AP Statistics Curriculum 2007 Bayesian Normal - Revision history

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

2020-03-03T18:10:56Z

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

Jenny: /* Motivational example */

2010-06-28T21:15:20Z

‎Motivational example

Jenny at 21:14, 28 June 2010

2010-06-28T21:14:40Z

IvoDinov: /* Motivational example */ minor table typo fix

2009-10-23T19:22:12Z

‎Motivational example: minor table typo fix

IvoDinov: corrected a minor table typo

2009-10-23T19:20:21Z

corrected a minor table typo

IvoDinov at 22:49, 22 October 2009

2009-10-22T22:49:21Z

IvoDinov at 22:48, 22 October 2009

2009-10-22T22:48:30Z

DaveZes at 16:04, 23 July 2009

2009-07-23T16:04:35Z

JayZzz at 05:03, 26 May 2009

2009-05-26T05:03:42Z

JayZzz at 02:32, 26 May 2009

2009-05-26T02:32:30Z

← Older revision		Revision as of 18:10, 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_Bayesian_Normal}}	+	"{{translate\|pageName=http://wiki.socr.umich.edu/index.php?title=AP_Statistics_Curriculum_2007_Bayesian_Normal}}

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 It is known that the speedometer that comes with a certain new sports car is not very accurate, which results in an estimate of the top speed of the car of 185 mph, with a standard deviation of 10 mph.  Knowing that his car is capable of much higher speeds, the owner take the car to the shop. After a checkup, the speedometer is replaced with a better one, which gives a new estimate of 220 mph with a standard deviation of 4 mph. The errors are assumed to be normally distributed.
-We can say that the owner '''S’s''' prior beliefs about the top speed of his car were represented by:
+We can say that the owner '''S’s''' prior beliefs about the top speed of his car are represented by:
 : <math>\mu \sim N \mu_0</math>, <math>\phi_0</math>, i.e., <math>\mu \sim N(185,10^2)</math>

@@ Line 2: / Line 2: @@
 ===Motivational example===
-It is known that the speedometer that comes with a certain new sports car is not very accurate, which results in an estimate of the top speed of the car of 185 mph, with a standard deviation of 10 mph.  Knowing that his car is capable of much higher speeds, the owner took the car to the shop. After a checkup, the speedometer was replaced with a better one, which gave a new estimate of 220 mph with a standard deviation of 4 mph. The errors are assumed to be normally distributed.
+It is known that the speedometer that comes with a certain new sports car is not very accurate, which results in an estimate of the top speed of the car of 185 mph, with a standard deviation of 10 mph.  Knowing that his car is capable of much higher speeds, the owner take the car to the shop. After a checkup, the speedometer is replaced with a better one, which gives a new estimate of 220 mph with a standard deviation of 4 mph. The errors are assumed to be normally distributed.
 We can say that the owner '''S’s''' prior beliefs about the top speed of his car were represented by:

@@ Line 48: / Line 48: @@
 {| class="wikitable" style="text-align:center; width:25%" border="1"
 |-
-! || Prior Distribution || Likelihood from Data || Posterior Distribution
+! Speed || Prior Distribution || Likelihood from Data || Posterior Distribution
 |-
 | S || <math>N(185 , 10^2)</math>  || || <math>N(218 , 4^2)</math>
 |-
 |-
-| || <math>N(220 , 4^2)</math> ||
+| || || <math>N(220 , 4^2)</math> ||
 |-
-| S’ || <math>N(200,30^2)</math>|| <math>N(224,4^2)</math>
+| S’ || <math>N(200,30^2)</math>|| || <math>N(224,4^2)</math>
 |}
 </center>
 ==See also==

@@ Line 50: / Line 50: @@
 ! || Prior Distribution || Likelihood from Data || Posterior Distribution
 |-
-| S || <math>N(185 , 10^2)</math>  || <math>N(218 , 4^2)</math>
+| S || <math>N(185 , 10^2)</math>  || || <math>N(218 , 4^2)</math>
 |-
-| S’ || <math>N(220 , 4^2)</math> || <math>N(200,30^2)</math>|| <math>N(224,4^2)</math>
+|-
 |}
 </center>

← Older revision		Revision as of 22:49, 22 October 2009
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−	==[[EBook \| Probability and Statistics Ebook]] - Normal Example	+	==[[EBook \| Probability and Statistics Ebook]] - Normal Example==

