AP Statistics Curriculum 2007 Normal Std - Revision history

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

2020-03-03T19:33:48Z

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

Dinov: /* Experiments */

2014-10-03T22:54:49Z

‎Experiments

WikiSysop: /* Experiments */

2014-07-25T22:15:20Z

‎Experiments

WikiSysop at 13:36, 25 July 2014

2014-07-25T13:36:43Z

IvoDinov: /* Standard Normal Distribution */

2012-02-08T16:34:23Z

‎Standard Normal Distribution

IvoDinov: /* Appendix */

2011-10-04T23:23:33Z

‎Appendix

IvoDinov: /* Appendix */

2011-10-04T22:37:19Z

‎Appendix

IvoDinov: /* Appendix */

2011-10-04T22:36:55Z

‎Appendix

IvoDinov: /* Appendix */

2011-10-04T22:34:50Z

‎Appendix

IvoDinov: /* Appendix */

2011-10-04T22:33:05Z

‎Appendix

← Older revision		Revision as of 19:33, 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_Normal_Std}}	+	"{{translate\|pageName=http://wiki.socr.umich.edu/index.php?title=AP_Statistics_Curriculum_2007_Normal_Std}}

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 ===Experiments===
 Suppose we decide to test the state of 100 used batteries. To do that, we connect each battery to a volt-meter by randomly attaching the positive (+) and negative (-) battery terminals to the corresponding volt-meter's connections. Electrical current always flows from + to -, i.e., the current goes in the direction of the voltage drop. Depending upon which way the battery is connected to the volt-meter we can observe positive or negative voltage recordings (voltage is just a difference, which forces current to flow from higher to the lower voltage.) Denote <math>X_i</math>={measured voltage for battery i} - this is random variable with mean of 0 and unitary variance. Assume the distribution of all <math>X_i</math> is Standard Normal, <math>X_i \sim N(0,1)</math>. Use the Normal Distribution (with mean=0 and variance=1) in the [http://socr.ucla.edu/htmls/SOCR_Distributions.html SOCR Distribution applet] to address the following questions. This [[Help_pages_for_SOCR_Distributions | Distributions help-page may be useful in understanding SOCR Distribution Applet]]. How many batteries, from the sample of 100, can we expect to have?
-$ * Absolute Voltage > 1? P(X>1) = 0.1586 $, thus we expect 15-16 batteries to have voltage exceeding 1.
+* Absolute Voltage > 1? $P(X>1) = 0.1586 $, thus we expect 15-16 batteries to have voltage exceeding 1.
 <center>[[Image:SOCR_EBook_Dinov_RV_Normal_013108_Fig1.jpg|500px]]</center>
-$ * |Absolute Voltage| > 1? P(|X|>1) = 1- 0.682689=0.3173 $, thus we expect 31-32 batteries to have absolute voltage exceeding 1.
+* |Absolute Voltage| > 1? $P(|X|>1) = 1- 0.682689=0.3173 $, thus we expect 31-32 batteries to have absolute voltage exceeding 1.
 <center>[[Image:SOCR_EBook_Dinov_RV_Normal_013108_Fig2.jpg|500px]]</center>
-$ * Voltage < -2? P(X<-2) = 0.0227 $, thus we expect 2-3 batteries to have voltage less than -2.
+* Voltage < -2? $P(X<-2) = 0.0227 $, thus we expect 2-3 batteries to have voltage less than -2.
 <center>[[Image:SOCR_EBook_Dinov_RV_Normal_013108 _Fig3.jpg|500px]]</center>
-$ * Voltage <= -2? P(X<=-2) = 0.0227 $, thus we expect 2-3 batteries to have voltage less than or equal to -2.
+* Voltage <= -2? $P(X<=-2) = 0.0227 $, thus we expect 2-3 batteries to have voltage less than or equal to -2.
 <center>[[Image:SOCR_EBook_Dinov_RV_Normal_013108_Fig3.jpg|500px]]</center>
-$ * -1.7537 < Voltage < 0.8465? P(-1.7537 < X < 0.8465)  = 0.761622 $, thus we expect 76 batteries to have voltage in this range.
+* -1.7537 < Voltage < 0.8465? $P(-1.7537 < X < 0.8465)  = 0.761622 $, thus we expect 76 batteries to have voltage in this range.
 <center>[[Image:SOCR_EBook_Dinov_RV_Normal_013108_Fig4.jpg|500px]]</center>

