SOCR - User contributions [en]

SOCR Courses 2007 2008 Stat13 1 Lab6

2007-11-13T08:33:08Z

TiffanyHead: /* Applications */

== [[SOCR_Courses_2007_2008_Stat13_1 | Stats 13.1]] - Laboratory Activity 6==

===Central Limit Theorem (CLT) Activity===

This activity represents a very general demonstration of the the [http://en.wikipedia.org/wiki/Central_limit_theorem Central Limit Theorem (CLT)]. The activity is based on the [http://socr.ucla.edu/htmls/SOCR_Experiments.html SOCR Sampling Distribution CLT Experiment]. This experiment builds upon a [http://www.ruf.rice.edu/~lane/stat_sim/sampling_dist/ RVLS CLT applet] by extending the applet functionality and providing the capability of sampling from any [[About_pages_for_SOCR_Distributions | SOCR Distribution]].
----

==Goals==
The aims of this activity are to:
* Provide intuitive notion of sampling from any process with a well-defined distribution.
* Motivate and facilitate learning of the [http://en.wikipedia.org/wiki/Central_limit_theorem central limit theorem].
* Empirically validate that sample-averages of random observations (most processes) follow approximately [http://en.wikipedia.org/wiki/Normal_distribution normal distribution].
* Empirically demonstrate that the ''sample-average'' is special and other [http://en.wikipedia.org/wiki/Point_estimator sample statistics] (e.g., median, variance, range, etc.) generally do not have distributions that are normal.
* Illustrate that the expectation of the sample-average equals the population mean (and the sample-average is typically a good measure of centrality for a population/process).
* Show that the variation of the sampling distribution of the mean rapidly decreases as the sample size increases (<math> ~1\over{\sqrt{n}}</math>).
* Reinforce the concepts of a native distribution, sample, sample distribution, sampling distribution, parameter estimator and data-driven numerical parameter estimate.

==The SOCR CLT Experiment==
To start the this Experiment, go to [http://www.socr.ucla.edu/htmls/SOCR_Experiments.html SOCR Experiments] and select the SOCR Sampling Distribution CLT Experiment from the drop-down list of experiments in the left panel. The image below shows the interface to this experiment. Notice the main control widgets on this image (boxed in blue and pointed to by arrows). The generic control buttons on the top allow you to do one or multiple steps/runs, stop and reset this experiment. The two tabs in the main frame provide graphical access to the results of the experiment (Histograms and Summaries) or the Distribution selection panel (Distributions). Remember that choosing sample-sizes <= 16 will animate the samples (second graphing row), whereas larger sample-sizes (N>20) will only show the updates of the sampling distributions (bottom two graphing rows).

<center>[[Image:SOCR_Activities_General_CLT_Dinov_012207_Fig1.jpg|400px]]</center>

===Experiment 1===
Expand your Experiment panel (right panel) by clicking/dragging the vertical split-pane bar. Choose the two sample sizes for the two statistics to be 10. Press the '''step'''-button a few of times (2-5) to see the experiment run several times. Notice how data is being sampled from the native population (the distribution of the process on the top). For each step, the process of sampling 2 samples of 10 observations will generate 2 sample statistics of the 2 parameters of interest (these are defaulted to ''mean'' and ''variance''). At each step, you can see the plots of all sample values, as well as the computed sample statistics for each parameter. The sample values are shown on the second row graph, below the distribution of the process, and the two sample statistics are plotted on the bottom two rows. If we run this experiment many times, the bottom two graphs/histograms become good approximations to the corresponding sampling distributions. If we did this infinitely many times these two graphs become the sampling distributions of the chosen sample statistics (as the observations/measurements are independent within each sample and between samples). Finally, press the '''Refresh Stats Table''' button on the top to see the sample summary statistics for the native population distribution (row 1), last sample (row 2) and the two sampling distributions, in this case ''mean'' and ''variance'' (rows 3 and 4).

