AP Statistics Curriculum 2007 Normal Prob - Revision history

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

2020-03-03T19:15:36Z

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

IvoDinov: /* Assessing Normality */

2013-05-03T15:54:27Z

‎Assessing Normality

IvoDinov: /* General Normal Distribution */

2012-02-10T16:43:24Z

‎General Normal Distribution

IvoDinov: /* General Normal Distribution */

2012-02-10T16:41:21Z

‎General Normal Distribution

IvoDinov: /* General Normal Distribution */

2012-02-09T23:47:14Z

‎General Normal Distribution

IvoDinov: /* General Normal Distribution */

2012-02-09T23:47:02Z

‎General Normal Distribution

IvoDinov: /* Sums and averages of independent Normal random variables */

2011-02-06T06:07:20Z

‎Sums and averages of independent Normal random variables

IvoDinov: /* Examples */ added additional examples for T=(X1+X2+X3) and A=(X1+X2+X3)/3

2011-02-06T06:04:26Z

‎Examples: added additional examples for T=(X1+X2+X3) and A=(X1+X2+X3)/3

IvoDinov: /* General Normal Distribution */ typo

2011-02-04T16:32:54Z

‎General Normal Distribution: typo

IvoDinov: added a link to the Problems set

2009-10-26T04:57:11Z

added a link to the Problems set

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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_Normal_Prob}}	+	"{{translate\|pageName=http://wiki.socr.umich.edu/index.php?title=AP_Statistics_Curriculum_2007_Normal_Prob}}

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 * Symmetry: Are the [[AP_Statistics_Curriculum_2007_EDA_Center |mean and median]] of the dataset equal (Mean = Median)? Use the [[AP_Statistics_Curriculum_2007_Distrib_MeanVar#Notable_Moments | skewness measure]].
 * Flatness: Is the data distribution as flat as the Normal distribution? Use the [[AP_Statistics_Curriculum_2007_Distrib_MeanVar#Notable_Moments |kurtosis measure]].
-* Do [[SOCR_EduMaterials_Activities_Histogram_Graphs | histogram]], [[SOCR_EduMaterials_Activities_BoxPlot | box-and-whisker]] or [[SOCR_EduMaterials_Activities_DotChart |dotplot]] and look for bias (skewness), asymmetry, outliers, etc.
+* Check the data [[SOCR_EduMaterials_Activities_Histogram_Graphs | histogram]], [[SOCR_EduMaterials_Activities_BoxPlot | box-and-whisker]] and [[SOCR_EduMaterials_Activities_DotChart |dotplot]] for bias (skewness), asymmetry, outliers, etc.
 * Empirical Rule - check the percent of data that falls within 1, 2 and 3 [[AP_Statistics_Curriculum_2007_EDA_Var | SD]]s from the mean (should be approximately 68%, 95% and 99.7%).
 * Or we can do a [[SOCR_EduMaterials_Activities_QQChart |Quantile-Quantile Probability plot]] comparing the quantiles of the data against their Normal distribution counterparts.

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 === General Normal Distribution===
-The (general) Normal Distribution, <math>N(\mu, \sigma^2)</math>, where <math>\mu</math> is the mean and <math>\sigma^2</math> is the variance, is a continuous distribution that has similar exact ''areas'', bound in terms of its mean and variance, like the [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal Distribution]] and the x-axis on the symmetric intervals around the origin:
+The (general) Normal Distribution, <math>N(\mu, \sigma^2)</math>, where <math>\mu</math> is the mean and <math>\sigma^2</math> is the variance, is a continuous distribution that has similar exact ''areas'' in terms of symmetric intervals around the origin on x-axis, relative to its mean and variance, as the [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal Distribution]]:
 * The area: <math>\mu -\sigma < x < \mu+\sigma = 0.8413 - 0.1587 = 0.6826</math>
 * The area: <math>\mu -2\sigma < x < \mu+2\sigma = 0.9772 - 0.0228 = 0.9544</math>

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 === General Normal Distribution===
-The (general) Normal Distribution, <math>N(\mu, \sigma^2)</math>, where <math>\mu</math> is the mean and <math>\sigma^2</math> is the variance, is a continuous distribution that has similar exact ''areas'', bound in terms of its mean, like the [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal Distribution]] and the x-axis on the symmetric intervals around the origin:
+The (general) Normal Distribution, <math>N(\mu, \sigma^2)</math>, where <math>\mu</math> is the mean and <math>\sigma^2</math> is the variance, is a continuous distribution that has similar exact ''areas'', bound in terms of its mean and variance, like the [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal Distribution]] and the x-axis on the symmetric intervals around the origin:
 * The area: <math>\mu -\sigma < x < \mu+\sigma = 0.8413 - 0.1587 = 0.6826</math>
 * The area: <math>\mu -2\sigma < x < \mu+2\sigma = 0.9772 - 0.0228 = 0.9544</math>

