AP Statistics Curriculum 2007 GLM Regress

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==[[AP_Statistics_Curriculum_2007 | General Advance-Placement (AP) Statistics Curriculum]] - Regression ==
==[[AP_Statistics_Curriculum_2007 | General Advance-Placement (AP) Statistics Curriculum]] - Regression ==
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=== Linear Modeling - Regression ===
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As we discussed in the [[AP_Statistics_Curriculum_2007_GLM_Corr |Correlation section]], many applications involve the analysis of relationships between two, or more, variables involved in the process of interest. Suppose we have bivariate data (''X'' and ''Y'') of a process and we are interested on determining the linear relation between X and Y (e.g., determining a straight line that best fits the pairs of data (''X,Y'')). A linear relationship between ''X'' and ''Y'' will give us the power to make predictions - i.e., given a value of ''X'' predict a corresponding ''Y'' response. Note that in this design, data consists of paired observations (''X,Y'') - for example, the [[SOCR_Data_Dinov_021708_Earthquakes | Longitude and Latitude of the SOCR Eathquake dataset]].
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Example on how to attach images to Wiki documents in included below (this needs to be replaced by an appropriate figure for this section)!
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<center>[[Image:AP_Statistics_Curriculum_2007_IntroVar_Dinov_061407_Fig1.png|500px]]</center>
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===Approach===
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===Lines in 2D===
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Models & strategies for solving the problem, data understanding & inference.  
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There are 3 types of lines in 2D planes - Vertical Lines, Horizontal Lines and Oblique Lines. In general, the mathematical representation of lines in 2D is given by equations like <math>aX + bY=c</math>, most frequently expressed as <math>Y=aX + b</math>, provides the line is not vertical.  
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* TBD
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Recall that there is a one-to-one correspondence between any line in 2D and (linear) equations of the form
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: If the line is '''vertical''' (<math>X_1 =X_2</math>): <math>X=X_1</math>;
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: If the line is '''horizontal''' (<math>Y_1 =Y_2</math>): <math>Y=Y_1</math>;
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: Otherwise ('''oblique''' line): <math>{Y-Y_1 \over Y_2-Y_1}= {X-X_1 \over X_2-X_1}</math>, (for <math>X_1\not=X_2</math> and <math>Y_1\not=Y_2</math>)
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where <math>(X_1,Y_1)</math> and <math>(X_2, Y_2)</math> are two points on the line of interest (2-distinct points in 2D determine a unique line).
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===Model Validation===
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* Try drawing the following lines manually and [http://www.pserc.cornell.edu/pserc/java/graph/examples/parse1d.html using this applet]:
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Checking/affirming underlying assumptions.  
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: Y=2X+1
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: Y=-3X-5
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=== Linear Modeling - Regression ===
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There are two contexts for regression:
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* Y is an observed variable and X is specified by the researcher - e.g., Y is hair growth after X months, for individuals at certain dose levels of hair growth cream.
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* TBD
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* X and Y are both observed variables - e.g., [[SOCR_Data_Dinov_020108_HeightsWeights | Height (Y) and weight (X)]] for 20 randomly selected individuals from the population.
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===Computational Resources: Internet-based SOCR Tools===
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Suppose we have ''n'' pairs ''(X,Y)'', {<math>X_1, X_2, X_3, \cdots, X_n</math>} and {<math>Y_1, Y_2, Y_3, \cdots, Y_n</math>}, of observations of the same process. If a [[SOCR_EduMaterials_Activities_ScatterChart |scatterplot]] of the data suggests a general linear trend, it would be reasonable to fit a line to the data. The main question is how to determine the best line?
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* TBD
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===Examples===
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====[[AP_Statistics_Curriculum_2007_GLM_Corr#Airfare_Example |Airfare Example]]====
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Computer simulations and real observed data.  
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We can see from the [[SOCR_EduMaterials_Activities_ScatterChart |scatterplot]] that greater distance is associated with higher airfare. In other words airports that tend to be further from Baltimore tend to be more expensive airfare. To decide on the best fitting line, we use the '''least-squares method''' to fit the least squares (regression) line.
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* TBD
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<center>[[Image:SOCR_EBook_Dinov_GLM_Regr_021708_Fig1.jpg|500px]]</center>
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===Hands-on activities===
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Step-by-step practice problems.  
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* TBD
 
<hr>
<hr>
===References===
===References===
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* TBD
 
<hr>
<hr>

Revision as of 02:37, 18 February 2008

Contents

General Advance-Placement (AP) Statistics Curriculum - Regression

As we discussed in the Correlation section, many applications involve the analysis of relationships between two, or more, variables involved in the process of interest. Suppose we have bivariate data (X and Y) of a process and we are interested on determining the linear relation between X and Y (e.g., determining a straight line that best fits the pairs of data (X,Y)). A linear relationship between X and Y will give us the power to make predictions - i.e., given a value of X predict a corresponding Y response. Note that in this design, data consists of paired observations (X,Y) - for example, the Longitude and Latitude of the SOCR Eathquake dataset.

Lines in 2D

There are 3 types of lines in 2D planes - Vertical Lines, Horizontal Lines and Oblique Lines. In general, the mathematical representation of lines in 2D is given by equations like aX + bY = c, most frequently expressed as Y = aX + b, provides the line is not vertical.

Recall that there is a one-to-one correspondence between any line in 2D and (linear) equations of the form

If the line is vertical (X1 = X2): X = X1;
If the line is horizontal (Y1 = Y2): Y = Y1;
Otherwise (oblique line): {Y-Y_1 \over Y_2-Y_1}= {X-X_1 \over X_2-X_1}, (for X_1\not=X_2 and Y_1\not=Y_2)

where (X1,Y1) and (X2,Y2) are two points on the line of interest (2-distinct points in 2D determine a unique line).

Y=2X+1
Y=-3X-5

Linear Modeling - Regression

There are two contexts for regression:

  • Y is an observed variable and X is specified by the researcher - e.g., Y is hair growth after X months, for individuals at certain dose levels of hair growth cream.

Suppose we have n pairs (X,Y), {X_1, X_2, X_3, \cdots, X_n} and {Y_1, Y_2, Y_3, \cdots, Y_n}, of observations of the same process. If a scatterplot of the data suggests a general linear trend, it would be reasonable to fit a line to the data. The main question is how to determine the best line?

Airfare Example

We can see from the scatterplot that greater distance is associated with higher airfare. In other words airports that tend to be further from Baltimore tend to be more expensive airfare. To decide on the best fitting line, we use the least-squares method to fit the least squares (regression) line.



References




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