@@ Line 18: / Line 18: @@
 ===Experiments===
 Suppose we decide to test the state of 100 used batteries. To do that, we connect each battery to a volt-meter by randomly attaching the positive (+) and negative (-) battery terminals to the corresponding volt-meter's connections. Electrical current always flows from + to -, i.e., the current goes in the direction of the voltage drop. Depending upon which way the battery is connected to the volt-meter we can observe positive or negative voltage recordings (voltage is just a difference, which forces current to flow from higher to the lower voltage.) Denote <math>X_i</math>={measured voltage for battery i} - this is random variable with mean of 0 and unitary variance. Assume the distribution of all <math>X_i</math> is Standard Normal, <math>X_i \sim N(0,1)</math>. Use the Normal Distribution (with mean=0 and variance=1) in the [http://socr.ucla.edu/htmls/SOCR_Distributions.html SOCR Distribution applet] to address the following questions. This [[Help_pages_for_SOCR_Distributions | Distributions help-page may be useful in understanding SOCR Distribution Applet]]. How many batteries, from the sample of 100, can we expect to have?
-* Absolute Voltage > 1? P(X>1) = 0.1586, thus we expect 15-16 batteries to have voltage exceeding 1.
+$ * Absolute Voltage > 1? P(X>1) = 0.1586 $, thus we expect 15-16 batteries to have voltage exceeding 1.
 <center>[[Image:SOCR_EBook_Dinov_RV_Normal_013108_Fig1.jpg|500px]]</center>
-* |Absolute Voltage| > 1? P(|X|>1) = 1- 0.682689=0.3173, thus we expect 31-32 batteries to have absolute voltage exceeding 1.
+$ * |Absolute Voltage| > 1? P(|X|>1) = 1- 0.682689=0.3173 $, thus we expect 31-32 batteries to have absolute voltage exceeding 1.
 <center>[[Image:SOCR_EBook_Dinov_RV_Normal_013108_Fig2.jpg|500px]]</center>
-* Voltage < -2? P(X<-2) = 0.0227, thus we expect 2-3 batteries to have voltage less than -2.
+$ * Voltage < -2? P(X<-2) = 0.0227 $, thus we expect 2-3 batteries to have voltage less than -2.
 <center>[[Image:SOCR_EBook_Dinov_RV_Normal_013108_Fig3.jpg|500px]]</center>
-* Voltage <= -2? P(X<=-2) = 0.0227, thus we expect 2-3 batteries to have voltage less than or equal to -2.
+$ * -1.7537 < Voltage < 0.8465? P(-1.7537 < X < 0.8465)  = 0.761622 $, thus we expect 76 batteries to have voltage in this range.
 <center>[[Image:SOCR_EBook_Dinov_RV_Normal_013108_Fig4.jpg|500px]]</center>

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 === Standard Normal Distribution===
 The Standard Normal Distribution is a continuous distribution with the following density:
-* Standard Normal ''density'' function <math>f(x)= {e^{-x^2 \over 2} \over \sqrt{2 \pi}}.</math>
+* Standard Normal ''density'' function $ f(x)= {e^{-x^2 \over 2} \over \sqrt{2 \pi}}. $
 * Standard Normal ''cumulative distribution'' function <math>\Phi(y)= \int_{-\infty}^{y}{{e^{-x^2 \over 2} \over \sqrt{2 \pi}} dx}.</math>
 * Why are these two functions, <math>f(x), \Phi(y)</math> well-defined density and distribution functions, i.e., <math>\int_{-\infty}^{\infty} {f(x)dx}=1</math>? [[AP_Statistics_Curriculum_2007_Normal_Std#Appendix|See the appendix below]].

