http://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_GLM_Predict&feed=atom&action=historyAP Statistics Curriculum 2007 GLM Predict - Revision history2024-03-28T14:58:42ZRevision history for this page on the wikiMediaWiki 1.15.1http://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_GLM_Predict&diff=10405&oldid=prevJenny at 20:54, 28 June 20102010-06-28T20:54:21Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>====Confidence Interval Estimating of the Slope and Intercept of Linear Model====</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>====Confidence Interval Estimating of the Slope and Intercept of Linear Model====</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>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><del class="diffchange diffchange-inline">, respectfully</del>.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>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>.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>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:</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>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:</div></td></tr>
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</table>Jennyhttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_GLM_Predict&diff=10303&oldid=prevJenny at 17:33, 28 June 20102010-06-28T17:33:59Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>=== Inference on Linear Models ===</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>=== Inference on Linear Models ===</div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>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. <del class="diffchange diffchange-inline">in </del>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?</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>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. <ins class="diffchange diffchange-inline">In </ins>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?</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>====[[AP_Statistics_Curriculum_2007_GLM_Corr#Airfare_Example |Airfare Example]]====</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>====[[AP_Statistics_Curriculum_2007_GLM_Corr#Airfare_Example |Airfare Example]]====</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>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 <del class="diffchange diffchange-inline">be </del>more expensive airfare. To decide on the best fitting line, we use the '''least-squares method''' to fit the least squares (regression) line.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>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 <ins class="diffchange diffchange-inline">have </ins>more expensive airfare. To decide on the best fitting line, we use the '''least-squares method''' to fit the least squares (regression) line.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><center>[[Image:SOCR_EBook_Dinov_GLM_Regr_021708_Fig1.jpg|500px]]</center></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><center>[[Image:SOCR_EBook_Dinov_GLM_Regr_021708_Fig1.jpg|500px]]</center></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>: <math> \hat{a} = \bar{y} - \hat{b} \bar{x} </math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>: <math> \hat{a} = \bar{y} - \hat{b} \bar{x} </math></div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>If the error terms are Normally distributed, the estimate of the slope coefficient has a normal distribution with mean <del class="diffchange diffchange-inline">equal </del>to '''b''' and '''standard error''' given by:</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>If the error terms are Normally distributed, the estimate of the slope coefficient has a normal distribution with mean <ins class="diffchange diffchange-inline">equals </ins>to '''b''' and '''standard error''' given by:</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>: <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>.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>: <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>.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math> [ \hat{b} - s_ \hat{b} t_{(\alpha/2, N-2)},\hat{b} + s_ \hat{b} t_{(\alpha/2, N-2)}] </math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>:<math> [ \hat{b} - s_ \hat{b} t_{(\alpha/2, N-2)},\hat{b} + s_ \hat{b} t_{(\alpha/2, N-2)}] </math></div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>In other words, if there is <del class="diffchange diffchange-inline">a </del>1 mile increase in distance the airfare will go up by between $0.054 and $0.180.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>In other words, if there is <ins class="diffchange diffchange-inline">an </ins>1 mile increase in distance the airfare will go up by between $0.054 and $0.180.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* '''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</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* '''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</div></td></tr>
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</table>Jennyhttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_GLM_Predict&diff=9524&oldid=prevIvoDinov: added a link to the Problems set2009-10-26T05:21:50Z<p>added a link to the Problems set</p>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>===<del class="diffchange diffchange-inline">References</del>===</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>===<ins class="diffchange diffchange-inline">[[EBook_Problems_GLM_Predict|Problems]]</ins>===</div></td></tr>
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</table>IvoDinovhttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_GLM_Predict&diff=9386&oldid=prevIvoDinov: /* Confidence Interval Estimating of the Slope and Intercept of Linear Model */2009-10-01T01:35:51Z<p><span class="autocomment">Confidence Interval Estimating of the Slope and Intercept of Linear Model</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* '''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</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* '''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</div></td></tr>
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<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>: Hypotheses: <math>H_o: b = 0</math> vs. <math>H_1: <del class="diffchange diffchange-inline">b1 </del>\not= 0</math> (or <math>H_1: b > 0</math> or <math>H_1: b < 0</math>).</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>: Hypotheses: <math>H_o: b = 0</math> vs. <math>H_1: <ins class="diffchange diffchange-inline">b </ins>\not= 0</math> (or <math>H_1: b > 0</math> or <math>H_1: b < 0</math>).</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>: 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]].</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>: 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]].</div></td></tr>
