AP Statistics Curriculum 2007 GLM Regress - Revision history

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

2020-03-03T17:48:33Z

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

IvoDinov at 19:25, 13 August 2012

2012-08-13T19:25:20Z

Jenny at 20:52, 28 June 2010

2010-06-28T20:52:57Z

Jenny at 17:32, 28 June 2010

2010-06-28T17:32:37Z

IvoDinov: /* Earthquake Example */ typo

2009-12-02T05:14:39Z

‎Earthquake Example: typo

IvoDinov: added a link to the Problems set

2009-10-26T05:21:11Z

added a link to the Problems set

IvoDinov: /* Regression Coefficients Inference */ typo

2008-12-10T23:24:26Z

‎Regression Coefficients Inference: typo

IvoDinov: /* Regression Coefficients Inference */ added the hypothesis test for the slope coefficient

2008-12-10T23:23:45Z

‎Regression Coefficients Inference: added the hypothesis test for the slope coefficient

IvoDinov: /* Regression Coefficients Inference */

2008-12-03T21:03:57Z

‎Regression Coefficients Inference

IvoDinov: /* Hands-on Example */

2008-12-03T20:59:26Z

‎Hands-on Example

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	* SOCR Home page: http://www.socr.ucla.edu		* SOCR Home page: http://www.socr.ucla.edu

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← Older revision		Revision as of 19:25, 13 August 2012
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	===[[EBook_Problems_GLM_Regress\|Problems]]===		===[[EBook_Problems_GLM_Regress\|Problems]]===
		+
		+	===See also===
		+	* [[SOCR_EduMaterials_AnalysisActivities_SLR\| SOCR Linear Regression Activity]]

	<hr>		<hr>

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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.
-* 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.
+* 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.
 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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 ====Estimating the Best Linear Fit====
-The parameters of the linear regression line, <math>Y = a + bX</math>, can be estimated using [http://en.wikipedia.org/wiki/Ordinary_Least_Squares  Least Squares]. The least squares technique finds the line that minimizes the sum of the squares of the regression '''residuals''', <math>\hat{\varepsilon_i}=\hat{y}_{i}-y_i</math>, <math> \sum_{i=1}^N {\hat{\varepsilon_i}^2} = \sum_{i=1}^N (\hat{y}_{i}-y_i)^2 </math>, where <math>y_i</math> and <math>\hat{y}_{i}=a+bx_i</math> are the observed and the predicted values of ''Y'' for <math>x_i</math>, respectfully.
+The parameters of the linear regression line, <math>Y = a + bX</math>, can be estimated using [http://en.wikipedia.org/wiki/Ordinary_Least_Squares  Least Squares]. The least squares technique finds the line that minimizes the sum of the squares of the regression '''residuals''', <math>\hat{\varepsilon_i}=\hat{y}_{i}-y_i</math>, <math> \sum_{i=1}^N {\hat{\varepsilon_i}^2} = \sum_{i=1}^N (\hat{y}_{i}-y_i)^2 </math>, where <math>y_i</math> and <math>\hat{y}_{i}=a+bx_i</math> are the observed and the predicted values of ''Y'' for <math>x_i</math>.
 The minimization problem can be solved using calculus, by finding the first order partial derivatives and setting them equal to zero. The solution gives the slope and y-intercept of the regressions line:

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 ==[[AP_Statistics_Curriculum_2007 | General Advance-Placement (AP) Statistics Curriculum]] - Regression ==
-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 Earthquake dataset]].
+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 in 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 Earthquake dataset]].
 ===Lines in 2D===
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 ====[[AP_Statistics_Curriculum_2007_GLM_Corr#Airfare_Example |Airfare Example]]====
-We can see from the [[SOCR_EduMaterials_Activities_ScatterChart |scatterplot]] that [[AP_Statistics_Curriculum_2007_GLM_Corr#Airfare_Example | 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.
+We can see from the [[SOCR_EduMaterials_Activities_ScatterChart |scatterplot]] that [[AP_Statistics_Curriculum_2007_GLM_Corr#Airfare_Example | greater distance is associated with higher airfare]]. In other words airports that tend to be further from Baltimore tend to have more expensive airfare. To decide on the best fitting line, we use the '''least-squares method''' to fit the least squares (regression) line.
 <center>[[Image:SOCR_EBook_Dinov_GLM_Regr_021708_Fig1.jpg|500px]]</center>