@@ Line 4: / Line 4: @@
 === General Normal Distribution===
-The (general) Normal Distribution, <math>N(\mu, \sigma^2)</math>, where <math>\mu</math> is the mean and <math>\sigma^2</math>is the variance, is a continuous distribution that has similar exact ''areas'', bound in terms of its mean, like the [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal Distribution]] and the x-axis on the symmetric intervals around the origin:
+The (general) Normal Distribution, <math>N(\mu, \sigma^2)</math>, where <math>\mu</math> is the mean and <math>\sigma^2</math> is the variance, is a continuous distribution that has similar exact ''areas'', bound in terms of its mean, like the [[AP_Statistics_Curriculum_2007_Normal_Std |Standard Normal Distribution]] and the x-axis on the symmetric intervals around the origin:
 * The area: <math>\mu -\sigma < x < \mu+\sigma = 0.8413 - 0.1587 = 0.6826</math>
 * The area: <math>\mu -2\sigma < x < \mu+2\sigma = 0.9772 - 0.0228 = 0.9544</math>

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 ====Sums and averages of independent Normal random variables====
-* Let X1, X2, and X3 represent the heights of 3 random individuals. Suppose the heights are Normally distributed with mean 170cm and standard deviation 20 cm (i.e., X1, X2, X3 ~N(\mu=170, \sigma=20). ''What is the probability that the total sum T=X1+X2+X3 is less than 500cm? That is, find P(T<500)''. As the [[AP_Statistics_Curriculum_2007_Limits_CLT|X variables are Normal and independent, the total sum, T, will be Normal]](<math>\mu_T, \sigma_T</math>) and we need to find the parameters <math>\mu_T, \sigma_T</math>.
+* Let X1, X2, and X3 represent the heights of 3 random individuals. Suppose the heights are Normally distributed with mean 170cm and standard deviation 20 cm (i.e., X1, X2, X3 ~<math>N(\mu=170, \sigma=20)</math>. ''What is the probability that the total sum T=X1+X2+X3 is less than 500cm? That is, find P(T<500)''. As the [[AP_Statistics_Curriculum_2007_Limits_CLT|X variables are Normal and independent, the total sum, T, will be Normal]](<math>\mu_T, \sigma_T</math>) and we need to find the parameters <math>\mu_T, \sigma_T</math>.
 ** <math>\mu_T=E(T)=E(X1+X2+X3) = E(X1)+E(X2)+E(X3)=3\times 170=510.</math>
 ** <math>\sigma_T^2 = Var(T) = Var(X1+X2+X3)=Var(X1)+Var(X2)+Var(X3)=</math><math>20^2+20^2+20^2=1,200</math>, and <math>\sigma_T=\sqrt{1,200}=34.64.</math>
-** Thus, <math>T~N(\mu_T=510,\sigma_T=34.64)</math>, and P(T<500)= 0.386380, which can be computed using the [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html SOCR Normal Distribution Calculator] or the [http://socr.ucla.edu/Applets.dir/Z-table.html SOCR Standard Normal Z Table] via the standardizing transformation.
+** Thus, T~<math>N(\mu_T=510,\sigma_T=34.64)</math>, and P(T<500)= 0.386380, which can be computed using the [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html SOCR Normal Distribution Calculator] or the [http://socr.ucla.edu/Applets.dir/Z-table.html SOCR Standard Normal Z Table] via the standardizing transformation.
-* If A is the average of the 3 heights, A=(X1+X2+X3)/3, ''what is the central 50-th percentile for the variable A''? That is, what are the lower (a1) and upper (a2) bounds that give P(a1<A<a2)=0.5, where a1 and a2 are symmetric with respect to the expected value of A, E(A)=\mu=170)?
+* If A is the average of the 3 heights, A=(X1+X2+X3)/3, ''what is the central 50-th percentile for the variable A''? That is, what are the lower (a1) and upper (a2) bounds that give P(a1<A<a2)=0.5, where a1 and a2 are symmetric with respect to the expected value of A, <math>E(A)=\mu=170</math>)?
 ** Note that it suffices to find one of the bounds (say a2) as these bounds are symmetric around the mean, 170. Thus, we are looking for a2, such that P(170<A<a2)=0.25, which is also the same as P(a1<A<170)=0.25.
 ** The [[AP_Statistics_Curriculum_2007_Limits_CLT|distribution of the average]] A~<math>N(\mu=170, \sigma)</math>.
 ** <math>Var(A) = Var((1/3)(X1+X2+X3))=(1/9)(Var(X1)+Var(X2)+Var(X3))=(1/9)(20^2+20^2+20^2)=133.33</math>, and <math>\sigma = 11.55</math>. Thus, A~<math>N(\mu=170, \sigma=11.55)</math>.
 ** As before, we can the [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html SOCR Normal Distribution Calculator] or the [http://socr.ucla.edu/Applets.dir/Z-table.html SOCR Standard Normal Z Table] via the standardizing transformation to compute a2=177.8, and <math>a1=\mu -(a2-\mu)=162.2</math>, as P(162.2<A<170)=P(170<A<177.8)=0.25.
 ====Systolic Arterial Pressure Example====

