http://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Infer_2Means_Dep&feed=atom&action=historyAP Statistics Curriculum 2007 Infer 2Means Dep - Revision history2024-03-28T21:35:54ZRevision history for this page on the wikiMediaWiki 1.15.1http://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Infer_2Means_Dep&diff=12290&oldid=prevIvoDinov: /* Paired vs. Independent Testing */2013-06-17T20:05:11Z<p><span class="autocomment">Paired vs. Independent Testing</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>: \(T_o = {\overline{x}-\overline{y} - \mu_o \over SE(\overline{x}+\overline{y})} \sim T(df=17)\)</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>: \(T_o = {\overline{x}-\overline{y} - \mu_o \over SE(\overline{x}+\overline{y})} \sim T(df=17)\)</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>: \(T_o = {\overline{x}-\overline{y} - \mu_o \over SE(\overline{x}+\overline{y})} = {0.682 -0.637- 0 \over \sqrt{SE^2(\overline{x})+SE^2(\overline{y})}}= \) \({0.682 -0.637\over \sqrt{{0.0742^2\over 10}+ {0.0709^2\over 10}}}={0.682 -0.637\over 0.0325}=1.38\)</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>: \(T_o = {\overline{x}-\overline{y} - \mu_o \over SE(\overline{x}+\overline{y})} = {0.682 -0.637- 0 \over \sqrt{SE^2(\overline{x})+SE^2(\overline{y})}}= \) \({0.682 -0.637\over \sqrt{{0.0742^2\over 10}+ {0.0709^2\over 10}}}={0.682 -0.637\over 0.0325}=1.38\)</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>: \(p-value=P(T>1.38)= 0.100449\) and we would have failed to reject the null-hypothesis ('''incorrect!''')</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>: \(p-value=P(T>1.38)= 0.100449\) and we would have failed to reject the null-hypothesis ('''<ins class="diffchange diffchange-inline">[[AP_Statistics_Curriculum_2007_Infer_2Means_Dep#Inference|</ins>incorrect!<ins class="diffchange diffchange-inline">]]</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: #eee; color:black; font-size: smaller;"><div>Similarly, had we incorrectly used the [[AP_Statistics_Curriculum_2007_Infer_2Means_Indep |independent design]] and constructed a corresponding Confidence interval, we would obtain an incorrect inference:</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Similarly, had we incorrectly used the [[AP_Statistics_Curriculum_2007_Infer_2Means_Indep |independent design]] and constructed a corresponding Confidence interval, we would obtain an incorrect inference:</div></td></tr>
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</table>IvoDinovhttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Infer_2Means_Dep&diff=12289&oldid=prevIvoDinov: /* Paired vs. Independent Testing */2013-06-17T20:04:04Z<p><span class="autocomment">Paired vs. Independent Testing</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>====Paired vs. Independent Testing====</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>====Paired vs. Independent Testing====</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Suppose we accidentally analyzed the groups independently (using the [[AP_Statistics_Curriculum_2007_Infer_2Means_Indep |independent T-test]]) rather than using this paired test (this would be an incorrect way of analyzing this ''before-after'' data). How would this change our results and findings?</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Suppose we accidentally analyzed the groups independently (using the [[AP_Statistics_Curriculum_2007_Infer_2Means_Indep |independent T-test]]) rather than using this paired test (this would be an incorrect way of analyzing this ''before-after'' data). How would this change our results and findings?</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>: <del class="diffchange diffchange-inline"><math></del>T_o = {\overline{x}-\overline{y} - \mu_o \over SE(\overline{x}+\overline{y})} \sim T(df=17)<del class="diffchange diffchange-inline"></math></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>: <ins class="diffchange diffchange-inline">\(</ins>T_o = {\overline{x}-\overline{y} - \mu_o \over SE(\overline{x}+\overline{y})} \sim T(df=17)<ins class="diffchange diffchange-inline">\)</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"><math></del>T_o = {\overline{x}-\overline{y} - \mu_o \over SE(\overline{x}+\overline{y})} = {0.682 -0.637- 0 \over \sqrt{SE^2(\overline{x})+SE^2(\overline{y})}}= {0.682 -0.637\over \sqrt{{0.0742^2\over 10}+ {0.0709^2\over 10}}}={0.682 -0.637\over 0.0325}=1.38<del class="diffchange diffchange-inline"></math></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>: <ins class="diffchange diffchange-inline">\(</ins>T_o = {\overline{x}-\overline{y} - \mu_o \over SE(\overline{x}+\overline{y})} = {0.682 -0.637- 0 \over \sqrt{SE^2(\overline{x})+SE^2(\overline{y})}}= <ins class="diffchange diffchange-inline">\) \(</ins>{0.682 -0.637\over \sqrt{{0.0742^2\over 10}+ {0.0709^2\over 10}}}={0.682 -0.637\over 0.0325}=1.38<ins class="diffchange diffchange-inline">\)</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"><math></del>p-value=P(T>1.38)= 0.100449<del class="diffchange diffchange-inline"></math> </del>and we would have failed to reject the null-hypothesis ('''incorrect!''')