AP Statistics Curriculum 2007 Limits Norm2Bin - Revision history

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

2020-03-03T16:44:10Z

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

IvoDinov: /* Normal Approximation to Binomial Distribution */

2011-02-12T02:46:19Z

‎Normal Approximation to Binomial Distribution

Jenny: /* Activities */

2010-06-28T20:20:30Z

‎Activities

IvoDinov: added a link to the Problems set

2009-10-26T05:01:50Z

added a link to the Problems set

PriscillaChui: /* Example */

2008-02-15T19:17:44Z

‎Example

IvoDinov: /* Normal Approximation to Binomial Distribution */

2008-02-04T05:06:55Z

‎Normal Approximation to Binomial Distribution

IvoDinov at 01:35, 3 February 2008

2008-02-03T01:35:37Z

IvoDinov: /* Example */

2008-02-03T01:27:29Z

‎Example

IvoDinov: /* Example */

2008-02-03T01:26:46Z

‎Example

IvoDinov: /* Example */

2008-02-03T01:26:17Z

‎Example

← Older revision		Revision as of 16:44, 3 March 2020
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	* SOCR Home page: http://www.socr.ucla.edu		* SOCR Home page: http://www.socr.ucla.edu

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 === Normal Approximation to Binomial Distribution===
-Suppose [[AP_Statistics_Curriculum_2007_Distrib_Binomial | <math>Y\sim Binomial(n, p)</math>]] and <math>Y=Y_1+ Y_2+ Y_3+\cdots+ Y_n</math>, where [[AP_Statistics_Curriculum_2007_Distrib_Binomial | <math>Y_k~Bernoulli(p)</math>]] , <math>E(Y_k)=p</math>  & <math>Var(Y_k)=p(1-p)</math>. Then   <math>E(Y)=np</math>,  <math>Var(Y)=np(1-p)</math> and <math>SD(Y)= \sqrt{np(1-p)}</math>. If we use the [[AP_Statistics_Curriculum_2007_Normal_Prob | Normal Standardization formula]] for ''Y'' we get <math>Z={Y-np\over \sqrt{np(1-p)}}</math>.
+Suppose [[AP_Statistics_Curriculum_2007_Distrib_Binomial | <math>Y\sim Binomial(n, p)</math>]] and <math>Y=Y_1+ Y_2+ Y_3+\cdots+ Y_n</math>, where [[AP_Statistics_Curriculum_2007_Distrib_Binomial | <math>Y_k \sim Bernoulli(p)</math>]] , <math>E(Y_k)=p</math>  & <math>Var(Y_k)=p(1-p)</math>. Then   <math>E(Y)=np</math>,  <math>Var(Y)=np(1-p)</math> and <math>SD(Y)= \sqrt{np(1-p)}</math>. If we use the [[AP_Statistics_Curriculum_2007_Normal_Prob | Normal Standardization formula]] for ''Y'' we get <math>Z={Y-np\over \sqrt{np(1-p)}}</math>.
 By [[AP_Statistics_Curriculum_2007_Limits_CLT | CLT]], <math>Z \sim N(0, 1)</math> and <math>Y \sim N [\mu=np, \sigma^2={np(1-p)}]</math>.

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 But we can also ''approximate'' this probability using the normal distribution.  We will need the mean and the standard deviation of this [[About_pages_for_SOCR_Distributions | normal distribution]].  These are
 <math> \mu=np=80\frac{4}{52}=6.154 </math> and
-<math> \sigma=\sqrt{80 \frac{4}{52}\frac{48}{52}}=2.383. </math>   Of course this can be obtained directly from the SOCR Binomial applet.  Now, all you need to do is to select the SOCR normal distribution applet and enter for the mean 6.154, and for the standard deviation 2.383.  To obtain the desire probability in the right cut-off box enter 7.5 (using the continuity correction for better approximation).  The approximate probability is <math> P(X \ge 8) \approx 0.2861 </math> (see figure below).
+<math> \sigma=\sqrt{80 \frac{4}{52}\frac{48}{52}}=2.383. </math>   Of course this can be obtained directly from the SOCR Binomial applet.  Now, all you need to do is to select the SOCR normal distribution applet and enter for the mean 6.154, and for the standard deviation 2.383.  To obtain the desire probability in the right cut-off box, enter 7.5 (using the continuity correction for better approximation).  The approximate probability is <math> P(X \ge 8) \approx 0.2861 </math> (see figure below).
 <center>[[Image: SOCR_Activities_ExploreDistributions_Christou_figure10.jpg|600px]]</center>
 <hr>
 ===[[EBook_Problems_Limits_Norm2Bin|Problems]]===