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</table>IvoDinovhttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_GLM_Predict&diff=6642&oldid=prevPriscillaChui: /* Confidence Interval Estimating of the Slope and Intercept of Linear Model */2008-02-19T21:02:28Z<p><span class="autocomment">Confidence Interval Estimating of the Slope and Intercept of Linear Model</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>In other words, if there is a 1 mile increase in distance the airfare will go up by between $0.054 and $0.180.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>In other words, if there is a 1 mile increase in distance the airfare will go up by between $0.054 and $0.180.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>* '''Significance testing''': If X is not useful for predicting Y <del class="diffchange diffchange-inline">this is like saying </del>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</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>* '''Significance testing''': If X is not useful for predicting Y<ins class="diffchange diffchange-inline">, then </ins>the true slope is zero. In a hypothesis test <ins class="diffchange diffchange-inline">,</ins>our status quo null hypothesis would be that there is no relationship between X and Y</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>: 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>).</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>: 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>).</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>: 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]].</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>: 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]].</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>====Example====</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>====Example====</div></td></tr>
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</table>PriscillaChuihttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_GLM_Predict&diff=6641&oldid=prevPriscillaChui: /* Airfare Example */2008-02-19T21:01:21Z<p><span class="autocomment"><a href="/socr/index.php/AP_Statistics_Curriculum_2007_GLM_Corr#Airfare_Example" title="AP Statistics Curriculum 2007 GLM Corr">Airfare Example</a></span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>====[[AP_Statistics_Curriculum_2007_GLM_Corr#Airfare_Example |Airfare Example]]====</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>====[[AP_Statistics_Curriculum_2007_GLM_Corr#Airfare_Example |Airfare Example]]====</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>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.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>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<ins class="diffchange diffchange-inline">, </ins>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.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><center>[[Image:SOCR_EBook_Dinov_GLM_Regr_021708_Fig1.jpg|500px]]</center></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><center>[[Image:SOCR_EBook_Dinov_GLM_Regr_021708_Fig1.jpg|500px]]</center></div></td></tr>
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</table>PriscillaChuihttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_GLM_Predict&diff=6640&oldid=prevPriscillaChui: /* Inference on Linear Models */2008-02-19T21:00:57Z<p><span class="autocomment">Inference on Linear Models</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>=== Inference on Linear Models ===</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>=== Inference on Linear Models ===</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>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 <del class="diffchange diffchange-inline">setion </del>we discussed how to fit a line to the data]]. The main question is how to determine the best line?</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>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 <ins class="diffchange diffchange-inline">section, </ins>we discussed how to fit a line to the data]]. The main question is how to determine the best line?</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>====[[AP_Statistics_Curriculum_2007_GLM_Corr#Airfare_Example |Airfare Example]]====</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>====[[AP_Statistics_Curriculum_2007_GLM_Corr#Airfare_Example |Airfare Example]]====</div></td></tr>
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</table>PriscillaChuihttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_GLM_Predict&diff=6639&oldid=prevPriscillaChui: /* Inference on Linear Models */2008-02-19T21:00:29Z<p><span class="autocomment">Inference on Linear Models</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>=== Inference on Linear Models ===</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>=== Inference on Linear Models ===</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>Suppose we have again <del class="diffchange diffchange-inline">an </del>''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?</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>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 setion we discussed how to fit a line to the data]]. The main question is how to determine the best line?</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>====[[AP_Statistics_Curriculum_2007_GLM_Corr#Airfare_Example |Airfare Example]]====</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>====[[AP_Statistics_Curriculum_2007_GLM_Corr#Airfare_Example |Airfare Example]]====</div></td></tr>
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</table>PriscillaChuihttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_GLM_Predict&diff=6578&oldid=prevIvoDinov at 17:58, 18 February 20082008-02-18T17:58:02Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==[[AP_Statistics_Curriculum_2007 | General Advance-Placement (AP) Statistics Curriculum]] - Variation and Prediction Intervals ==</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>==[[AP_Statistics_Curriculum_2007 | General Advance-Placement (AP) Statistics Curriculum]] - Variation and Prediction Intervals ==</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>=== Linear <del class="diffchange diffchange-inline">Modeling - Variation and Prediction Intervals </del>===</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>=== <ins class="diffchange diffchange-inline">Inference on </ins>Linear <ins class="diffchange diffchange-inline">Models </ins>===</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">Example on how to attach images to Wiki documents in included below (this needs to be replaced by an appropriate figure for this section)!</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline"><center>[[Image:AP_Statistics_Curriculum_2007_IntroVar_Dinov_061407_Fig1.png|500px]]</center></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">===Approach===</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">Suppose we have again an ''n'' pairs ''(X</ins>,<ins class="diffchange diffchange-inline">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 </ins>data<ins class="diffchange diffchange-inline">]]</ins>. <ins class="diffchange diffchange-inline">The main question is how to determine the best line?</ins></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">Models & strategies for solving the problem</del>, data <del class="diffchange diffchange-inline">understanding & inference</del>. </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">* TBD</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">====[[AP_Statistics_Curriculum_2007_GLM_Corr#Airfare_Example |Airfare Example]]====</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">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.</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">===Model Validation===</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline"><center>[[Image:SOCR_EBook_Dinov_GLM_Regr_021708_Fig1</ins>.<ins class="diffchange diffchange-inline">jpg|500px]]</center></ins></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">Checking/affirming underlying assumptions</del>. </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">* TBD</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">====Confidence Interval Estimating of the Slope and Intercept of Linear Model====</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">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.