</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>: <ins class="diffchange diffchange-inline">\(</ins>p-value=P(T>1.38)= 0.100449<ins class="diffchange diffchange-inline">\) </ins>and we would have failed to reject the null-hypothesis ('''incorrect!''')</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>Similarly, had we incorrectly used the [[AP_Statistics_Curriculum_2007_Infer_2Means_Indep |independent design]] and constructed a corresponding Confidence interval, we would obtain an incorrect inference:</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Similarly, had we incorrectly used the [[AP_Statistics_Curriculum_2007_Infer_2Means_Indep |independent design]] and constructed a corresponding Confidence interval, we would obtain an incorrect inference:</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>: <del class="diffchange diffchange-inline"><math></del>CI: {\overline{x}-\overline{y} - \mu_o \pm t_{(df=17, \alpha/2)} \times SE(\overline{x}+\overline{y})} = 0.045 \pm 1.740\times 0.0325 = [-0.0116 ; 0.1016]<del class="diffchange diffchange-inline"></math></del></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>: <ins class="diffchange diffchange-inline">\(</ins>CI: {\overline{x}-\overline{y} - \mu_o \pm t_{(df=17, \alpha/2)} \times SE(\overline{x}+\overline{y})} = <ins class="diffchange diffchange-inline">\) \(</ins>0.045 \pm 1.740\times 0.0325 = [-0.0116 ; 0.1016]<ins class="diffchange diffchange-inline">.\)</ins></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><hr></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><hr></div></td></tr>
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</table>IvoDinovhttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Infer_2Means_Dep&diff=12288&oldid=prevIvoDinov: /* Paired vs. Independent Testing */2013-06-17T20:02:29Z<p><span class="autocomment">Paired vs. Independent Testing</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Suppose we accidentally analyzed the groups independently (using the [[AP_Statistics_Curriculum_2007_Infer_2Means_Indep |independent T-test]]) rather than using this paired test (this would be an incorrect way of analyzing this ''before-after'' data). How would this change our results and findings?</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Suppose we accidentally analyzed the groups independently (using the [[AP_Statistics_Curriculum_2007_Infer_2Means_Indep |independent T-test]]) rather than using this paired test (this would be an incorrect way of analyzing this ''before-after'' data). How would this change our results and findings?</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>: <math>T_o = {\overline{x}-\overline{y} - \mu_o \over SE(\overline{x}+\overline{y})} \sim T(df=17)</math></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>: <math>T_o = {\overline{x}-\overline{y} - \mu_o \over SE(\overline{x}+\overline{y})} \sim T(df=17)</math></div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>: <math>T_o = {\overline{x}-\overline{y} - \mu_o \over SE(\overline{x}+\overline{y})} = {0.682 -0.<del class="diffchange diffchange-inline">687</del>- 0 \over \sqrt{SE^2(\overline{x})+SE^2(\overline{y})}}= {0.682 -0.<del class="diffchange diffchange-inline">687</del>\over \sqrt{{0.0742^2\over 10}+ {0.0709^2\over 10}}}={0.682 -0.<del class="diffchange diffchange-inline">687</del>\over 0.0325}=1.38</math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>: <math>T_o = {\overline{x}-\overline{y} - \mu_o \over SE(\overline{x}+\overline{y})} = {0.682 -0.<ins class="diffchange diffchange-inline">637</ins>- 0 \over \sqrt{SE^2(\overline{x})+SE^2(\overline{y})}}= {0.682 -0.<ins class="diffchange diffchange-inline">637</ins>\over \sqrt{{0.0742^2\over 10}+ {0.0709^2\over 10}}}={0.682 -0.<ins class="diffchange diffchange-inline">637</ins>\over 0.0325}=1.38</math></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>: <math>p-value=P(T>1.38)= 0.100449</math> and we would have failed to reject the null-hypothesis ('''incorrect!''')</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>: <math>p-value=P(T>1.38)= 0.100449</math> and we would have failed to reject the null-hypothesis ('''incorrect!''')