Statistics
5.4 Sampling Distributions and the Central Limit Theorem
LEQ: How do you interpret the Central Limit Theorem?
Procedure:
1. Sampling Distributions:
a. Definition 1: A ________________________________ is the probability
distribution of a sample statistic that is formed when samples of size 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.
b. Properties of Sampling Distributions of Sample Means:
i. The mean of the sample means  x is equal to the population mean µ.
ii. The standard deviation of the same means is equal to the population
standard deviation divided by the square root of n.
The standard deviation of the sampling distribution of the sample means is
called the standard error of the mean.
2. The Central Limit Theorem:
a. The Central Limit Theorem:
i. If samples of size n, where n  30 , are drawn from any population with a
mean  and a standard deviation  , then the sampling distribution of
sample means approximates a normal distribution. The greater the sample
size, the better the approximation.
ii. If the population itself is normally distributed, the sampling distribution of
sample means is normally distributed for any sample size n.
In either case, the sampling distribution of sample means has a mean equal
to the population mean.
The sampling distribution of sample means has a variance equal to 1/n times
the variance of the population and a standard deviation equal to the
population standard deviation divided by the square root of n.
The standard deviation of the sampling distribution of the sample means is
also called the standard error of the mean.
b. Example 1: Interpreting the central limit theorem:
Phone bills for residents of a city 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 this population, and the mean of each sample is determined. Find the
mean and the standard error of the mean of the sampling distribution. Then
sketch a graph of the sampling distribution of sample means.
c. Example 2: Interpreting the central limit theorem:
The heights of fully grown white oak trees are normally distributed, with a mean of
90 feet and standard deviation of 3.5 feet, as shown in the following graph.
Random samples of size 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 the graph of the sampling distribution of
sample means.
3. Probability and Central Limit Theorem:
a. Example 3: Finding probabilities for sampling distributions:
The mean room and board expense per year at a four-year college is $5850. You
randomly select 9 four-year colleges. What is the probability that the mean room
and board is less than $6180? Assume that the room and board expenses are
normally distributed, with a standard deviation of $1125.
b. Example 4: Finding probabilities for sampling distributions:
The average sales price of a single-family house in the U.S. is $243,756. You
randomly select 12 single-family houses. What is the probability that the mean
sales price is more than $200,000? Assume that the sales prices are normally
distributed with a standard deviation of $44,000.
c. Example 5: Finding probabilities:
A bank auditor claims that credit card balances are normally distributed, with a
mean of $2870 and a standard deviation of $900.
1. What is the probability that a randomly selected credit card holder has a
credit card balance less than $2500?
2. You randomly select 25 credit card holders. What is the probability that
their mean credit card balance is less than $2500?
3. Compare probabilities from (1) and (2) and interpret your answer in
terms of the auditor’s claim.
4. HW: p. 254 (1 – 8, 14 – 18 evens, 22 – 34 evens)