<center>[[Image:SOCR_Activities_General_CLT_Dinov_012207_Fig2.jpg|400px]]</center>

===Experiment 2===
For this experiment we'll look at the mean, standard deviation, skewness and kurtosis of the sample-average and the sample-variance (these two sample data-driven statistical estimates). Choose sample-sizes of 50, for both estimates (mean and variance). Select the '''Fit Normal Curve''' check-boxes for both sample distributions. '''Step''' through the experiment a few times (by clicking the Run button) and then click '''Refresh Stats Table''' button on the top to see the sample summary statistics. Try to understand and relate these sample-distribution statistics to their analogues from the native population (on the top row). For example, the mean of the multiple sample-averages is about the same as the mean of the native population, but the standard deviation of the sampling distribution of the average is about <math>\sigma\over{\sqrt{n}}</math>, where <math>\sigma</math> is the standard deviation of the original native process/distribution.

<center>[[Image:SOCR_Activities_General_CLT_Dinov_012207_Fig3.jpg|400px]]</center>

===Experiment 3===
Now let's select any of the [[About_pages_for_SOCR_Distributions | SOCR Distributions]], sample from it repeatedly and see if the central limit theorem is valid for the process we have selected. Try Normal, Poisson, Beta, Gamma, Cauchy and other continuous or discrete distributions. Are our empirical results in agreement with the CLT? Go to the '''Distributions''' tab on the top of the graphing panel. Reset the experiments panel (button on the top). Select a distribution from the drop-down list of distributions in this list. Choose appropriate parameters for your distribution, if any, and click the '''Sample from this Current Distribution''' button to send this distribution to the graphing panel in the '''Histograms and Summaries''' tab. Go to this panel and again run the experiment several times. Notice how we now sample from a Non-Normal Distribution for the first time. In this case we had chosen the Beta distribution (<math>\alpha=6.7, \beta=0.5</math>).
<center>[[Image:SOCR_Activities_General_CLT_Dinov_012207_Fig4.jpg|300px]]
[[Image:SOCR_Activities_General_CLT_Dinov_012207_Fig5.jpg|300px]]</center>

===Experiment 4===
Suppose the distribution we want to sample from is not included in the list of [[About_pages_for_SOCR_Distributions | SOCR Distributions]], under the '''Distributions''' tab. We can then draw a shape for a hypothetical distribution by clicking and dragging the mouse in the top graphing canvas (Histograms and Summaries tab panel). This away you can construct contiguous and discontinuous, symmetric and asymmetric, unimodal and multi-modal, leptokurtic and mesokurtic and other [http://en.wikipedia.org/wiki/Probability_distribution types of distributions]. In the figure below, we had demonstrated this functionality to study differences between two data-driven estimates for the population center - sample [http://en.wikipedia.org/wiki/Mean mean] and sample [http://en.wikipedia.org/wiki/Median median]. Look how the sampling distribution of the sample-average is very close to Normal, where as the sampling distribution of the sample median is not.
<center>[[Image:SOCR_Activities_General_CLT_Dinov_012207_Fig6.jpg|300px]]</center>

==Questions==
* What effects will asymmetry, gaps and continuity of the native distribution have on the applicability of the CLT, or on the asymptotic distribution of various sample statistics?
* When can we reasonably expect statistics, other than the sample mean, to have CLT properties?
* If a native process has <math>\sigma_{X}=10</math> and we take a sample of size 10, what will be <math>\sigma_{\overline{X}}</math>? Does it depend on the shape of the original process? How large should the sample-size be so that <math>\sigma_{\overline{X}}={2\over 3}\sigma_{X}</math>?

==Applications==
The second part of this SOCR Activity demonstrates the [[SOCR_EduMaterials_Activities_GCLT_Applications | applications of the Central Limit Theorem]].

SOCR Courses 2007 2008 Stat13 1 Lab6

2007-11-13T08:32:38Z

TiffanyHead: New page: == Stats 13.1 - Laboratory Activity 6== ===Central Limit Theorem (CLT) Activity=== This activity represents a very general demonstration of the the ...

