Download 5.4 The Central Limit Theorem

The Central Limit Theorem
Honors Statistics
Lesson 7.4
Objectives/Assignment
• How to find sampling distributions and
verify their properties
• How to interpret the Central Limit Theorem
• How to apply the Central Limit Theorem to
find the probability of a sample mean
Introduction
• In previous sections, you studied the relationship between the mean
of a population and values of random variable. In this section, you
will study the relationship between a population mean and the
means of samples taken from the population.
• Definition: A sampling distribution is the probability distribution of a
sample statistic that is formed when samples of sizes n are
repeatedly taken from a population. If the sample statistic is the
sample mean, then the distribution is the sampling distribution of
sample means.
Sampling Distributions
•
For instance, consider the following Venn diagram. The rectangle
represents a large population, and each circle represents a sample of size
n. Because the sample entries can differ, the sample means can also differ.
The mean of Sample 1 is x1, the mean of Sample 2 is x2, and so on. The
sampling distribution of the sample means of size n for this population
consists of x1, x2, x3, and so on. If the samples are drawn with replacement,
an infinite number of samples can be drawn from the population.
Ex. 1: A Sampling Distribution of Sample
Means
• You write the population values {1, 3, 5, 7} on slips of
paper and put them in a box. Then you randomly
choose two slips of paper, with replacement. List all
possible samples of size n = 2 and calculate the mean of
each. These means form the sampling distribution of the
sample means. Find the mean, variance and standard
deviation of the sample means. Compare your result
with the mean = 4, variance 2 = 5, and standard
deviation of  = √5 = 2.236 of the population.
Solution
•
List all 16 samples of size 2 from the population and the mean of each
sample.
Relative frequency histogram of
population
Relative Frequency Distribution of
Sample Means
Relative Histogram of Sampling
Distribution
Solution continued
• After constructing a relative frequency distribution of the sample
means, you can graph the sampling distribution by using a relative
histogram as shown. Notice the shape of the histogram is bell
shaped and symmetric, similar to a normal curve. The mean,
variance and standard deviation of the 16 sample means are:
The Central Limit Theorem
• The Central Limit Theorem is one of the most important
and useful theorems in statistics. This theorem forms
the foundation for the inferential branch of statistics. The
Central Limit Theorem describes the relationship
between the sampling distribution of sample means and
the population that the samples are taken from.
Insight
• The distribution of sample means has the same mean as
the population. But its standard deviation is less than
the standard deviation of the population. This tells you
that the distribution of sample means has the same
center as the population, but it is not as spread out.
Moreover, the distribution of the sample means becomes
less and less spread out (tighter concentration about the
mean) as the sample size n increases.
Ex. 2: Interpreting the Central Limit
Theorem
• Phone bills for residents of Cincinnati have a mean of $64 and a
standard deviation of $9, as shown in the following graph. Random
samples of 36 phone bills are drawn from the population and the
mean of each sample is determined. Find the mean and standard
error of the mean of the sampling distribution. Then sketch a graph
of the sampling distribution.
Solution
• The mean of the sampling distributions is equal to the population
mean, and the standard error of the mean is equal to the population
standard deviation divided by √n. So,
Solution continued
• From the Central Limit Theorem, because the sample
size is greater than 30, the sampling distribution can
be approximated by a normal distribution with  = $64
and  = $1.50
Ex. 3: Interpreting the Central Limit
Theorem
• The heights of fully grown white oak trees are normally distributed,
with a mean of 90 feet and a standard deviation of 3.5 feet.
Random samples of 4 are drawn from this population, and the
mean of each sample is determined. Find the mean and standard
error of the mean of the sampling distribution. Then sketch a graph
of the sampling distribution.
Solution
• The mean of the sampling distribution is equal to the
population mean and the standard error of the mean is
equal to the population standard deviation divided by √n.
So,
Solution
• From the Central Limit Theorem, because the population
is normally distributed, the sampling distribution is also
normally distributed.
Probability and the Central Limit Theorem
• In sections 5.2 and 5.3, you learned how to find the
probability that a random variable, x, will fall in a given
interval of population values. In a similar manner, you
can find the probability that a sample mean, x bar will fall
in a given interval of the x bar sampling distribution. To
transform x bar to a z-score, you can use the following
equation.
Ex. 4: Finding Probabilities for Sampling
Distributions
• The graph at the right lists
the length of time adults
spend reading
newspapers. You
randomly select 50 adults
ages 18 to 24. What is the
probability that the mean
time they spend reading
the newspaper is between
8.7 and 9.5 minutes?
Assume that  = 1.5
minutes
Ex. 4: Solution
• The graph of this
distribution is shown right
with a shaded area
between 8.7 and 9.5
minutes.
Ex. 4: Solution
• Because the sample size is greater than 30, you can use
the Central Limit Theorem to conclude that the
distribution of sample means is approximately normal
with a mean and a standard deviation of:
Ex. 4: Solution
• The z-scores that correspond to sample means of 8.7
and 9.5 minutes are
• So, the probability that the mean time the adults spend
reading the newspaper is between 8.7 and 9.5 is:
• So, 91.16% of adults aged 18 to 24 spend between 8.7
and 9.5 minutes reading the newspaper.
z-score distribution of sample
means for Ex. 4
Ex. 5: Finding Probabilities for Sampling
Distributions
• The mean rent of an apartment
in a professionally managed
apartment building is $780.
You randomly select 9
professionally managed
apartments. What is the
probability that the mean rent
is less than $825? Assume
that the rents are normally
distributed with a mean of
$780 and a standard deviation
of $150.
Ex. 5 Solution
• Because the population is normally distributed, you can
use the Central Limit Theorem to conclude that the
distribution of sample means is normally distributed with
a mean of $780 and a standard deviation of $150.
• The graph of this
distribution is shown.
The area to the left of
$825 is shaded. The zscore that corresponds
to $825 is:
Ex. 5 Solution
• The graph of this distribution is shown. The
area to the left of $825 is shaded. The zscore that corresponds to $825 is:
Ex. 6: Finding probabilities for x and x bar
1. In this case, you are asked to find the probability associated with a
certain value of the random variable, x. The z-score that
corresponds to x = $2500 is
Ex. 6: Finding probabilities for x and x bar
2. Here, you are asked to find the probability associated with a
sample mean x bar. The z-score that corresponds to x bar =
$2500 is
Ex. 6: Finding probabilities for x and x bar
3. Where there is a 34% chance that an individual will have a
balance of less than $2500, there is only a 2% chance that the
mean of a sample of 25 will have a balance of less than $2500
OR