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 <hr>
-===References===
+===[[EBook_Problems_Limits_Norm2Bin|Problems]]===
 <hr>

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 ===Example===
-The [[SOCR_EduMaterials_Activities_RouletteExperiment | Roulette Game]] has 38 slots 18red 18 black and 2 neutral. Suppose we play 100 games betting on red each time. Would observing 58 wins in the 100 games be considered atypical (susspicious)?
+The [[SOCR_EduMaterials_Activities_RouletteExperiment | Roulette Game]] has 38 slots: 18 red, 18 black and 2 neutral. Suppose we play 100 games betting on red each time. Would observing 58 wins in the 100 games be considered atypical (suspicious)?
 To answer this question, we need to compute the probability <math>P(Y\geq58)</math>,  where <math>Y\sim Binomial(100, 0.47)</math>, as ''P(Win)=P(Red outcome) = 18/38=0.47''.

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 : <math>np\geq 10</math>
 : <math>n(1-p)> 10</math>
 ===Example===

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 To answer this question, we need to compute the probability <math>P(Y\geq58)</math>,  where <math>Y\sim Binomial(100, 0.47)</math>, as ''P(Win)=P(Red outcome) = 18/38=0.47''.
-Since <math>np=47 \geq 10</math> and <math>n(1-p)=53>10</math>, Normal approximation is justified. Let <math>Z={(Y-np)\over \sqrt{np(1-p)}} ={(58-100*0.47)\over \sqrt{100*0.47*0.53}}=2.2</math>. Thus we can approximate <math>P(Y\geq 58) \approx P(Z\geq 2.2) = 0.0139</math>. The last equation can be directly computed using the [http://socr.ucla.edu/htmls/SOCR_Distributions.html SOCR Normal Distribution Calculator]. You can also use the [http://socr.ucla.edu/htmls/SOCR_Distributions.html SOCR Binomial Distribution Calculator] to compute the exact probability <math>P(Y\geq 58) = 0.0177</math>.
+Since <math>np=47 \geq 10</math> and <math>n(1-p)=53>10</math>, Normal approximation is justified. Let <math>Z={(Y-np)\over \sqrt{np(1-p)}} ={(58-100*0.47)\over \sqrt{100*0.47*0.53}}=2.2</math>. Thus we can '''approximate''' <math>P(Y\geq 58) \approx P(Z\geq 2.2) = 0.0139</math>. The last equation can be directly computed using the [http://socr.ucla.edu/htmls/SOCR_Distributions.html SOCR Normal Distribution Calculator]. You can also use the [http://socr.ucla.edu/htmls/SOCR_Distributions.html SOCR Binomial Distribution Calculator] to compute the '''exact probability''' <math>P(Y\geq 58) = 0.0177</math>.
 So why is the [[AP_Statistics_Curriculum_2007_Normal_Std | Normal]] approximation to [[AP_Statistics_Curriculum_2007_Distrib_Binomial | Binomial]] distribution necessary in practice?

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 Since <math>np=47 \geq 10</math> and <math>n(1-p)=53>10</math>, Normal approximation is justified. Let <math>Z={(Y-np)\over \sqrt{np(1-p)}} ={(58-100*0.47)\over \sqrt{100*0.47*0.53}}=2.2</math>. Thus we can approximate <math>P(Y\geq 58) \approx P(Z\geq 2.2) = 0.0139</math>. The last equation can be directly computed using the [http://socr.ucla.edu/htmls/SOCR_Distributions.html SOCR Normal Distribution Calculator]. You can also use the [http://socr.ucla.edu/htmls/SOCR_Distributions.html SOCR Binomial Distribution Calculator] to compute the exact probability <math>P(Y\geq 58) = 0.0177</math>.
-So why is the [[AP_Statistics_Curriculum_2007_Normal |_Std Normal]] approximation to [[AP_Statistics_Curriculum_2007_Distrib_Binomial | Binomial]] distribution necessary in practice?
+So why is the [[AP_Statistics_Curriculum_2007_Normal_Std | Normal]] approximation to [[AP_Statistics_Curriculum_2007_Distrib_Binomial | Binomial]] distribution necessary in practice?
 Binomial approximation by Normal distribution is useful when no access to the online SOCR resources is available (but we can use printed version of the [http://socr.ucla.edu/Applets.dir/Z-table.html Normal Table] or when N is large and binomial probabilities are difficult to compute precisely!