</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">===Computational Resources: Internet</del>-<del class="diffchange diffchange-inline">based SOCR Tools===</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">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</ins>-<ins class="diffchange diffchange-inline">intercept of the regressions line:</ins></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">* TBD</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>===<del class="diffchange diffchange-inline">Examples</del>===</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">* Regression line Slope:</ins></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">Computer simulations and real observed data</del>. </div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">: <math> \hat{b} </ins>= <ins class="diffchange diffchange-inline">\frac {\sum_{i</ins>=<ins class="diffchange diffchange-inline">1}^{N} (x_{i} - \bar{x})(y_{i} - \bar{y}) } {\sum_{i</ins>=<ins class="diffchange diffchange-inline">1}^{N} (x_{i} - \bar{x}) ^2} </math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">: <math> \hat{b} </ins>= <ins class="diffchange diffchange-inline">\frac {\sum_{i</ins>=<ins class="diffchange diffchange-inline">1}^{N} {(x_{i}y_{i})} - N \bar{x} \bar{y}} {\sum_{i</ins>=<ins class="diffchange diffchange-inline">1}^{N} (x_{i})^2 - N \bar{x}^2} = \rho_{X,Y} \frac {s_y}{s_x} </math>, where [[AP_Statistics_Curriculum_2007_GLM_Corr |<math>\rho_{X,Y}</math> is the correlation coefficient]]</ins>.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>* <del class="diffchange diffchange-inline">TBD </del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>* <ins class="diffchange diffchange-inline">Y-intercept:</ins></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline"> </del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">: <math> \hat{a} </ins>= <ins class="diffchange diffchange-inline">\bar{y} </ins>- <ins class="diffchange diffchange-inline">\hat{b} \bar{x} </math></ins></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>=<del class="diffchange diffchange-inline">==Hands-on activities===</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del class="diffchange diffchange-inline">Step-by</del>-<del class="diffchange diffchange-inline">step practice problems. </del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>* <del class="diffchange diffchange-inline">TBD</del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">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:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">: <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>.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>* <ins class="diffchange diffchange-inline">A '''confidence interval''' for ''b'' can be created using a [[AP_Statistics_Curriculum_2007_StudentsT | T-distribution with N-2 degrees of freedom]]:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">:<math> [ \hat{b} - s_ \hat{b} t_{(\alpha/2, N-2)},\hat{b} + s_ \hat{b} t_{(\alpha/2, N-2)}] </math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">In other words, if there is a 1 mile increase in distance the airfare will go up by between $0.054 and $0.180.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">* '''Significance testing''': If X is not useful for predicting Y this is like saying 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</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">: 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>).</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">: 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]].</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">====Example====</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">For the [[AP_Statistics_Curriculum_2007_GLM_Corr#Airfare_Example | distance vs. airfare example]], we can compute the standard error of the slope coefficient (''b''), SE(b)</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">: <math>SE(b)={37.83 \over \sqrt{1786499}}=0.0283</math>.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">* Then a 95% confidence interval for '''b''' is given by:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">: CI(b): <math>b \pm t_{(\alpha/2, df=10)}SE(b)=0.11738 \pm 2.228\times 0.02832=[0.054 , 0.180].</math></ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">* Significance testing:</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">: <math>t_o={b-0\over SE(b)}={0.11738-0 \over 0.02832}=4.145</math> and <math>p-value =0.002</math>.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">===Earthquake Example===</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">Use the [[SOCR_Data_Dinov_021708_Earthquakes | SOCR Earthquake Dataset]] to formulate and test a research hypothesis about the slope of 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. 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 and make inference on the slope.</ins></div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div> </div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline"><center>[[Image:SOCR_EBook_Dinov_GLM_Regr_021708_Fig2.jpg|500px]]</center></ins></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===References===</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===References===</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div><del style="color: red; font-weight: bold; text-decoration: none;">* TBD</del></div></td><td colspan="2"> </td></tr>
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</table>IvoDinovhttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_GLM_Predict&diff=4135&oldid=prevIvoDinov at 19:15, 14 June 20072007-06-14T19:15:00Z<p></p>
<p><b>New page</b></p><div>==[[AP_Statistics_Curriculum_2007 | General Advance-Placement (AP) Statistics Curriculum]] - Variation and Prediction Intervals ==<br />
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=== Linear Modeling - Variation and Prediction Intervals ===<br />
Example on how to attach images to Wiki documents in included below (this needs to be replaced by an appropriate figure for this section)!<br />
<center>[[Image:AP_Statistics_Curriculum_2007_IntroVar_Dinov_061407_Fig1.png|500px]]</center><br />
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===Approach===<br />
Models & strategies for solving the problem, data understanding & inference. <br />
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* TBD<br />
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===Model Validation===<br />
Checking/affirming underlying assumptions. <br />
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* TBD<br />
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===Computational Resources: Internet-based SOCR Tools===<br />
* TBD<br />
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===Examples===<br />
Computer simulations and real observed data. <br />
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* TBD <br />
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===Hands-on activities===<br />
Step-by-step practice problems. <br />
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* TBD<br />
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===References===<br />
* TBD<br />
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* SOCR Home page: http://www.socr.ucla.edu<br />
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{{translate|pageName=http://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_GML_Predict}}</div>IvoDinov