</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>Similarly, had we incorrectly used the [[AP_Statistics_Curriculum_2007_Infer_2Means_Indep |independent design]] and constructed a corresponding Confidence interval, we would obtain an incorrect inference:</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Similarly, had we incorrectly used the [[AP_Statistics_Curriculum_2007_Infer_2Means_Indep |independent design]] and constructed a corresponding Confidence interval, we would obtain an incorrect inference:</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>: <math>CI: {\overline{x}-\overline{y} - \mu_o \pm t_{(df=17, \alpha/2)} \times SE(\overline{x}+\overline{y})} = 0.<del class="diffchange diffchange-inline">682 -0.687 </del>\pm 1.740\times 0.0325 = [-0.0116 ; 0.1016]</math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>: <math>CI: {\overline{x}-\overline{y} - \mu_o \pm t_{(df=17, \alpha/2)} \times SE(\overline{x}+\overline{y})} = 0.<ins class="diffchange diffchange-inline">045 </ins>\pm 1.740\times 0.0325 = [-0.0116 ; 0.1016]</math></div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><hr></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><hr></div></td></tr>
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</table>IvoDinovhttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Infer_2Means_Dep&diff=12287&oldid=prevIvoDinov: /* Paired vs. Independent Testing */2013-06-17T19:57:55Z<p><span class="autocomment">Paired vs. Independent Testing</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Similarly, had we incorrectly used the [[AP_Statistics_Curriculum_2007_Infer_2Means_Indep |independent design]] and constructed a corresponding Confidence interval, we would obtain an incorrect inference:</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Similarly, had we incorrectly used the [[AP_Statistics_Curriculum_2007_Infer_2Means_Indep |independent design]] and constructed a corresponding Confidence interval, we would obtain an incorrect inference:</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>: <math>CI: {\overline{x}-\overline{y} - \mu_o \pm t_{(df=17, \alpha/2)} \times SE(\overline{<del class="diffchange diffchange-inline">x_1</del>}+\overline{<del class="diffchange diffchange-inline">x_1</del>})} = 0.682 -0.687 \pm 1.740\times 0.0325 = [-0.0116 ; 0.1016]</math></div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>: <math>CI: {\overline{x}-\overline{y} - \mu_o \pm t_{(df=17, \alpha/2)} \times SE(\overline{<ins class="diffchange diffchange-inline">x</ins>}+\overline{<ins class="diffchange diffchange-inline">y</ins>})} = 0.682 -0.687 \pm 1.740\times 0.0325 = [-0.0116 ; 0.1016]</math></div></td></tr>
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</table>IvoDinovhttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Infer_2Means_Dep&diff=12286&oldid=prevIvoDinov: /* Test Statistics */2013-06-17T19:55:52Z<p><span class="autocomment">Test Statistics</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>====Test Statistics====</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>====Test Statistics====</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* If the two populations that the {<math>X_i</math>} and {<math>Y_i</math>} samples were drawn from are approximately Normal, then the [http://en.wikipedia.org/wiki/Hypothesis_testing#Common_test_statistics Test Statistics] is: </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* If the two populations that the {<math>X_i</math>} and {<math>Y_i</math>} samples were drawn from are approximately Normal, then the [http://en.wikipedia.org/wiki/Hypothesis_testing#Common_test_statistics Test Statistics] is: </div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>: <math>T_o = {\overline{d} - \mu_o \over SE(\overline{d})} = {\overline{<del class="diffchange diffchange-inline">x</del>} - \mu_o \over {{1\over \sqrt{n}} \sqrt{\sum_{i=1}^n{(d_i-\overline{d})^2\over n-1}}}} \sim T_{(df=n-1)}</math>.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>: <math>T_o = {\overline{d} - \mu_o \over SE(\overline{d})} = {\overline{<ins class="diffchange diffchange-inline">d</ins>} - \mu_o \over {{1\over \sqrt{n}} \sqrt{\sum_{i=1}^n{(d_i-\overline{d})^2\over n-1}}}} \sim T_{(df=n-1)}</math>.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>====Effects of Ignoring the Pairing====</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>====Effects of Ignoring the Pairing====</div></td></tr>
