Unit 9: Sampling Distributions

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Data Analysis & 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 "Java-enabled 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?