STA220- Section 4.9
̅ and the Central Limit Theorem
The Sampling Distribution of 𝒙
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.
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.
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.
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 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.
Probability and the Central Limit Theorem
In sections 4.3 and 4.7, 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. 5: Finding Probabilities for Sampling Distributions
The mean rent of an apartment in a professionally managed apartment building is $780. You randomly
select nine 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. 6: Finding probabilities for x and 𝒙