AP Statistics Curriculum 2007 Limits CLT - Revision history

Tdlee: Text replacement - "{{translate|pageName=http://wiki.stat.ucla.edu/socr/" to ""{{translate|pageName=http://wiki.socr.umich.edu/"

2020-03-03T19:24:12Z

Text replacement - "{{translate|pageName=http://wiki.stat.ucla.edu/socr/" to ""{{translate|pageName=http://wiki.socr.umich.edu/"

Jenny: /* Symbolic Statement of the Central Limit Theorem */

2010-06-28T20:17:58Z

‎Symbolic Statement of the Central Limit Theorem

Jenny: /* Motivation */

2010-06-28T20:17:18Z

‎Motivation

Jenny at 17:04, 28 June 2010

2010-06-28T17:04:52Z

IvoDinov: /* References */

2010-05-24T15:24:22Z

‎References

IvoDinov: added a link to the Problems set

2009-10-26T04:58:34Z

added a link to the Problems set

IvoDinov: /* CLT Applications */ added U-quadratic note

2009-05-22T17:54:37Z

‎CLT Applications: added U-quadratic note

IvoDinov: /* General Statement of the Central Limit Theorem */ typo

2009-05-20T17:09:44Z

‎General Statement of the Central Limit Theorem: typo

IvoDinov: typo

2008-05-16T17:04:41Z

typo

IvoDinov at 22:34, 18 March 2008

2008-03-18T22:34:16Z

← Older revision		Revision as of 19:24, 3 March 2020
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	* SOCR Home page: http://www.socr.ucla.edu		* SOCR Home page: http://www.socr.ucla.edu

−	{{translate\|pageName=http://wiki.~~stat~~.~~ucla~~.edu~~/socr~~/index.php?title=AP_Statistics_Curriculum_2007_Limits_CLT}}	+	"{{translate\|pageName=http://wiki.socr.umich.edu/index.php?title=AP_Statistics_Curriculum_2007_Limits_CLT}}

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 ===Symbolic Statement of the Central Limit Theorem===
-The [http://en.wikipedia.org/wiki/Central_limit_theorem formal statement of the CLT is described here], however a more appropriate statement for many undergraduate and graduate classes uses the following statement of the central limit theorem:
+The [http://en.wikipedia.org/wiki/Central_limit_theorem formal statement of the CLT is described here]. However undergraduate and graduate classes uses the following statement of the central limit theorem:
 Let {<math>X_1,X_2, \cdots, Xn</math>} be a random sample (IID) from a (native) distribution with well-defined and finite mean <math>\mu_X</math> and variance <math>\sigma_X^2</math>. Then as n increases, the sampling distributions of the sample average <math>\overline{X_n}={1\over n}\sum_{i=1}^n{X_i}</math> and the total sum  <math>\overline{T_n}=\sum_{i=1}^n{X_i}</math> approach Normal distributions with corresponding means and variances:

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 Suppose there are 10 world renowned laboratories, each is charged with conducting the same experiment (e.g., sequencing the genome [http://en.wikipedia.org/wiki/Drosophila Drosophila, the fruit fly]), using the same protocols. One of the outcomes of this study that the sponsor agency is interested could be the ''average rate of occurrence'' of the [http://lists.informatics.jax.org/searches/GO.cgi?id=GO:0033434 ATC codon] in 10,000 base-pairs in the Drosophila genome. After completing the sequencing of the genome, each lab selects a random segment of 1,000,000 base-pairs and counts the number of ATC codons in every segment of 10,000 base-pairs (there are 100 such segments). Finally they compute the average of the 100 counts they obtained. The funding/sponsoring agency receives 10 average counts from the 10 distinct laboratories. Most likely there will be differences between these averages.
-The funding agency poses the most important question: Can we predict how much variation (i.e., discrepancy) there will be among the 10 lab averages, if we had only been able to fund/conduct 1 experiment at one site (due to resource/budgetary limitations)? In other words, if the sponsoring organization could only support one lab to carry out the experiment, can they estimate what are the possible errors that may be committed by using the sample average (of 100 samples) obtained from the chosen lab?
+The funding agency poses the most important question: Can we predict how much variation (i.e., discrepancy) there will be among the 10 lab averages, if we had only been able to fund/conduct one experiment at one site (due to resource/budgetary limitations)? In other words, if the sponsoring organization could only support one lab to carry out the experiment, can they estimate what are the possible errors that may be committed by using the sample average (of 100 samples) obtained from the chosen lab?
 The answer is yes. They can accurately estimate the real count of ATC codons in the Drosophila genome from a single lab experiment as the sampling distribution of the average (across labs) is known to be (approximately) Normal!

