m11186Clt simulation, college study notes - Clt simulation | Study notes Mathematical Modeling and Simulation

Connexions module: m11186 1

CLT Simulation

∗

David Lane

This work is produced by The Connexions Project and licensed under the

Creative Commons Attribution License

†

1 General Instructions

This simulation demonstrates the eect of sample size on the shape of the sampling distribution of the mean.

Depicted on the top graph is the population which is sometimes referred to as the parent distributoin.

Two sampling distributions of the mean, associated with their respective sample size will be created on the

second and third graphs.

For both the population distribution and the sampling distributions, their mean and the standard de-

viation are depicted graphically on the frequency distribution itself. The blue-colored vertical bar below

the X-axis indicates where the mean value falls. The red line starts from this mean value and extends one

standard deviation in length in both directions. The values of both the mean and the standard deviation are

also given to the left of the graph. Notice that the numeric form of a property matches its graphical form

in color. In additon, the skew and the kurtosis of each distribution are also provided to the left. These two

variables are determined by the shape of distribution. The skew and kurtosis for a normal distribution are

both 0.

In this simulation, you need to rst specify a population (the default is uniform distribution). Take

note of the skew and kurtosis of the population. Then pick two dierent sample sizes (the defaults are

N= 2

and

N= 10

), and sample a suciently large number of samples until the sampling distributions

change relatively little with additional samples (about 50,000 samples.) Observe the overall shape of the two

sampling distributions, and further compare their means, standard deviations, skew and kurtosis. Change

the sample sizes and repeat the process a few times. Do you observe a general rule regarding the eect of

sample size on the shape of the sampling distribution?

You may also test the eect of sample size with populations of other shape (uniform, skewed or customed

ones).

2 Step by Step Instructions

Show Questions

1. With the default setting (uniform population, sample sizes set at 2 and 5, respectively), click the

button "5 Samples" a couple of times. Notice how the sample means accumulate at the bottom two graphs.

Then click the button "5000 Samples" multiple times until the total number of samples exceeds 50,000.

Observe the shape of the two distributions, and compare their variance, skew and kurtosis. Write these

numbers down on a piece of paper for future reference. (Square the standard deviation to get the variance).

∗

Version 1.3: Jul 14, 2003 2:48 pm GMT-5

†

http://creativecommons.org/licenses/by/1.0

"Central Limit Theorem Demo" <http://cnx.org/content/m11185/latest/>

http://cnx.org/content/m11186/1.3/

Partial preview of the text

Download m11186Clt simulation, college study notes - Clt simulation and more Study notes Mathematical Modeling and Simulation in PDF only on Docsity!

Connexions module: m11186 1

CLT Simulation

David Lane

This work is produced by The Connexions Project and licensed under the Creative Commons Attribution License †

1 General Instructions

This simulation demonstrates the eect of sample size on the shape of the sampling distribution of the mean. Depicted on the top graph is the population which is sometimes referred to as the parent distributoin. Two sampling distributions of the mean, associated with their respective sample size will be created on the second and third graphs. For both the population distribution and the sampling distributions, their mean and the standard de- viation are depicted graphically on the frequency distribution itself. The blue-colored vertical bar below the X-axis indicates where the mean value falls. The red line starts from this mean value and extends one standard deviation in length in both directions. The values of both the mean and the standard deviation are also given to the left of the graph. Notice that the numeric form of a property matches its graphical form in color. In additon, the skew and the kurtosis of each distribution are also provided to the left. These two variables are determined by the shape of distribution. The skew and kurtosis for a normal distribution are both 0. In this simulation, you need to rst specify a population (the default is uniform distribution). Take note of the skew and kurtosis of the population. Then pick two dierent sample sizes (the defaults are N = 2 and N = 10), and sample a suciently large number of samples until the sampling distributions change relatively little with additional samples (about 50,000 samples.) Observe the overall shape of the two sampling distributions, and further compare their means, standard deviations, skew and kurtosis. Change the sample sizes and repeat the process a few times. Do you observe a general rule regarding the eect of sample size on the shape of the sampling distribution? You may also test the eect of sample size with populations of other shape (uniform, skewed or customed ones).

2 Step by Step Instructions

Show Questions^1

With the default setting (uniform population, sample sizes set at 2 and 5, respectively), click the button "5 Samples" a couple of times. Notice how the sample means accumulate at the bottom two graphs. Then click the button "5000 Samples" multiple times until the total number of samples exceeds 50,000. Observe the shape of the two distributions, and compare their variance, skew and kurtosis. Write these numbers down on a piece of paper for future reference. (Square the standard deviation to get the variance).

∗Version 1.3: Jul 14, 2003 2:48 pm GMT- †http://creativecommons.org/licenses/by/1. (^1) "Central Limit Theorem Demo"

http://cnx.org/content/m11186/1.3/

Connexions module: m11186 2

Set the sample sizes to be 10 and 15, respectively. Sample 50,000 times for each sample size. Observe the shape of the two distributions, and compare their variance, skew and kurtosis. Write them down for future reference. Repeat for samples size 25.
Review the data you have written down. Answer the following question: How does sample size aect the shape of the sampling distribution of the mean? What is the eect of sample size on the variance. What is the eect on the variance of doubling the sample size (Compare N = 5 to N = 10). What is the eect of tripling the sample size? How does sample size aect skew and kurtosis?
Set the population to be "Normal", set the sample size to be 2, 5, 10, 15, 25, respectively. Sample 50,000 times in each case. Write down the variance associated with each sample size on a piece of paper. Does the rule you found with the uniform population hold here?
Set the population to be "Skewed" and repeat steps 1-3.
Set the population to be "Custom", click and drag mouse in the top graph to construct a distribution of your own., then repeat steps 1-3. This is a Java Applet. To view, please see http://cnx.org/content/m11186/latest/

3 Summary

Skew and kurtosis are statistics that reect the shape of a distribution. The shape of a sampling distribution of the mean is aected by the sample size. As sample size increases, the sampling distribution of the mean approaches a normal distribution. This is an important part of the "Central Limit Theorem".

http://cnx.org/content/m11186/1.3/

m11186Clt simulation, college study notes - Clt simulation, Study notes of Mathematical Modeling and Simulation

Related documents

Partial preview of the text

Download m11186Clt simulation, college study notes - Clt simulation and more Study notes Mathematical Modeling and Simulation in PDF only on Docsity!

CLT Simulation

David Lane

1 General Instructions

2 Step by Step Instructions

3 Summary