Download The Central Limit Theorem:

The Central Limit Theorem:
Properties of x bar Distribution, Assuming x has Normal Distribution
Theorem 7.1: Let x be a random variable with a normal distribution whose mean
is μ and standard deviation is σ. Let x bar be the sample mean corresponding to random
samples of size n taken from the x distribution. Then the following are true:
a) The x bar distribution is a normal distribution.
b) The mean of the x bar distribution is μ.
c) The standard deviation of the x bar distribution is σ/√n.
We conclude from the above theorem that when x has a normal distribution, the x bar
distribution will be normal for any sample size n. Furthermore, we can convert the x bar
distribution to the standard normal z distribution using the following formulas.
μx bar = μ
σx bar = σ/√n
z = _x bar - μx bar_ = _x bar – μ_ or _√n(x bar – μ)_
σx bar
σ/√n
σ
Where n is the sample size, μ is the mean of the x distribution, and σ is the standard
deviation of the x distribution.
***To see the theorem in action, view example 2 on text p. 409 – 411 and Figure 7-2 on
text p.411.
Standard Error of the Mean
The term standard error is widely used in statistical literature. Theorem 7.1
describes the distribution of a statistic, namely, the distribution of sample means x bar.
The standard deviation of a statistic is referred to as the standard error of that statistic.
For the x bar sampling distribution, the standard error of x bar is σx bar or σ/√n. In other
words, the standard error of the mean is σ/√n.
Central Limit Theorem
The central limit theorem tells us what to expect when we don’t have enough
information about the shape of the original x distribution.
Theorem 7.2: If x possesses any distribution with mean μ and standard deviation
σ, then the sample mean x bar based on a random sample of size n will have a
distribution that approaches the distribution of a normal random variable with mean μ and
standard deviation σ/√n as n increases without limit.
This theorem says that x can have any distribution, but as the sample size gets
larger and larger, the distribution of x bar will approach a normal distribution. From this
relation, we begin to appreciate the scope and significance of the normal distribution.
How large should the sample size be if we want to apply the central limit
theorem? Statisticians agree that if n is 30 or larger, the x bar distribution will appear to
be normal and the central limit theorem will apply.
In practice, it is a good idea, when possible, to make a histogram of sample x
values. If the histogram is approximately mound-shaped and if it is more or less
symmetrical, then we may be assured that, for all practical purposes, the x bar
distribution will be well approximated by a normal distribution and the central limit
theorem will apply when the sample size is 30 or larger.
Summary
For almost all x distributions, if we use a random sample size of 30 or larger, the
x bar distribution will be approximately normal, and the larger the sample size becomes,
the closer the x bar distribution gets to normal. Furthermore, we may convert the x bar
distribution to a standard normal distribution using the formulas shown below.
μx bar = μ
σx bar = σ/√n
z = _x bar - μx bar_ = _x bar – μ_ or _√n(x bar – μ)_
σx bar
σ/√n
σ
Where n is the sample size (n > 30), μ is the mean of the x distribution, and σ is the
standard deviation of the x distribution.
*** View Guided Exercise 2, Example 3, and Guided Exercise 3 on text p. 413 – 416.
Complete text p. 416 – 422.