SOCR Courses 2007 2008 Stat13 1

2007-11-13T08:26:54Z

TiffanyHead: /* SOCR 2007-2008 Courses - [http://www.stat.ucla.edu/%7Edinov/courses_students.dir/07/Fall/STAT13.1.dir/STAT13.html Stats 13.1] Laboratory Activities */

== [[SOCR_Courses_2007_2008 | SOCR 2007-2008 Courses]] - [http://www.stat.ucla.edu/%7Edinov/courses_students.dir/07/Fall/STAT13.1.dir/STAT13.html Stats 13.1] Laboratory Activities==

* [[SOCR_Courses_2007_2008_Stat13_1_Lab1 | Lab 1]]

* [[SOCR_Courses_2007_2008_Stat13_1_Lab2 | Lab 2]]

* [[SOCR_Courses_2007_2008_Stat13_1_Lab3 | Lab 3]]

* [[SOCR_Courses_2007_2008_Stat13_1_Lab4 | Lab 4]]

* [[SOCR_Courses_2007_2008_Stat13_1_Lab5 | Lab 5]]

* [[SOCR_Courses_2007_2008_Stat13_1_Lab6 | Lab 6]]

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2007_2008_Stat13_1}}

SOCR Courses 2007 2008 Stat13 1 Lab2

2007-10-09T04:10:46Z

TiffanyHead: /* Coin Die Experiment */

== [[SOCR_Courses_2007_2008_Stat13_1 | Stats 13.1]] - Laboratory Activity 2==

===Coin Die Experiment===

Go to <nowiki>http://www.socr.ucla.edu/htmls/SOCR Experiments.html</nowiki> and use the scroll bar to find the Coin Die Experiment. Once you find it, click on the About button and read about the experiment. Answer the following questions with references to graphs where appropriate.

====1)====

There are two random variables (X,Y ), one parameter (p) for the coin, and the probability distribution that governs each die (green or red) involved in this experiment. Describe what each one represents.

* Choose fair coin (p = 0.5), and fair dice (green and red).
* Perform 10 runs and take a snapshot
* Perform 100 runs and take a snapshot

====2)====

Construct the probability distribution of Y

====3)====

Comment on the empirical and theoretical distributions how the sample size affects them.

* Reset
* Choose p = 0.8, 1-6 flat for the green die, 3-4 flat for the red die
* Perform 100 runs and take a snapshot

====4)====

Construct the probability distribution of Y

====5)====

Compute the mean of Y

====6)====

Compute the standard deviation of Y

====7)====

Verify that these numbers agree with the ones in the applet.
* Reset
* Choose p = 0.2, green fair die, 2-5 flat for red die
* Perform 100 runs and take a snapshot

====8)====

Construct the probability distribution of Y

====9)====

Comment on the empirical and theoretical distributions.

===Monte Hall Activity (Let's Make a Deal)===

The Monte Hall Activity can be found here: [[SOCR EduMaterials Activities MontyHall]]

SOCR Courses 2007 2008 Stat13 1 Lab2

2007-10-09T04:09:59Z

TiffanyHead: /* Coin Die Experiment */

== [[SOCR_Courses_2007_2008_Stat13_1 | Stats 13.1]] - Laboratory Activity 2==

===Coin Die Experiment===

Go to [http://www.socr.ucla.edu/htmls/SOCR Experiments.html] and use the scroll bar to find the Coin Die Experiment. Once you find it, click on the About button and read about the experiment. Answer the following questions with references to graphs where appropriate.

====1)====

There are two random variables (X,Y ), one parameter (p) for the coin, and the probability distribution that governs each die (green or red) involved in this experiment. Describe what each one represents.

* Choose fair coin (p = 0.5), and fair dice (green and red).
* Perform 10 runs and take a snapshot
* Perform 100 runs and take a snapshot

====2)====

Construct the probability distribution of Y

====3)====

Comment on the empirical and theoretical distributions how the sample size affects them.

* Reset
* Choose p = 0.8, 1-6 flat for the green die, 3-4 flat for the red die
* Perform 100 runs and take a snapshot

====4)====

Construct the probability distribution of Y

====5)====

Compute the mean of Y

====6)====

Compute the standard deviation of Y

====7)====

Verify that these numbers agree with the ones in the applet.
* Reset
* Choose p = 0.2, green fair die, 2-5 flat for red die
* Perform 100 runs and take a snapshot

====8)====

Construct the probability distribution of Y

====9)====

Comment on the empirical and theoretical distributions.

