means are presented in the stem-and-leaf plots
in Tables 11.1, 11.2, and 11.3. From these stem-and-leaf plots we constructed the
relative frequency histograms of Figures 11.2, 11.3, and 11.4.
Let’s make some observations about the results of our sampling as we look at
these stem-and-leaf plots and the associated relative frequency histograms.
1. The “grand” mean of the 100 sample means is in all three cases close to the
population mean of 63.5. For example, the mean of the 100 sample means of
size 4 is 63.3, as can be computed from Table 11.1. This is what the central limit
theorem tells us to expect because it says that the relative frequency histogram
of 100 Xs will be close to a normal density with mean ␮ .
2. As the sample size increases from n ⳱ 4, a number much too small for the
central limit theorem to apply, to n ⳱ 16, a number for which the central limit
theorem should begin to hold, the relative frequency histogram goes from a
very non-normal shape in Figure 11.2 to a fairly normal shape in Figure 11.3.
Similarly, Figure 11.3 appears somewhat more bell-shaped. This is what the
central limit theorem tells us to expect.
3. As the sample size increases, the variation among the sample means decreases.
More specifically, as the sample size increases, the standard error (the standard
deviation of the sample means) decreases from 5.7 to 2.9 to 2.0 as n increases
from 4 to 16 to 36. This is precisely the kind of behavior that ␴ /冪n in the central
limit theorem predicts. Indeed, the calculated value of the standard error found
by computing the standard deviation of the 100 sample means for each sample
size is close to the value of ␴ /冪n asserted for the normal density in the central
limit theorem. For example, the calculated standard error from Table 11.1 is 5.7,
compared with ␴ / 冪n ⳱ 6.0, while the calculated standard error from Table
11.3 is 2.0, compared with ␴ /冪n ⳱ 2.0—an exact (and lucky) match.
The central limit theorem is a very powerful tool in statistical analysis.
In the present context we will use the central limit theorem to calculate
confidence intervals for population means, for proportions, and for the
difference between two means.
Sample Means of
Table 11.1
100 Samples of Size 4 (Population
Values: ␮ ⳱ 63.5, ␴ ⳱ 12.0)
Stem
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
Leaf
8
7
1,5,6
2,3,3,5,6,7
4,6,6,6,9
1,1,1,3,8,9
1,3,5,8,8,9
1,1,3,4,5,5,6,8
3
3
2,2,5,5,8
2,2,4,5,5,5,7,9
0,0,0,2,4,7,9,9
1,1,3,3,4,7
1,3,5,5,6,6,9,9
0,2,7,9
0,2,3,5
3,5,5,6,6,8
8
1,2,4,5,6,6,8,9
2,9
1
3
Frequency
0
0
1
1
3
0
6
5
6
6
8
1
1
5
8
8
6
8
4
4
6
1
8
0
2
1
1
0
0
Total 100
Sample values:
n⳱4
mean ⳱ 63.3
standard error ⳱ 5.7
Table 11.2
Sample Means of 100 Samples of Size 16
(Population Values: ␮ ⳱ 63.5, ␴ ⳱ 12.0)
Stem
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
Leaf
Frequency
0
0
1
1
0
8
6
7
11
25
10
8
6
6
8
1
1
0
1
0
0
9
7
0,2,2,5,6,6,7,9
0,2,5,7,7,7
1,2,5,8,9,9,9
0,1,3,3,4,5,5,6,7,8,8
2,2,2,2,3,3,3,3,3,4,4,4,4,4,5,6,6,7,7,8,8,9,9,9,9
2,2,3,4,4,5,6,6,7,8
0,0,0,3,3,3,5,7
0,1,2,4,4,5
0,4,4,5,6,7
2,3,3,4,5,6,6,9
5
2
1
Sample values:
n ⳱ 16
mean ⳱ 63.9
standard error ⳱ 2.9
Total 100
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
50
55
60
65
70
75
Figure 11.2
Relative frequency histogram of sample means
of 100 samples of size 4.
80
Table 11.3
Sample Means of 100 Samples of Size 36
(Population Values: ␮ ⳱ 63.5, ␴ ⳱ 12.0)
Stem
56
57
58
59
60
61
62
63
64
65
66
67
68
69
Leaf
Frequency
1,6,9
7,8,9
0,0,3,6,7,7,7,8,8,9
0,1,2,4,5,8,9
0,0,2,2,2,2,4,5,6,6,7,7,7,8,8,8
1,1,1,1,2,3,3,4,4,5,5,7,8,9
0,0,1,1,2,2,2,3,3,4,4,4,4,4,4,5,5,6,6,6,7,8,8
0,0,0,2,3,3,4,4,4,4,5,5,8,9
0,3,4,5,5,8,8
1,2,4
0
0
3
3
10
7
16
14
23
14
7
3
0
0
Total 100
Sample values:
n ⳱ 36
mean ⳱ 63.4
standard error ⳱ 2.0
0.19
0.18
0.16
0.14
0.12
0.10
0.08
0.06
0.04
0.02
50
60
70
Figure 11.3
Relative frequency histogram of sample means
of 100 samples of size 16.
80
0.25
0.20
0.15
0.10
0.05
50
60
70
80
Figure 11.4
Relative frequency histogram of sample means of
100 samples of size 36.
SECTION 11.4 EXERCISES
1. Suppose the mean weight of all male Americans is 175 pounds with a standard deviation
of 15 pounds. What are the estimated theoretical mean and standard deviation of a sample
mean from this population based on 50 observations? How is the sample mean distributed?
2. Suppose the sample mean in Exercise 1 is
based on 10 observations instead of 50. Can
you still apply the central limit theorem to
answer the questions in Exercise 1? Explain
your answer.
3. Which relative frequency histogram will look
more like a normal distribution—the histogram of sample means based on 20 observations per mean or the histogram based on
40 observations per mean? (Assume that the
samples are taken from the same population).
4. What are the mean and standard deviation
of a sample mean of 40 observations based
on a population with mean 0 and standard
deviation 1? What is the distribution of the
sample mean approximately like?