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</table>IvoDinovhttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Infer_2Means_Dep&diff=11526&oldid=prevIvoDinov: /* Test Statistics */2012-03-06T05:45:39Z<p><span class="autocomment">Test Statistics</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>====Test Statistics====</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>====Test Statistics====</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* If the two populations that the {<math>X_i</math>} and {<math>Y_i</math>} samples were drawn from are approximately Normal, then the [http://en.wikipedia.org/wiki/Hypothesis_testing#Common_test_statistics Test Statistics] is: </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>* If the two populations that the {<math>X_i</math>} and {<math>Y_i</math>} samples were drawn from are approximately Normal, then the [http://en.wikipedia.org/wiki/Hypothesis_testing#Common_test_statistics Test Statistics] is: </div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>: <math>T_o = {\overline{d} - \mu_o \over SE(\overline{d})} = {\overline{x} - \mu_o \over {{1\over \sqrt{n}} \sqrt{\sum_{i=1}^n{(d_i-\overline{d})^2\over n-1}}}<del class="diffchange diffchange-inline">)</del>} \sim T_{(df=n-1)}</math>.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>: <math>T_o = {\overline{d} - \mu_o \over SE(\overline{d})} = {\overline{x} - \mu_o \over {{1\over \sqrt{n}} \sqrt{\sum_{i=1}^n{(d_i-\overline{d})^2\over n-1}}}} \sim T_{(df=n-1)}</math>.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>====Effects of Ignoring the Pairing====</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>====Effects of Ignoring the Pairing====</div></td></tr>
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</table>IvoDinovhttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Infer_2Means_Dep&diff=10399&oldid=prevJenny: /* Effects of Ignoring the Pairing */2010-06-28T20:46:33Z<p><span class="autocomment">Effects of Ignoring the Pairing</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>====Effects of Ignoring the Pairing====</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>====Effects of Ignoring the Pairing====</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>The SE estimate will be '''smaller''' for correctly paired data. If we look <del class="diffchange diffchange-inline">at the data </del>within each sample we notice variation from one subject to the next. This information gets incorporated into the SE for the independent t-test via <math>s_1</math> and <math>s_2</math>. The original reason we paired was to try to control for some of this inter-subject variation, which is not of interest in the paired design. Notice that the inter-subject variation has no influence on the SE for the paired test, because only the differences were used in the calculation. The price of pairing is smaller degrees of freedom of the T-test. However, this can be compensated with a smaller SE if we had paired correctly.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>The SE estimate will be '''smaller''' for correctly paired data. If we look <ins class="diffchange diffchange-inline"> </ins>within each sample <ins class="diffchange diffchange-inline">at the data, </ins>we notice variation from one subject to the next. This information gets incorporated into the SE for the independent t-test via <math>s_1</math> and <math>s_2</math>. The original reason we paired was to try to control for some of this inter-subject variation, which is not of interest in the paired design. Notice that the inter-subject variation has no influence on the SE for the paired test, because only the differences were used in the calculation. The price of pairing is smaller degrees of freedom of the T-test. However, this can be compensated with a smaller SE if we had paired correctly.</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>'''Pairing''' is used to ''reduce'' bias and ''increase'' precision in our inference. By '''matching/blocking''' we can control variation due to extraneous variables.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''Pairing''' is used to ''reduce'' bias and ''increase'' precision in our inference. By '''matching/blocking''' we can control variation due to extraneous variables.</div></td></tr>
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</table>Jennyhttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Infer_2Means_Dep&diff=10398&oldid=prevJenny: /* Analysis Protocol for Paired Designs */2010-06-28T20:44:44Z<p><span class="autocomment">Analysis Protocol for Paired Designs</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Now we can clearly see that the group effect (group differences) is directly represented in the {<math>d_i</math>} sequence. The [[AP_Statistics_Curriculum_2007_Hypothesis_S_Mean#.28Approximately.29_Nornal_Process_with_Unknown_Variance | one-sample T test]] is the proper strategy to analyze the difference sample {<math>d_i</math>}, if the <math>X_i</math> and <math>Y_i</math> samples come from [[AP_Statistics_Curriculum_2007#Chapter_V:_Normal_Probability_Distribution |Normal distributions]].