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 ===Symbolic Statement of the Central Limit Theorem===
-The [http://en.wikipedia.org/wiki/Central_limit_theorem formal statement of the CLT is described here], however a more appropriate statement for many undergraduate and graduate classes use the following statement of the central limit theorem:
+The [http://en.wikipedia.org/wiki/Central_limit_theorem formal statement of the CLT is described here], however a more appropriate statement for many undergraduate and graduate classes uses the following statement of the central limit theorem:
 Let {<math>X_1,X_2, \cdots, Xn</math>} be a random sample (IID) from a (native) distribution with well-defined and finite mean <math>\mu_X</math> and variance <math>\sigma_X^2</math>. Then as n increases, the sampling distributions of the sample average <math>\overline{X_n}={1\over n}\sum_{i=1}^n{X_i}</math> and the total sum  <math>\overline{T_n}=\sum_{i=1}^n{X_i}</math> approach Normal distributions with corresponding means and variances:

← Older revision		Revision as of 15:24, 24 May 2010
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	===References===		===References===
		+	* Dinov, ID, Christou, N, and Sanchez, J (2008) ''Central Limit Theorem: New SOCR Applet and Demonstration Activity''. [http://www.amstat.org/publications/jse/v16n2/dinov.html Journal of Statistics Education, Volume 16, Number 2].

	<hr>		<hr>

← Older revision		Revision as of 04:58, 26 October 2009
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	* In the [http://socr.ucla.edu/htmls/exp/Sampling_Distribution_CLT_Experiment.html Sampling Distribution CLT Experiment], select [[UQuadraticDistribuionAbout \| Q-quadratic distribution]] (under the distribution tab). Set the sample sizes (n1 and n2) first to 2 and then to 4. Observe the shape of the sampling distribution -- it will become first tri-modal (n=2) and then five-modal (for n=4), respectively. As the sample sizes exceed 5, these multiple modes will merge into one, and the sampling distribution will become unimodal. Of course, the CLT guarantees that the sampling distribution of the average will ultimately become Normal, as the sample size increases.		* In the [http://socr.ucla.edu/htmls/exp/Sampling_Distribution_CLT_Experiment.html Sampling Distribution CLT Experiment], select [[UQuadraticDistribuionAbout \| Q-quadratic distribution]] (under the distribution tab). Set the sample sizes (n1 and n2) first to 2 and then to 4. Observe the shape of the sampling distribution -- it will become first tri-modal (n=2) and then five-modal (for n=4), respectively. As the sample sizes exceed 5, these multiple modes will merge into one, and the sampling distribution will become unimodal. Of course, the CLT guarantees that the sampling distribution of the average will ultimately become Normal, as the sample size increases.

		+	===[[EBook_Problems_Limits_CLT\|Problems]]===
		+
	<hr>		<hr>

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

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 === General Statement of the Central Limit Theorem===
-The Central Limit Theorem (CLT) argues that the distribution of the sum or average of independent observations from the same random process (with final mean and variance), will be approximately Normally distributed (i.e., bell-shaped curve). That is, the CLT expresses the fact that any sum or average of (many) independent and identically-distributed random variables will tend to be distributed according to a particular ''Attractor Distribution''; the Normal Distribution effectively represents the core of the universe of all (nice) distributions.
+The Central Limit Theorem (CLT) argues that the distribution of the sum or average of independent observations from the same random process (with finite mean and variance), will be approximately Normally distributed (i.e., bell-shaped curve). That is, the CLT expresses the fact that any sum or average of (many) independent and identically-distributed random variables will tend to be distributed according to a particular ''Attractor Distribution''; the Normal Distribution effectively represents the core of the universe of all (nice) distributions.
 ===Symbolic Statement of the Central Limit Theorem===

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 You can see [[SOCR_EduMaterials_Activities_GCLT_Applications | a number of applications of the Central Limit Theorem here]].
-=== The Central Limit Theorem===
+=== General Statement of the Central Limit Theorem===
 The Central Limit Theorem (CLT) argues that the distribution of the sum or average of independent observations from the same random process (with final mean and variance), will be approximately Normally distributed (i.e., bell-shaped curve). That is, the CLT express the fact that any sum or average of (many) independent and identically-distributed random variables will tend to be distributed according to a particular ''Attractor Distribution''; the Normal Distribution effectively represents the core of the universe of all (nice) distributions.
-===The Central Limit Theorem===
+===Symbolic Statement of the Central Limit Theorem===
 The [http://en.wikipedia.org/wiki/Central_limit_theorem formal statement of the CLT is described here], however a more appropriate statement for many undergraduate and graduate classes use the following statement of the central limit theorem: