ELEMENTARY STATISTICS 3E
William Navidi and Barry Monk
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Sampling Distributions and
The Central Limit Theorem
Section 7.3
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Objectives
1. Construct the sampling distribution of a sample mean
2. Use the Central Limit Theorem to compute probabilities
involving sample means
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Objective 1
Construct the sampling distribution of a sample mean
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Sampling Distribution of the Sample Mean
In real situations, statistical studies involve sampling several
individuals then computing numerical summaries of the samples.
Most often the sample mean, 𝑥,ҧ is computed.
If several samples are drawn from a population, they are likely to
have different values for 𝑥.ҧ Because the value of 𝑥ҧ varies each time a
ഥ is a random variable. For each value of the
sample is drawn, 𝒙
random variable, 𝑥,ҧ we can compute a probability. The probability
distribution of 𝑥ҧ is called the sampling distribution of 𝑥.ҧ
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An Example of a Sampling Distribution
Tetrahedral dice are shaped like a pyramid with
four faces. Each face corresponds to a number
between 1 and 4. Tossing a tetrahedral die is like
sampling a value from the population 1, 2, 3, 4 .
We can easily find the population mean, 𝜇 = 2.5,
and the population standard deviation 𝜎 = 1.118.
Suppose that a tetrahedral die is tossed three times. The sequence of
three numbers observed may be thought of as a sample of size 3. We
may get samples such as [1, 1, 1], [1, 1, 2], [1, 1, 3], and so on.
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An Example of a Sampling Distribution (Continued)
The table displays all possible samples of size 3 and the sample mean 𝑥ҧ of
each. The mean of all of values of 𝑥ҧ is 𝜇𝑥ҧ = 2.5 and the standard deviation of
all values of 𝑥ҧ is 𝜎𝑥ҧ = 0.6455.
Next, we compare
these values to the
population mean
(2.5) and population
standard deviation
(1.118).
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Mean and Standard Deviation of a Sampling Distribution
The mean of the sampling distribution is 𝜇𝑥ҧ = 2.5, which is the same as the
mean of the population, 𝜇 = 2.5. This relation always holds.
The mean of the sampling distribution is denoted by 𝝁𝒙 and equals the
mean of the population:
𝝁𝒙 = 𝝁
The standard deviation of the sampling distribution is 𝜎𝑥ҧ = 0.6455, which is
less than the population standard deviation 𝜎 = 1.118. It is not obvious how
1.118
𝜎
these two quantities are related. Note, that 𝜎𝑥ҧ = 0.6455 =
= . Recall
that the sample size is 𝑛 = 3, which suggests that 𝜎𝑥ҧ =
𝜎
.
𝑛
3
3
The standard deviation of the sampling distribution, sometimes called the
standard error, is denoted by 𝝈𝒙 and equals the standard deviation of the
population divided by the square root of the sample size:
𝝈𝒙 =
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𝝈
𝒏
Example: Sampling Distribution
Among students at a certain college, the mean number of hours of
television watched per week is 𝜇 = 10.5, and the standard deviation is
𝜎 = 3.6. A simple random sample of 16 students is chosen for a study
of viewing habits. Let 𝑥ҧ be the mean number of hours of TV watched
by the sampled students. Find the mean 𝜇𝑥 and the standard deviation
𝜎𝑥 of 𝑥ҧ .
Solution:
The mean of 𝑥ҧ is:
𝜇𝑥 = 𝜇 = 10.5
The sample size is 𝑛 = 16. Therefore, the standard deviation of 𝑥ҧ is:
𝜎𝑥 =
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𝜎
𝑛
=
3.6
16
= 0.9
Sampling Distribution for Sample of Size 3
Consider again the tetrahedral die example. The
sampling distribution for 𝑥ҧ can be determined
from the table of all possible values of 𝑥.ҧ
The probability that the sample mean is 1.00 is
1
, or 0.016, because out of the 64 possible
64
samples, only 1 has a sample mean equal to
3
1.00. Similarly, the probability that 𝑥ҧ =1.33 is ,
64
or 0.047, because there are 3 samples whose
sample mean is 1.33.
The remaining probabilities are as follows.
ഥ
𝒙
1.00
1.33
1.67
2.00
2.33
2.67
3.00
3.33
3.67
4.00
𝑷(ഥ
𝒙)
0.016
0.047
0.094
0.156
0.188
0.188
0.156
0.094
0.047
0.016
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Probability Histogram for a Sampling Distribution
In the tetrahedral die example, the population is
1, 2, 3, 4 . When a die is rolled, each number has the
1
same chance of appearing, or 0.25.
4
The probability histogram for the sampling distribution
of 𝑥ҧ with sample size 3 is obtained from the sampling
distribution on the previous slide.
The probability histogram for the sampling distribution
looks a lot like the normal curve, whereas the
probability histogram for the population does not.
Remarkably, it is true that, for any population, if the sample size is large
enough, the sample mean 𝑥ҧ will be approximately normally distributed. For a
symmetric population like the tetrahedral die population, the sample mean is
approximately normally distributed even for a small sample size like 𝑛 = 3.
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Sampling Distribution from a Skewed Population
If a population is skewed, a larger sample size is
necessary for the sampling distribution of 𝑥ҧ to be
approximately normal. Consider the following
probability distribution.
Below are the probability histograms for the sampling distribution of 𝑥ҧ for
samples of size 3, 10, and 30. Note that the shapes of the distributions
begin to approximate a normal curve as the sample size increases.
The size of the sample needed to obtain approximate normality depends
mostly on the skewness of the population. In practice, a sample of size
𝑛 > 30 is large enough.
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The Central Limit Theorem
The remarkable fact that the sampling distribution of 𝑥ҧ is
approximately normal for a large sample from any distribution is part
of one of the most used theorems in Statistics, the Central Limit
Theorem.
Let 𝑥ҧ be the mean of a large (𝑛 > 30) simple random sample from a
population with mean 𝜇 and standard deviation 𝜎. Then 𝑥ҧ has an
approximately normal distribution, with mean 𝜇𝑥 = 𝜇 and standard
𝜎
deviation 𝜎𝑥 = .
𝑛
The Central Limit Theorem applies for all populations. However, for
symmetric populations, a smaller sample size may suffice. If the
ഥ will be normal for
population itself is normal, the sample mean 𝒙
any sample size.
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Example 1: The Central Limit Theorem
A sample of size 45 will be drawn from a population
with mean 𝜇 = 15 and standard deviation 𝜎 = 3.5. Is it
appropriate to use the normal distribution to find
probabilities for 𝑥?
ҧ
Solution:
Yes, by The Central Limit Theorem, since 𝑛 > 30, 𝑥ҧ has
an approximately normal distribution.
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Example 2: The Central Limit Theorem
A sample of size 8 will be drawn from a normal population
with mean 𝜇 = –60 and standard deviation 𝜎 = 5. Is it
appropriate to use the normal distribution to find
probabilities for 𝑥?
ҧ
Solution:
Yes, since the population itself is approximately normal, 𝑥ҧ
has an approximately normal distribution.
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Example 3: The Central Limit Theorem
A sample of size 24 will be drawn from a population
with mean 𝜇 = 35 and standard deviation 𝜎 = 1.2. Is it
appropriate to use the normal distribution to find
probabilities for 𝑥?
ҧ
Solution:
No, since the population is not known to be normal and
𝑛 is not greater than 30, we cannot be certain that 𝑥ҧ
has an approximately normal distribution.
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Objective 2
Use the Central Limit Theorem to compute
probabilities involving sample means
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Example 1: Using the Central Limit Theorem
Recent data from the U.S. Census indicates that the mean age of college
students is 𝜇 = 25 years, with a standard deviation of 𝜎 = 9.5 years. A simple
random sample of 125 students is drawn. What is the probability that the
sample mean age of the students is greater than 26 years?
Solution:
The sample size is 125, which is greater than 30.
Therefore, we may use the normal curve.
We compute 𝜇𝑥ҧ and 𝜎𝑥ҧ
𝜇𝑥 = 𝜇 = 25 and 𝜎𝑥 =
𝜎
𝑛
=
9.5
125
= 0.85
Find the area under the normal curve.
The probability that the sample mean age of the students is greater than 26
years is approximately 0.1197.
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Example 2: Using the Central Limit Theorem
Hereford cattle are one of the most popular breeds of cattle. Based on
data from the Hereford Cattle Society, the mean weight of a one-yearold Hereford bull is 1135 pounds, with a standard deviation of 97
pounds. Would it be unusual for the mean weight of 100 head of cattle
to be less than 1100 pounds?
Solution:
The sample size is 100, which is greater than 30.
Therefore, we may use the normal curve.
We compute 𝜇𝑥ҧ and 𝜎𝑥ҧ
𝜎
97
𝜇𝑥 = 𝜇 = 1135 and 𝜎𝑥 = =
= 9.7
𝑛
100
Find the area under the normal curve.
This probability is less than 0.05, so it would be unusual for the sample
mean to be less than 1100.
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You Should Know . . .
• How to construct the sampling distribution of a
sample mean
• How to find the mean and standard deviation of a
sampling distribution of 𝑥ҧ
• The Central Limit Theorem
• How to use the Central Limit Theorem to compute
probabilities involving sample means (Calculator)
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