AP Statistics Curriculum 2007 GLM Predict - Revision history

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

2020-03-03T20:34:52Z

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

Jenny at 20:54, 28 June 2010

2010-06-28T20:54:21Z

Jenny at 17:33, 28 June 2010

2010-06-28T17:33:59Z

IvoDinov: added a link to the Problems set

2009-10-26T05:21:50Z

added a link to the Problems set

IvoDinov: /* Confidence Interval Estimating of the Slope and Intercept of Linear Model */

2009-10-01T01:35:51Z

‎Confidence Interval Estimating of the Slope and Intercept of Linear Model

PriscillaChui: /* Confidence Interval Estimating of the Slope and Intercept of Linear Model */

2008-02-19T21:02:28Z

‎Confidence Interval Estimating of the Slope and Intercept of Linear Model

PriscillaChui: /* Airfare Example */

2008-02-19T21:01:21Z

‎Airfare Example

PriscillaChui: /* Inference on Linear Models */

2008-02-19T21:00:57Z

‎Inference on Linear Models

PriscillaChui: /* Inference on Linear Models */

2008-02-19T21:00:29Z

‎Inference on Linear Models

IvoDinov at 17:58, 18 February 2008

2008-02-18T17:58:02Z

← Older revision		Revision as of 20:34, 3 March 2020
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	* SOCR Home page: http://www.socr.ucla.edu		* SOCR Home page: http://www.socr.ucla.edu

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 ====Confidence Interval Estimating of the Slope and Intercept of Linear Model====
-The parameters (''a'' and ''b'') of the linear regression line, <math>Y = a + bX</math>, are 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 (''a'' and ''b'') of the linear regression line, <math>Y = a + bX</math>, are 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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 === Inference on Linear Models ===
-Suppose we have again ''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. in the [[AP_Statistics_Curriculum_2007_GLM_Regress |previous section, we discussed how to fit a line to the data]]. The main question is how to determine the best line?
+Suppose we have again ''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. In the [[AP_Statistics_Curriculum_2007_GLM_Regress |previous section, we discussed how to fit a line to the data]]. The main question is how to determine the best line?
 ====[[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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 : <math> \hat{a} = \bar{y} - \hat{b} \bar{x} </math>
-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:
+If the error terms are Normally distributed, the estimate of the slope coefficient has a normal distribution with mean equals to '''b''' and '''standard error''' given by:
 : <math> 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>.
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 :<math> [ \hat{b} - s_ \hat{b} t_{(\alpha/2, N-2)},\hat{b} + s_ \hat{b} t_{(\alpha/2, N-2)}] </math>
-In other words, if there is a 1 mile increase in distance the airfare will go up by between $0.054 and $0.180.
+In other words, if there is an 1 mile increase in distance the airfare will go up by between $0.054 and $0.180.
 * '''Significance testing''':  If X is not useful for predicting Y, then the true slope is zero. In a hypothesis test ,our status quo null hypothesis would be that there is no relationship between X and Y

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

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 * '''Significance testing''':  If X is not useful for predicting Y, then the true slope is zero. In a hypothesis test ,our status quo null hypothesis would be that there is no relationship between X and Y
-: Hypotheses: <math>H_o: b = 0</math> vs. <math>H_1: b1 \not= 0</math> 	(or <math>H_1: b > 0</math>  or <math>H_1: b < 0</math>).
+: Hypotheses: <math>H_o: b = 0</math> vs. <math>H_1: b \not= 0</math> 	(or <math>H_1: b > 0</math>  or <math>H_1: b < 0</math>).
 : Test-statistics: <math>t_o={b-0\over SE(b)}</math>, where <math>t_o \sim t_{(df=n-2)}</math> is the [[AP_Statistics_Curriculum_2007_StudentsT |T-Distribution]].

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 ==[[AP_Statistics_Curriculum_2007 | General Advance-Placement (AP) Statistics Curriculum]] - Variation and Prediction Intervals ==
-=== Linear Modeling - Variation and Prediction Intervals ===
+=== Inference on Linear Models ===
-===Approach===
+Suppose we have again an ''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. in the [[AP_Statistics_Curriculum_2007_GLM_Regress |previous setion we discussed how to fit a line to the data]]. The main question is how to determine the best line?
-* TBD
+====[[AP_Statistics_Curriculum_2007_GLM_Corr#Airfare_Example |Airfare Example]]====
-===Model Validation===
+<center>[[Image:SOCR_EBook_Dinov_GLM_Regr_021708_Fig1.jpg|500px]]</center>
-* TBD
+====Confidence Interval Estimating of the Slope and Intercept of Linear Model====
-===Computational Resources: Internet-based SOCR Tools===
+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:
-===Examples===
+* Regression line Slope:
-Computer simulations and real observed data.
+: <math> \hat{b} = \frac {\sum_{i=1}^{N}  (x_{i} - \bar{x})(y_{i} - \bar{y}) }  {\sum_{i=1}^{N} (x_{i} - \bar{x}) ^2} </math>
-* TBD
+* Y-intercept:
+: <math> \hat{a} = \bar{y} - \hat{b} \bar{x} </math>
-* TBD
+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:
 <hr>
 ===References===
 <hr>