</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Now we can clearly see that the group effect (group differences) is directly represented in the {<math>d_i</math>} sequence. The [[AP_Statistics_Curriculum_2007_Hypothesis_S_Mean#.28Approximately.29_Nornal_Process_with_Unknown_Variance | one-sample T test]] is the proper strategy to analyze the difference sample {<math>d_i</math>}, if the <math>X_i</math> and <math>Y_i</math> samples come from [[AP_Statistics_Curriculum_2007#Chapter_V:_Normal_Probability_Distribution |Normal distributions]].</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">Because </del>we are focusing on the differences, we can use the same reasoning as we did in the [[AP_Statistics_Curriculum_2007_Hypothesis_S_Mean#.28Approximately.29_Nornal_Process_with_Unknown_Variance |single sample case]] to calculate the standard error (i.e., the standard deviation of the sampling distribution of <math>\overline{d}</math>) of <math>\overline{d}={1\over n}\sum_{i=1}^n{d_i}</math>.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div><ins class="diffchange diffchange-inline">Since </ins>we are focusing on the differences, we can use the same reasoning as we did in the [[AP_Statistics_Curriculum_2007_Hypothesis_S_Mean#.28Approximately.29_Nornal_Process_with_Unknown_Variance |single sample case]] to calculate the standard error (i.e., the standard deviation of the sampling distribution of <math>\overline{d}</math>) of <math>\overline{d}={1\over n}\sum_{i=1}^n{d_i}</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>Thus, the standard error of <math>\overline{d}</math> is given by <math>{{1\over \sqrt{n}} \sqrt{\sum_{i=1}^n{(d_i-\overline{d})^2\over n-1}}}</math>, where <math>d_i=X_i-Y_i, \forall 1\leq i\leq n</math>.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Thus, the standard error of <math>\overline{d}</math> is given by <math>{{1\over \sqrt{n}} \sqrt{\sum_{i=1}^n{(d_i-\overline{d})^2\over n-1}}}</math>, where <math>d_i=X_i-Y_i, \forall 1\leq i\leq n</math>.</div></td></tr>
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</table>Jennyhttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Infer_2Means_Dep&diff=10397&oldid=prevJenny: /* Inferences About Two Means: Dependent Samples */2010-06-28T20:44:09Z<p><span class="autocomment">Inferences About Two Means: Dependent Samples</span></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>=== Inferences About Two Means: Dependent Samples===</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>=== Inferences About Two Means: Dependent Samples===</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>In all study designs, it is always critical to clearly identify whether samples we compare come from dependent or independent populations. There is a general formulation for the significance testing when the samples are independent. The fact that there may be uncountable many different types of dependencies prevents us from having a similar analysis protocol for ''all'' dependent sample cases. However, in one specific case - paired samples - we have a theory to generalize the significance testing analysis protocol. Two populations (or samples) are ''dependent because of pairing'' (or paired) if they are linked in some way, usually by a direct relationship. For example, measure the weight of subjects before and after a six month diet.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>In all study designs, it is always critical to clearly identify whether samples we compare come from dependent or independent populations. There is a general formulation for the significance testing when the samples are independent. The fact that there may be uncountable many different types of dependencies <ins class="diffchange diffchange-inline">that </ins>prevents us from having a similar analysis protocol for ''all'' dependent sample cases. However, in one specific case - paired samples - we have a theory to generalize the significance testing analysis protocol. Two populations (or samples) are ''dependent because of pairing'' (or paired) if they are linked in some way, usually by a direct relationship. For example, measure the weight of subjects before and after a six month diet.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===Paired Designs===</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===Paired Designs===</div></td></tr>