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 ===Earthquake Example===
-Use the [[SOCR_Data_Dinov_021708_Earthquakes | SOCR Earthquake Dataset]] to determine the best-leaner fit between the [http://nationalatlas.gov/articles/mapping/a_latlong.html Longitude] and the [http://nationalatlas.gov/articles/mapping/a_latlong.html Latitude] of the California Earthquakes since 1900. What is the interpretation of this regression line ([http://en.wikipedia.org/wiki/San_Andreas_Fault San Andreas fault]?). You can see the [http://socr.ucla.edu/docs/resources/SOCR_Data/SOCR_Earthquake5Data_GoogleMap.html SOCR Geomap of these Earthquakes]. The image below shows how to use the [[SOCR_EduMaterials_AnalysisActivities_SLR |Simple Linear regression]] in [http://socr.ucla.edu/htmls/SOCR_Analyses.html SOCR Analyses] to calculate the regression line:
+Use the [[SOCR_Data_Dinov_021708_Earthquakes | SOCR Earthquake Dataset]] to determine the best-linear fit between the [http://nationalatlas.gov/articles/mapping/a_latlong.html Longitude] and the [http://nationalatlas.gov/articles/mapping/a_latlong.html Latitude] of the California Earthquakes since 1900. What is the interpretation of this regression line ([http://en.wikipedia.org/wiki/San_Andreas_Fault San Andreas fault]?). You can see the [http://socr.ucla.edu/docs/resources/SOCR_Data/SOCR_Earthquake5Data_GoogleMap.html SOCR Geomap of these Earthquakes]. The image below shows how to use the [[SOCR_EduMaterials_AnalysisActivities_SLR |Simple Linear regression]] in [http://socr.ucla.edu/htmls/SOCR_Analyses.html SOCR Analyses] to calculate the regression line:
 : <math>Latitude = -40.149 -0.6457\times Longitude</math>
 <center>[[Image:SOCR_EBook_Dinov_GLM_Regr_021708_Fig2.jpg|500px]]</center>
 ===Hands-on Example===

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 <hr>
-===References===
+===[[EBook_Problems_GLM_Regress|Problems]]===
 <hr>

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 * The corresponding test for the regression slope coefficient ''b'' is analogously computed (<math>H_o: b=b_o</math> vs. <math>H_a: b \not= b_0</math>, frequently <math>b_0 = 0</math>)
-::<math> t_o = {\hat{b} - b_o \ over s_ \hat{b} } \sim T_{df=N-2}. </math>
+::<math> t_o = {\hat{b} - b_o \over s_ \hat{b} } \sim T_{df=N-2}. </math>
 ===Hands-on Example===

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 If the error terms are Normally distributed, the estimate of the slope coefficient has a normal distribution with mean equal to '''b''' and ''standard error'' given by:
-: <math> SE(\hat{b}) = s_ \hat{b} = \sqrt { {1\over (N-2)} \frac {\sum_{i=1}^N \hat{\varepsilon_i}^2} {\sum_{i=1}^N (x_i - \bar{x})^2} }</math>.
+:: <math> SE(\hat{b}) = s_ \hat{b} = \sqrt { {1\over (N-2)} \frac {\sum_{i=1}^N \hat{\varepsilon_i}^2} {\sum_{i=1}^N (x_i - \bar{x})^2} }</math>.
-A confidence interval for ''b'' can be created using a [[AP_Statistics_Curriculum_2007_StudentsT | T-distribution with N-2 degrees of freedom]]:
+* A confidence interval for ''b'' can be created using a [[AP_Statistics_Curriculum_2007_StudentsT | T-distribution with N-2 degrees of freedom]]:
-:<math> [ \hat{b} - s_ \hat{b} t_{(\alpha/2, N-2)},\hat{b} + s_ \hat{b} t_{(\alpha/2, N-2)}] </math>
+::<math> [ \hat{b} - s_ \hat{b} t_{(\alpha/2, N-2)},\hat{b} + s_ \hat{b} t_{(\alpha/2, N-2)}] </math>
 ===Hands-on Example===

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 | 4 || 2 || 2 || 1.5 || 4 || 2.25 || 3
 |-
-| <math>\bar{x}</math>  || <math>\bar{y}</math> ||  ||  ||  ||  ||   ||
+| <math>\bar{x}</math>  || <math>\bar{y}</math> ||  ||  ||  ||  ||
 |}
 </center>