	===Motivational example===		===Motivational example===

@@ Line 1: / Line 1: @@
-''Normal Example:''
+==[[EBook | Probability and Statistics Ebook]] - Normal Example
 It is known that the speedometer that comes with a certain new sports car is not very accurate, which results in an estimate of the top speed of the car of 185 mph, with a standard deviation of 10 mph.  Knowing that his car is capable of much higher speeds, the owner took the car to the shop. After a checkup, the speedometer was replaced with a better one, which gave a new estimate of 220 mph with a standard deviation of 4 mph. The errors are assumed to be normally distributed.
 We can say that the owner '''S’s''' prior beliefs about the top speed of his car were represented by:
-<div style="text-align: center;"> µ ~ N(<math>\mu_0</math>, <math>\phi_0</math>) = µ ~ N(185,<math>10^2</math>) </div>
+: <math>\mu \sim N \mu_0</math>, <math>\phi_0</math>, i.e., <math>\mu \sim N(185,10^2)</math>
 We could then say that the measurements using the new speedometer result in a measurement of:
-<div style="text-align: center;">''' x ~ N(<math>\mu</math>, <math>\phi</math>) = x ~ N(µ,<math> 4^2</math>)''' </div>
+:<math>x \sim N(\mu, \phi)</math>, i.e., <math> x \sim N(\mu, 4^2)</math>
 We note that the observation '''x''' turned out to be 210, and we see that '''S’s''' posterior beliefs about '''µ''' should be represented by:
-<div style="text-align: center;"> '''µ | x ~ N(<math>\mu_1</math>, <math>\phi_1</math>)''' </div>
+:<math>\mu | x \sim N(\mu_1, \phi_1)</math>
 where (rounded)
-<div style="text-align: center;"> '''<math>\phi_1</math> = <math>(10^{-2} + 4^{-2})^{-1}</math> = 14 = <math>4^2</math>''' </div>
+:<math>\phi_1 = (10^{-2} + 4^{-2})^{-1} = 14 = 4^2</math>
-<div style="text-align: center;"> '''<math>\mu_1</math> = <math>14(185/10^2 + 220/4^2) = 218</math>''' </div>
+:<math>\mu_1 = 14(185/10^2 + 220/4^2) = 218</math>.
 Therefore, the posterior for the top speed is:
-<div style="text-align: center;"> '''<math>\mu</math> | x ~ N(<math>218,4^2</math>)''' </div>
+:<math>\mu | x \sim N(218,4^2)</math>. Meaning 218 +/- 4 mph.
-Meaning 218 +/- 4 mph.
+If the new speedometer measurements were considered by another person '''S’''' who had no knowledge of the readings from the first speedometer, but still had a vague idea (from knowledge of the stock speedometer) that the top speed was about 200 +/- 30 mph. Then:
-If the new speedometer measurements were considered by another person '''S’''' who had no knowledge of the readings from the first speedometer, but still had a vague idea (from knowledge of the stock speedometer) that the top speed was about 200 +/- 30 mph,
+:<math>\mu \sim N(200,30^2)</math>.
 Then '''S’''' would have a posterior variance:
-<div style="text-align: center;"> '''<math>\phi_1 = (30^{-2} + 4^{-2})^{-1} = 16 = 4^2</math>''' </div>
+: <math>\phi_1 = (30^{-2} + 4^{-2})^{-1} = 16 = 4^2</math>.
 '''S’''' would have a posterior mean of:
-<div style="text-align: center;"> '''<math>\mu_1 = 16(200/30^2 + 220/4^2) = 224</math>'''</div>
+:<math>\mu_1 = 16(200/30^2 + 220/4^2) = 224</math>.
 Therefore, the distribution of '''S’''' would be:
-<div style="text-align: center;"> '''<math>\mu</math> | x ~ N<math>(224,4^2)</math>''' </div>
+:<math>\mu | x \sim N(224,4^2)</math>. Meaning 224 +/- 4 mph.
 This calculation has been carried out assuming that the prior information we have is rather vague, and therefore the posterior is almost entirely determined by the data.
 The situation is summarized as follows:
+==See also==
-          '''Prior Distribution     Likelihood from Data     Posterior Distribution'''
+* [[EBook#Chapter_III:_Probability |Probability Chapter]]
-     '''S     N<math>(185 , 10^2)</math>                                        N<math>(218 , 4^2)</math>'''
+==References==
-                                    '''N<math>(220 , 4^2)</math>'''
+<hr>
-    ''' S’    N<math>(200 , 30^2)</math>				            N<math>(224 , 4^2)</math>'''
+{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Bayesian_Normal}}