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</table>Jennyhttp://wiki.stat.ucla.edu/socr/index.php?title=AP_Statistics_Curriculum_2007_Infer_2Means_Dep&diff=10297&oldid=prevJenny at 17:28, 28 June 20102010-06-28T17:28:10Z<p></p>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>=== Inferences About Two Means: Dependent Samples===</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>=== Inferences About Two Means: Dependent Samples===</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>In all study designs, it is always critical to clearly identify whether samples we compare come from dependent or independent populations. There is a general formulation for the significance testing when the samples are independent. The fact that there may be uncountable many different types of dependencies <del class="diffchange diffchange-inline">prevent </del>us from having a similar analysis protocol for ''all'' dependent sample cases. However, in one specific case - paired samples - we have <del class="diffchange diffchange-inline">the </del>theory to generalize the significance testing analysis protocol. Two populations (or samples) are ''dependent because of pairing'' (or paired) if they are linked in some way, usually by a direct relationship. For example, measure the weight of subjects before and after a six month diet.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>In all study designs, it is always critical to clearly identify whether samples we compare come from dependent or independent populations. There is a general formulation for the significance testing when the samples are independent. The fact that there may be uncountable many different types of dependencies <ins class="diffchange diffchange-inline">prevents </ins>us from having a similar analysis protocol for ''all'' dependent sample cases. However, in one specific case - paired samples - we have <ins class="diffchange diffchange-inline">a </ins>theory to generalize the significance testing analysis protocol. Two populations (or samples) are ''dependent because of pairing'' (or paired) if they are linked in some way, usually by a direct relationship. For example, measure the weight of subjects before and after a six month diet.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===Paired Designs===</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>===Paired Designs===</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>To study paired data, we would like to examine the differences between each pair. Suppose {<math>X_1, X_2, X_3, \cdots , X_n</math>} and {<math>Y_1, Y_2, Y_3, \cdots , Y_n</math>} represent the 2 paired samples. Then we want to study the difference sample {<math>d_1=X_1-Y_1, d_2=X_2-Y_2, d_3=X_3-Y_3, \cdots , d_n=X_n-Y_n</math>}. Notice the effect of the pairings of each <math>X_i</math> and <math>Y_i</math>. </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>To study paired data, we would like to examine the differences between each pair. Suppose {<math>X_1, X_2, X_3, \cdots , X_n</math>} and {<math>Y_1, Y_2, Y_3, \cdots , Y_n</math>} represent the 2 paired samples. Then we want to study the difference sample {<math>d_1=X_1-Y_1, d_2=X_2-Y_2, d_3=X_3-Y_3, \cdots , d_n=X_n-Y_n</math>}. Notice the effect of the pairings of each <math>X_i</math> and <math>Y_i</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: #ffa; color:black; font-size: smaller;"><div>Now we can clearly see that the group effect (group differences) <del class="diffchange diffchange-inline">are </del>directly represented in the {<math>d_i</math>} sequence. The [[AP_Statistics_Curriculum_2007_Hypothesis_S_Mean#.28Approximately.29_Nornal_Process_with_Unknown_Variance | one-sample T test]] is the proper strategy to analyze the difference sample {<math>d_i</math>}, if the <math>X_i</math> and <math>Y_i</math> samples come from [[AP_Statistics_Curriculum_2007#Chapter_V:_Normal_Probability_Distribution |Normal distributions]].</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Now we can clearly see that the group effect (group differences) <ins class="diffchange diffchange-inline">is </ins>directly represented in the {<math>d_i</math>} sequence. The [[AP_Statistics_Curriculum_2007_Hypothesis_S_Mean#.28Approximately.29_Nornal_Process_with_Unknown_Variance | one-sample T test]] is the proper strategy to analyze the difference sample {<math>d_i</math>}, if the <math>X_i</math> and <math>Y_i</math> samples come from [[AP_Statistics_Curriculum_2007#Chapter_V:_Normal_Probability_Distribution |Normal distributions]].</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>Because we are focusing on the differences, we can use the same reasoning as we did in the [[AP_Statistics_Curriculum_2007_Hypothesis_S_Mean#.28Approximately.29_Nornal_Process_with_Unknown_Variance |single sample case]] to calculate the standard error (i.e., the standard deviation of the sampling distribution of <math>\overline{d}</math>) of <math>\overline{d}={1\over n}\sum_{i=1}^n{d_i}</math>.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>Because we are focusing on the differences, we can use the same reasoning as we did in the [[AP_Statistics_Curriculum_2007_Hypothesis_S_Mean#.28Approximately.29_Nornal_Process_with_Unknown_Variance |single sample case]] to calculate the standard error (i.e., the standard deviation of the sampling distribution of <math>\overline{d}</math>) of <math>\overline{d}={1\over n}\sum_{i=1}^n{d_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>====Effects of Ignoring the Pairing====</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>====Effects of Ignoring the Pairing====</div></td></tr>
