Activity
In this simulation lab from the Virtual Statistics Lab at Rice
University, you will investigate distributions of sample statistics.
The investigation asks you to focus on the mean of samples from several
populations and then to repeat the simulation and focus on the standard
deviation and then the third repeat asks you to focus on the shape of
the distribution. Rather than focus on one element (center, spread, or
shape) at a time as you move through various parent population
distributions, you may want to consider all three elements in one
sequence of simulations as all are elements used to answer the larger
question “What does the sampling distribution look like?” While
some students will prefer focusing on one element of the sampling
distribution at a time and how it behaves across different
distributions, others will prefer to examine all qualities of the
sampling distribution from a specific parent population at once and
then move to seeing what would be different if the starting
distribution were changed.
The lab is located at www.ruf.rice.edu/~lane/rvls.html
From Simulations/Demonstrations, open the Sampling Distributions
Simulation, read the directions and then press Begin. (You need a
"Javaenabled browser." The best way to know whether you have one is to
go to this site and see if the demonstration begins. Be patient;
sometimes it takes a few seconds. Works best with Internet Explore or
Netscape.)
Mean of the Sample Means
1. First we use a normal distribution for graph 1, select sample
size n = 25, and show the distribution of sample means on graph 3 (this
is the default). Animate several reps to see what the simulation is
doing and then do several thousand simulations.
a) Where is the center of the distribution of sample means?
(Notice that the mean is shown in red in graph 3.)
b) What is the center or mean of the original normal distribution?
(Notice that the mean is shown in blue to the left of the distribution.)
c) Repeat the simulation, changing the sample size to n = 16 and
then to other sample sizes.
d) Make a conjecture about the mean of the distribution of sample
means and the mean of the original population.
e) Now we will change the original population from normal to
uniform and repeat the simulation with both sample sizes.
f) How does the center of the distribution of sample means
compare to the mean of the original uniform distribution?
g) Finally, we repeat the simulations drawing our own “weirdly”
shaped distribution in graph 1 and observing the distribution of sample
means in graph 3. You can draw your own distribution by selecting
“Custom” and then clicking and dragging your mouse along the axis to
draw in any shape that you choose. Try something that is strongly
skewed or has particularly heavy tails.
h) How does the center of the distribution of sample means
compare to the mean of the original weird distribution?
Standard Deviation of the Sample Means
2. Return to the normal distribution in graph 1, sample size n = 25,
and make a distribution of sample means on graph 3.
a) What is the standard deviation of the distribution of sample
means?
(Shown in blue on graph 3)
b) Repeat the simulation changing the sample sizes to 16, 10, 5,
and 2.
c) Complete the table and calculate the ratio s /sx.
n 25
16 10
5 2
s/ sx
d) Use the table to write a relationship between s and sx.
e) Now change from the normal distribution to a uniform
distribution with standard deviation of 9.52. Investigate the standard
deviation of the distribution of sample means by changing the size of
the samples and recording the results in an appropriate table.
f) Finally repeat the simulations using your own weirdly shaped
distribution in graph 1.
Shape of the Distribution of Sample Means
3. Now we consider the shape of the distribution of sample means
when the original population is normal, uniform, or weird. What do you
observe about the shape of the distribution of sample means?
Doing this activity you should have made an observation about the mean
of sample means. (How does it relate to the population
mean?) You should also have made an observation about the
standard deviation of sample means. (How does it relate to the
population standard deviation?) Finally, you should have made a
very important observation about the shape of the distribution of
sample means, which is formalized in what is called The Central Limit
Theorem. What about the distributions of other statistics? The
simulation can help us investigate the distribution of sample medians,
variances, or standard deviations. Do the earlier observations hold for
these measures?