===Monte Hall Activity (Let's Make a Deal)===

The Monte Hall Activity can be found here: [[SOCR EduMaterials Activities MontyHall]]

SOCR Courses 2007 2008 Stat13 1 Lab2

2007-10-09T02:27:38Z

TiffanyHead: /* Stats 13.1 - Laboratory Activity 2 */

== [[SOCR_Courses_2007_2008_Stat13_1 | Stats 13.1]] - Laboratory Activity 2==

===Coin Die Experiment===

Go to http://www.socr.ucla.edu/htmls/SOCR Experiments.html and use the scroll bar to find the Coin Die Experiment. Once you find it, click on the About button and read about the experiment. Answer the following questions with references to graphs where appropriate.

====1)====

There are two random variables (X,Y ), one parameter (p) for the coin, and the probability distribution that governs each die (green or red) involved in this experiment. Describe what each one represents.

* Choose fair coin (p = 0.5), and fair dice (green and red).
* Perform 10 runs and take a snapshot
* Perform 100 runs and take a snapshot

====2)====

Construct the probability distribution of Y

====3)====

Comment on the empirical and theoretical distributions how the sample size affects them.

* Reset
* Choose p = 0.8, 1-6 flat for the green die, 3-4 flat for the red die
* Perform 100 runs and take a snapshot

====4)====

Construct the probability distribution of Y

====5)====

Compute the mean of Y

====6)====

Compute the standard deviation of Y

====7)====

Verify that these numbers agree with the ones in the applet.
* Reset
* Choose p = 0.2, green fair die, 2-5 flat for red die
* Perform 100 runs and take a snapshot

====8)====

Construct the probability distribution of Y

====9)====

Comment on the empirical and theoretical distributions.

===Monte Hall Activity (Let's Make a Deal)===

The Monte Hall Activity can be found here: [[SOCR EduMaterials Activities MontyHall]]

SOCR Courses 2007 2008 Stat13 1 Lab2

2007-10-09T02:19:33Z

TiffanyHead:

SOCR Courses 2007 2008 Stat13 1

2007-10-09T02:09:59Z

TiffanyHead: /* SOCR 2007-2008 Courses - [http://www.stat.ucla.edu/%7Edinov/courses_students.dir/07/Fall/STAT13.1.dir/STAT13.html Stats 13.1] Laboratory Activities */

SOCR Courses 2007 2008 Stat13 1 Lab1

2007-10-01T20:17:50Z

TiffanyHead: /* Stats 13.1 - Laboratory Activity 1 */

== [[SOCR_Courses_2007_2008_Stat13_1 | Stats 13.1]] - Laboratory Activity 1==

===Histogram Activity===

This is an exploratory data analysis SOCR activity that illustrates the generation and interpretation of the histogram of quantitative data. In a nutshell, a histogram of a dataset is a graphical visualization of tabulated frequencies or counts of data within equispaced partition of the range of the data. A histogram shows what proportion of measurements fall into each of the categories defined by the partition of the data range space.

Go to www.socr.ucla.edu and click on the Charts tab at the top of the page. Once the page comes up, go to the left side of he page and drag the grey bar to the right.

====1. Histogram from Categories and Frequencies====

* In the area on the left, click the arrow next to Bar Charts then XYPlots. Then click on HistogramChartDemo3
* Click on the DATA tab to view the default data. Notice that the chart requires the user to enter the counts/frequencies of observations within each of the range categories (in this default data case, year).
* Using the SHOW ALL tab you can see all three (graph, data and mapping) in the same view.
* Try revising some of the numbers in the second (frequency) column and click UPDATE button to see the effect of these changes on the histogram. Do this several times. Take a SNAPSHOT and print off one new histogram.