← Older revision		Revision as of 06:04, 6 February 2011
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	====[[SOCR_EduMaterials_Activities_Normal_Probability_examples \| A large number of Normal distribution examples using SOCR tools]]====		====[[SOCR_EduMaterials_Activities_Normal_Probability_examples \| A large number of Normal distribution examples using SOCR tools]]====
		+
		+	====Sums and averages of independent Normal random variables====
		+
		+	* Let X1, X2, and X3 represent the heights of 3 random individuals. Suppose the heights are Normally distributed with mean 170cm and standard deviation 20 cm (i.e., X1, X2, X3 ~N(\mu=170, \sigma=20). ''What is the probability that the total sum T=X1+X2+X3 is less than 500cm? That is, find P(T<500)''. As the [[AP_Statistics_Curriculum_2007_Limits_CLT\|X variables are Normal and independent, the total sum, T, will be Normal]](<math>\mu_T, \sigma_T</math>) and we need to find the parameters <math>\mu_T, \sigma_T</math>.
		+	** <math>\mu_T=E(T)=E(X1+X2+X3) = E(X1)+E(X2)+E(X3)=3\times 170=510.</math>
		+	** <math>\sigma_T^2 = Var(T) = Var(X1+X2+X3)=Var(X1)+Var(X2)+Var(X3)=</math><math>20^2+20^2+20^2=1,200</math>, and <math>\sigma_T=\sqrt{1,200}=34.64.</math>
		+	** Thus, <math>T~N(\mu_T=510,\sigma_T=34.64)</math>, and P(T<500)= 0.386380, which can be computed using the [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html SOCR Normal Distribution Calculator] or the [http://socr.ucla.edu/Applets.dir/Z-table.html SOCR Standard Normal Z Table] via the standardizing transformation.
		+
		+	* If A is the average of the 3 heights, A=(X1+X2+X3)/3, ''what is the central 50-th percentile for the variable A''? That is, what are the lower (a1) and upper (a2) bounds that give P(a1<A<a2)=0.5, where a1 and a2 are symmetric with respect to the expected value of A, E(A)=\mu=170)?
		+	** Note that it suffices to find one of the bounds (say a2) as these bounds are symmetric around the mean, 170. Thus, we are looking for a2, such that P(170<A<a2)=0.25, which is also the same as P(a1<A<170)=0.25.
		+	** The [[AP_Statistics_Curriculum_2007_Limits_CLT\|distribution of the average]] A~<math>N(\mu=170, \sigma)</math>.
		+	** <math>Var(A) = Var((1/3)(X1+X2+X3))=(1/9)(Var(X1)+Var(X2)+Var(X3))=(1/9)(20^2+20^2+20^2)=133.33</math>, and <math>\sigma = 11.55</math>. Thus, A~<math>N(\mu=170, \sigma=11.55)</math>.
		+	** As before, we can the [http://socr.ucla.edu/htmls/dist/Normal_Distribution.html SOCR Normal Distribution Calculator] or the [http://socr.ucla.edu/Applets.dir/Z-table.html SOCR Standard Normal Z Table] via the standardizing transformation to compute a2=177.8, and <math>a1=\mu -(a2-\mu)=162.2</math>, as P(162.2<A<170)=P(170<A<177.8)=0.25.

	====Systolic Arterial Pressure Example====		====Systolic Arterial Pressure Example====

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 * The area: <math>\mu -2\sigma < x < \mu+2\sigma = 0.9772 - 0.0228 = 0.9544</math>
 * The area: <math>\mu -3\sigma < x < \mu +3\sigma= 0.9987 - 0.0013 = 0.9974</math>
-* Note that the [http://en.wikipedia.org/wiki/Inflection_point inflection points] (<math>f ''(x)=0</math>)of the (general) Normal density function are <math>\pm \sigma</math>.
+* Note that the [http://en.wikipedia.org/wiki/Inflection_point inflection points] (<math>f ''(x)=0</math>) of the (general) Normal density function are <math>\pm \sigma</math>.
 <center>[[Image:SOCR_EBook_Dinov_RV_Normal_013108_Fig6.jpg|500px]]</center>

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	* [http://en.wikipedia.org/wiki/Standard_normal_distribution#Normality_tests There are also a number of quantitative methods (statistical tests) to assess normality].		* [http://en.wikipedia.org/wiki/Standard_normal_distribution#Normality_tests There are also a number of quantitative methods (statistical tests) to assess normality].
		+
		+	===[[EBook_Problems_Normal_Prob\|Problems]]===

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