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 * Standard Normal ''density'' function <math>f(x)= {e^{-x^2 \over 2} \over \sqrt{2 \pi}}.</math>
 * Standard Normal ''cumulative distribution'' function <math>\Phi(y)= \int_{-\infty}^{y}{{e^{-x^2 \over 2} \over \sqrt{2 \pi}} dx}.</math>
-* Why are these two functions, <math>f(x), \Phi(y)</math> well-defined density and distribution functions, i.e., <math>\int_{-\infty}^{\infty} {f(x)dx}=1</math>? See the appendix below.
+* Why are these two functions, <math>f(x), \Phi(y)</math> well-defined density and distribution functions, i.e., <math>\int_{-\infty}^{\infty} {f(x)dx}=1</math>? [[AP_Statistics_Curriculum_2007_Normal_Std#Appendix|See the appendix below]].
 Note that the following exact ''areas'' are bound between the Standard Normal Density Function and the x-axis on these symmetric intervals around the origin:

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 :: <math>x=r\cos(\theta)</math>
 :: <math>x=r\cos(\theta)</math>, <math>0\le \theta\le 2\pi</math>
-:: Hence, <math>x^2+w^2=r^2</math>, <math>e^{-x^2 \over 2} \times e^{-w^2 \over 2} =e^{-r^2 \over 2}</math>, and
+:: Hence,
-:: <math>dx=\cos(\theta)dr</math>
+::: <math>x^2+w^2=r^2</math>,
-:: <math>dy=r\cos(\theta)dr</math>
+::: <math>e^{-x^2 \over 2} \times e^{-w^2 \over 2} =e^{-r^2 \over 2}</math>,
 : Therefore, <math>A^2=\int_{0}^{\infty} {\int_{0}^{2\pi} {{e^{-r^2 \over 2} \over 2 \pi}\cos^2(\theta)rdrd\theta}}</math>, and
 : <math>A^2=\int_{0}^{\infty} {\int_{0}^{2\pi} {{e^{-r^2 \over 2} \over 2 \pi}d{\frac{r^2}{2}}d\theta}}=\int_{0}^{\infty} {{e^{-r^2 \over 2} \over \pi}d{\frac{r^2}{2}}} \times \int_{0}^{2\pi} {\cos^2(\theta)d\theta}=1</math>, since

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 :: <math>dy=r\cos(\theta)dr</math>
 : Therefore, <math>A^2=\int_{0}^{\infty} {\int_{0}^{2\pi} {{e^{-r^2 \over 2} \over 2 \pi}(-\cos^2(\theta))rdrd\theta}}</math>, and
-: <math>A^2=-\int_{0}^{\infty} {\int_{0}^{2\pi} {{e^{-r^2 \over 2} \over 2 \pi}d{\frac{r^2}{2}}d\theta}}=-\int_{0}^{\infty} {{e^{-r^2 \over 2} \over \pi}d{\frac{r^2}{2}}} \times \int_{0}^{2\pi} {\cos^2(\theta)d\theta}=1</math>, since
+: <math>A^2=\int_{0}^{\infty} {\int_{0}^{2\pi} {{e^{-r^2 \over 2} \over 2 \pi}d{\frac{r^2}{2}}d\theta}}=\int_{0}^{\infty} {{e^{-r^2 \over 2} \over \pi}d{\frac{r^2}{2}}} \times \int_{0}^{2\pi} {\cos^2(\theta)d\theta}=1</math>, since
 :: <math>\int_{0}^{2\pi} {\cos^2(\theta)d\theta}=\int_{0}^{2\pi} {\frac{1+\cos(2\theta)}{2}d\theta}=\pi</math>, and
 :: <math>\int_{0}^{\infty} {e^{-w}dw}=1</math>.