<tr><td class='diff-marker'>-</td><td style="background: #ffa; color:black; font-size: smaller;"><div>The SE estimate will be '''smaller''' for correctly paired data. If we look at the within each sample <del class="diffchange diffchange-inline">at the data </del>we notice variation from one subject to the next. This information gets incorporated into the SE for the independent t-test via <math>s_1</math> and <math>s_2</math>. The original reason we paired was to try to control for some of this inter-subject variation, which is not of interest in the paired design. Notice that the inter-subject variation has no influence on the SE for the paired test, because only the differences were used in the calculation. The price of pairing is smaller degrees of freedom of the T-test. However, this can be compensated with a smaller SE if we had paired correctly.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>The SE estimate will be '''smaller''' for correctly paired data. If we look at the <ins class="diffchange diffchange-inline">data </ins>within each sample we notice variation from one subject to the next. This information gets incorporated into the SE for the independent t-test via <math>s_1</math> and <math>s_2</math>. The original reason we paired was to try to control for some of this inter-subject variation, which is not of interest in the paired design. Notice that the inter-subject variation has no influence on the SE for the paired test, because only the differences were used in the calculation. The price of pairing is smaller degrees of freedom of the T-test. However, this can be compensated with a smaller SE if we had paired correctly.</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>'''Pairing''' is used to ''reduce'' bias and ''increase'' precision in our inference. By '''matching/blocking''' we can control variation due to extraneous variables.</div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>'''Pairing''' is used to ''reduce'' bias and ''increase'' precision in our inference. By '''matching/blocking''' we can control variation due to extraneous variables.</div></td></tr>
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<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>: <math>p-value=P(T_{(df=9)}>T_o=5.4022)=0.000216</math> for this (one-sided) test. </div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div>: <math>p-value=P(T_{(df=9)}>T_o=5.4022)=0.000216</math> for this (one-sided) test. </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>Therefore, we '''can reject''' the null hypothesis at <math>\alpha=0.05</math>! The left white area at the tails of the T(df=9) distribution <del class="diffchange diffchange-inline">depict </del>graphically the probability of interest, which represents the strength of the evidence (in the data) against the Null hypothesis. In this case, this area is 0.000216, which is much smaller than the initially set [[AP_Statistics_Curriculum_2007_Hypothesis_Basics | Type I]] error <math>\alpha = 0.05</math> and we reject the null hypothesis.</div></td><td class='diff-marker'>+</td><td style="background: #cfc; color:black; font-size: smaller;"><div>Therefore, we '''can reject''' the null hypothesis at <math>\alpha=0.05</math>! The left white area at the tails of the T(df=9) distribution <ins class="diffchange diffchange-inline">depicts </ins>graphically the probability of interest, which represents the strength of the evidence (in the data) against the Null hypothesis. In this case, this area is 0.000216, which is much smaller than the initially set [[AP_Statistics_Curriculum_2007_Hypothesis_Basics | Type I]] error <math>\alpha = 0.05</math> and we reject the null hypothesis.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><center>[[Image:SOCR_EBook_Dinov_Infer_2Means_Dep_020908_Fig4.jpg|600px]]</center></div></td><td class='diff-marker'> </td><td style="background: #eee; color:black; font-size: smaller;"><div><center>[[Image:SOCR_EBook_Dinov_Infer_2Means_Dep_020908_Fig4.jpg|600px]]</center></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>
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</table>Jenny