====2. Simple Histogram from Raw Data====

* Click on HistogramChartDemo
* Scroll down to find the Bin Size adjustment bar
* Change the bin size

====3. Histogram from Simulated Data====

* Lets first get some data: Go to the Modeler tab at the top of the page. It is best if you open a new page for this.
* Click on the Data Generation button in the center of the screen. From the drop down bar choose the Normal Distribution and change the number of samples to 20 and the standard deviation to 100. Make sure the Raw Data box on the left is checked.
* Hit Sample then click on the Data tab to see the data you generated.
* Copy these data using the Copy button at the top. (select all using Apple+a)
* Go back to the data section of the histogram chart and replace the current data with the data you just copied. Make sure to use the PASTE button on the left.
* Click UPDATE CHART
* Change the bin size smaller and larger and observe how the graph changes
* Repeat but change the number of samples to 100
* Take a SNAPSHOT and print off one new histogram

====4. Questions====

Answer each question fully using appropriate terminology and references to snapshots if
appropriate.

1) What is the effect of the width/size of the histogram bin on the shape of the resulting histogram? Would the shape of the histogram change significantly if we alter the bin-size?

2) How do the sample size and bin size interact?

3) Would you expect the shape of the sample histogram to look like the shape of the population distribution the data sample came from?

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2007_2008_Stat13_1_Lab1}}

SOCR Courses 2007 2008 Stat13 1 Lab1

2007-10-01T06:32:05Z

TiffanyHead: /* Stats 13.1 - Laboratory Activity 1 */

== [[SOCR_Courses_2007_2008_Stat13_1 | Stats 13.1]] - Laboratory Activity 1==

===Histogram Activity===

This is an exploratory data analysis SOCR activity that illustrates the generation and
interpretation of the histogram of quantitative data. In a nutshell, a histogram of a dataset
is a graphical visualization of tabulated frequencies or counts of data within equispaced
partition of the range of the data. A histogram shows what proportion of measurements
that fall into each of the categories defined by the partition of the data range space.

Go to www.socr.ucla.edu and click on the Charts tab at the top of the page. Once the page comes up, go to the left side of he page and drag the grey bar to the right.

====1. Histogram from Categories and Frequencies====

* In the area on the left, click the arrow next to Bar Charts then XYPlots. Then click on HistogramChartDemo3
* Click on the DATA tab to view the default data. Notice that the chart requires the user to enter the counts/frequencies of observations within each of the range categories (in this default data case, year).
* Using the SHOW ALL tab you can see all three (graph, data and mapping) in the same view.
* Try revising some of the numbers in the second (frequency) column and click UPDATE button to see the effect of these changes on the histogram. Do this several times. Take a SNAPSHOT and print off one new histogram.

====2. Simple Histogram from Raw Data====

* Click on HistogramChartDemo
* Scroll down to find the Bin Size adjustment bar
* Change the bin size

====3. Histogram from Simulated Data====

* Lets first get some data: Go to the Modeler tab at the top of the page. It is best if you open a new page for this.
* Click on the Data Generation button in the center of the screen. From the drop down bar choose the Normal Distribution and change the number of samples to 20 and the standard deviation to 100. Make sure the Raw Data box on the left is checked.
* Hit Sample then click on the Data tab to see the data you generated.
* Copy these data using the Copy button at the top. (select all using Apple+a)
* Go back to the data section of the histogram chart and replace the current data with the data you just copied. Make sure to use the PASTE button on the left.
* Click UPDATE CHART
* Change the bin size smaller and larger and observe how the graph changes
* Repeat but change the number of samples to 100
* Take a SNAPSHOT and print off one new histogram

====4. Questions====

Answer each question fully using appropriate terminology and references to snapshots if
appropriate.

1) What is the effect of the width/size of the histogram bin on the shape of the resulting histogram? Would the shape of the histogram change significantly if we alter the bin-size?

2) How do the sample size and bin size interact?

3) Would you expect the shape of the sample histogram to look like the shape of the population distribution the data sample came from?

<hr>
* SOCR Home page: http://www.socr.ucla.edu

{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=SOCR_Courses_2007_2008_Stat13_1_Lab1}}