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Unformatted text preview: Œ##############z###,4# þ###Œ4#########$Ý4#Œ¸¦Œ= #Œ###Example 3 The following is a scatterplot of the final exam score versus midterm score for 11 sections of an introductory statistics class:#### ############Œ######.#####z###y## x## ###### ###$P¤z=P¤z=#########Œ##########z###,4# þ###Œ4#####$Ý4#Œ¸¦Œ= ####MORE EXAMPLES##############Œ######,#####z###y## x## ###### ###$P¤z=P¤z=#########Œ##########z###,4# þ###Œ4#####$Ý4#Œ¸¦Œ= ##### ##########Œ### ###*#####z###y## x## ###### ###$P¤z=P¤z=#########Œ##########z###,4# þ###Œ4#####$Ý4#Œ¸¦Œ= ##### ##########Œ### ###(#####z###y## x## ###### ###$P¤z=P¤z=#########Œ####@#####z###,4# þ###Œ4#####$Ý4#Œ¸¦Œ= ##### ##########Œ### ###&#####z###y## x## ###### ###$P¤z=P¤z=#########Œ####@###$###@#####z###,4# þ###Œ4#########$Ý4#Œ¸¦Œ= #ú###The fifty-first points would not be as influential, since the pattern is more established by 50 points than it was by 4. In order to be as influential as (10, 0) an additional point would have to be much further out along the x-axis, maybe (100, 0).####ã###ä############Œ############z###y## x## ###### ###$P¤z=P¤z=#########Œ##########z###,4# þ###Œ4#####$Ý4#Œ¸¦Œ= ##### ##########Œ### ###!#####z###y## x## ###### ###$P¤z=P¤z=#########Œ##########z###,4# þ###Œ4#####$Ý4#Œ¸¦Œ= #Œ###E) Originally there were only four points here. Suppose instead that we had started with 50 points clustered in essentially the same region and displaying an association of roughly the same strength and direction. Would our fifth points sill be as influential? Where would you locate one additional point so influential that it changed the line as dramatically as (10, 0) did above?#########Œ### #########z###y## x## ###### ###$P¤z=P¤z=#########Œ##########z###,4# þ###Œ4#####$Ý4#Œ¸¦Œ= ##### ##########Œ### #########z###y## x## ###### ###$P¤z=P¤z=#########Œ########V#########z###,4# þ###Œ4#########$Ý4#Œ¸¦Œ= #b###If removing the point gives a very different model, we say that the point is an influential point.####P###a############Œ############z###y## x## ###### ###$P¤z=P¤z=#########Œ##########z###,4# þ###Œ4#####$Ý4#Œ¸¦Œ= ##### ##########Œ### #########z###y## x## ###### ###$P¤z=P¤z=#########Œ##########z###,4# þ###Œ4#####$Ý4#Œ¸¦Œ= ##### ##########Œ##z#######R###Ú#######À## #### ###P¤z=######" ####C#a#l#i#b#r#i#########*#########ÿ####Œ##########z###,4# þ###Œ4#####$Ý4#Œ¸¦Œ= #Œ###Influential points are more easily seen in scatterplots of the original data or by finding the regression model with and without the points.#######Œ################z###y## x## ###### ###$&##$P¤z=P¤z=#################Œ################z###y## x## ###### ##$###$P¤z=P¤z=#################Œ####@#####z###,4# þ###Œ4#####$Ý4#Œ¸¦Œ= ##### ##########Œ##z#######R###Ú#######À## #### ###P¤z=######" ####C#a#l#i#b#r#i#########*#########ÿ####Œ##########z###,4# þ###Œ4#####$Ý4#Œ¸¦Œ= #[###Points with high leverage pull the line close to them, so they often have small residuals. ########Œ################z###y## x## ###### ###$&##$P¤z=P¤z=#################Œ################z###y## x## ###### ##$###$P¤z=P¤z=#################Œ######## #######@#####z###,4# þ###Œ4#########$Ý4#Œ¸¦Œ= #?### is large, or data points where the x-value is far from the mean. A way to visualize this is to look at the linear model as a lever with the fulcrum at the mean point,. The farther a point is from the fulcrum, the more leverage it has in the model. These points have the potential to make a big impact on the model. ###$###%###}###############Œ############z###y## x## ###### ###$P¤z=P¤z=#########Œ####V#######û#####z###,4# þ###"##########$Ý4#Œ¸¦Œ= #\###H#i#g#h# #L#e#v#e#r#a#g#e# #P#o#i#n#t#s# ## #a#r#e# #d#a#t#a# #p#o#i#n#t#s# #w#h#e#r#e# ######################Œ######ÿ#####z###y## x## ###### ###$P¤z=ŒÎv=#########Œ##########z###,4# þ###Œ4#####$Ý4#Œ¸¦Œ= ##### ##########Œ### ###ý#####z###y## x## ###### ###$P¤z=P¤z=#########Œ####V###û#######@#######@#####z###,4# þ###"##########$Ý4#Œ¸¦Œ= #l###O#u#t#l#i#e#r#s# ## #a#r#e# #d#a#t#a# #p#o#i#n#t#s# #w#h#i#c#h# #a#r#e# #n#o#t#i#c#e#a#b#l#y# #d#i#f#f#e#r#e#n#t# #f#r#o#m# #t#h#e# #m#a#j#o#r#i#t#y# #o#f# #t#h#e# #s#a#m#p#l#e#.# # #E#x#a#m#p#l#e#s#:# #c#o#u#l#d# #b#e# #w#h#i#l#e# #l#o#o#k#i#n#g# #a#t# #h#e#i#g#h#t# #v#s#.# #w#e#i#g#h#t#,# #a#n# #8#-#f#o#o#t#-#t#a#l#l# #p#e#r#s#o#n# #w#h#o# #w#e#i#g#h#s# #1#2#0# #p#o#u#n#d#s#;# #o#r# #w#h#i#l#e# #l#o#o#k#i#n#g# #a#t# #d#a#t#a# #o#f# #t#e#m#p#e#r#a#t#u#r#e#s# #i#n# #d#i#f#f#e#r#e#n#t# #m#o#n#t#h#s#,# #a# #d#a#y# #i#n# #D#e#c#e#m#b#e#r# #w#h#e#r#e# #t#h#e# #t#e#m#p#e#r#a#t#u#r#e# #i#s# #1#0#0# #d#e#g#r#e#e#s#.####### ###Z###f###n###o##########Œ######ú#####z###y## x## ###### ###$P¤z=ŒÎv=#########Œ##########z###,4# þ###Œ4#####$Ý4#Œ¸¦Œ= #####WANDERING POINTS:##########Œ######ø#####z###y## x## ###### ###$P¤z=P¤z=#########Œ####¶###ö###¶#####z###,4# þ###Œ4#########$Ý4#Œ¸¦Œ= #C###To spot influential points in a scatterplot, look for points that depart from the overall pattern, especially those points that are outliers in the explanatory, or x, direction. DO NOT look for points with large residuals. Influential points change the slope of the regression lines, so they often have small residuals. ####¤###¥###########Œ######õ#####z###y## x## ###### ###$P¤z=P¤z=##...
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This note was uploaded on 12/12/2012 for the course MATH 1681 taught by Professor Staff during the Fall '11 term at North Texas.

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