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MATH 214 Lab 7: Sampling Distribution
In today’s lab we will be studying the central limit theorem. We will be developing
data to demonstrate that as the sample size increases, the distribution of x :
a. tends to become more like the normal distribution.
b. has a smaller standard deviation.
We will use an applet called the Central Limit Theorem experiment to select random
samples from population having different distributions. This applet will allow us to store
the mean of each sample and then graph them. We can choose to generate 100, 1000, or
10,000 samples. (In theory we should consider all possible samples of a given size).
Go to http://faculty.salisbury.edu/~bawainwright/applets/home.html
Click on Central Limit Theorem Experiment.
I.
The Normal distribution
First setting of the applet has normal distribution for population.
A.
Reset n=1, set the mean, μ = 0 and standard deviation, σ = 1.
Note: The graph on the left is the theoretical distribution of a random variable having
normal distribution with mean μ = 0. The graph on the right is the theoretical
distribution of the sample means obtained by selecting samples of size 1 (Sampling
distribution).
Explain why these are the same.
Slowly increase the sample size from 1 to 50
The graph on the right now represents the sampling distribution of the sample means
from samples of that particular size.
What happens to the shape of this graph as n is increased?
What happens to the mean and standard deviation of the sample means (the sampling
distribution) as n is increased?
Reset the sample size to n=10.
Click on Stop 10 and choose 1000. This enables us to select 1000 samples of size 10
from the normal distribution on the left. The mean of each of these samples will be
plotted over the graph at the right.
To run this simulation, click on the box with the double arrow, ►►.
Discuss how the histogram of the sample means (in red) compares to the theoretical
curve (in blue). Analyze the descriptive statistics. How do the mean and standard
deviation of the 1000 sample means compare to what is expected?
B. Repeat for samples with 50 members.
To do this you only need to change the sample size.
Comment on what happens to the shape of the histogram and to the descriptive
statistics as the sample size is increased. (As part of your answer, be sure to consider
the mean and the standard deviation of the population.)
II.
The Uniform Distribution
A. Reset the sample size to 1. Note: the values of d and c are the minimum and
maximum of the distribution, respectively. The mean of the distribution is (d+c)/2.
Slowly change the sample size and notice how the sampling distribution changes.
Discuss how this change compares to that when sampling from the exponential and
gamma.
B. Select 1000 samples of size 10 and 50.
Compare the shape, the mean and the standard deviation (standard error) of the
resulting sampling distributions. What effect, if any, does sample size have on the
sampling distribution?
III.
The Exponential distribution Distribution
Click drop down menu by box containing normal.
Select exponential.
Adjust mean to 1 by sliding the arrow bar.
A.
Now slowly increase the sample size. Discuss what happens to the shape of the
graph of the theoretical sampling distribution of the mean.
B.
Select 1000 samples of size 10 and then 1000 samples of size 50 from exponential
distribution.
Compare the shape, the mean and the standard deviation (standard error) of the
resulting sampling distributions. What effect, if any, does sample size have on the
sampling distribution?
IV.
The Gamma Distribution
A. Notice the theoretical distribution to the left. The mean of a gamma distribution is kb.
Move the slide bar to vary the values of k and b. Notice what happens to the shape of the
distribution as we change k and b.
Now slowly increase the sample size. Discuss what happens to the shape of the graph
of the theoretical sampling distribution of the mean.
B. Select 1000 samples of size 10 and then 1000 samples of size 50 from a gamma
distribution. Compare the shape, the mean and the standard deviation (standard error)
of the resulting sampling distributions. What effect, if any, does sample size have on
the sampling distribution?
V.
The Cauchy Distribution
Select Cauchy distribution. Set n=1 and slowly increase n to 50. Note: t is the location
parameter (median).
Select 1000 samples of size 50 from a Cauchy with t=0 and s=1. Compare the red
histogram to the blue theoretical graph. Also observe the actual mean and standard
deviation of the 1000 sample means
Does the Central Limit Theorem seem to apply here?
Note: the blue graph is NOT the theoretical sampling distribution of the mean. Click the
drop down menu by mean and Select median. Run the simulation again selecting 1000
samples of size 50. This time the sample medians will be plotted rather than the sample
means.
What does the sampling distribution of the medians look like? Compare the actual
mean and standard deviation of the 1000 medians to those of the means.
When sampling from the Cauchy distribution, which estimator (mean or median)
would provide a better estimate for the location parameter? Why?