假设检验之对于成对数据的检验

朴灿烈づ我的快乐病毒、 2023-06-25 10:07 17阅读 0赞

<table> 
 <thead> 
  <tr> 
   <th>假设</th> 
   <th>何时拒绝<span><span><span> H 0 H_0 </span><span><span><span style="height: 0.83333em; vertical-align: -0.15em;"></span><span><span style="margin-right: 0.08125em;">H</span><span><span><span><span style="height: 0.301108em;"><span style="top: -2.55em; margin-left: -0.08125em; margin-right: 0.05em;"><span style="height: 2.7em;"></span><span><span>0</span></span></span></span><span></span></span><span><span style="height: 0.15em;"><span></span></span></span></span></span></span></span></span></span></span></th> 
  </tr> 
 </thead> 
 <tbody> 
  <tr> 
   <td><span><span><span> H 0 : μ D = 0 , H 1 : μ D ≠ 0 H_0:\mu_{D}=0,H_1:\mu_D\not=0 </span><span><span><span style="height: 0.83333em; vertical-align: -0.15em;"></span><span><span style="margin-right: 0.08125em;">H</span><span><span><span><span style="height: 0.301108em;"><span style="top: -2.55em; margin-left: -0.08125em; margin-right: 0.05em;"><span style="height: 2.7em;"></span><span><span>0</span></span></span></span><span></span></span><span><span style="height: 0.15em;"><span></span></span></span></span></span></span><span style="margin-right: 0.277778em;"></span><span>:</span><span style="margin-right: 0.277778em;"></span></span><span><span style="height: 0.625em; vertical-align: -0.19444em;"></span><span><span>μ</span><span><span><span><span style="height: 0.328331em;"><span style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span style="height: 2.7em;"></span><span><span><span style="margin-right: 0.02778em;">D</span></span></span></span></span><span></span></span><span><span style="height: 0.15em;"><span></span></span></span></span></span></span><span style="margin-right: 0.277778em;"></span><span>=</span><span style="margin-right: 0.277778em;"></span></span><span><span style="height: 0.87777em; vertical-align: -0.19444em;"></span><span>0</span><span>,</span><span style="margin-right: 0.166667em;"></span><span><span style="margin-right: 0.08125em;">H</span><span><span><span><span style="height: 0.301108em;"><span style="top: -2.55em; margin-left: -0.08125em; margin-right: 0.05em;"><span style="height: 2.7em;"></span><span><span>1</span></span></span></span><span></span></span><span><span style="height: 0.15em;"><span></span></span></span></span></span></span><span style="margin-right: 0.277778em;"></span><span>:</span><span style="margin-right: 0.277778em;"></span></span><span><span style="height: 0.88888em; vertical-align: -0.19444em;"></span><span><span>μ</span><span><span><span><span style="height: 0.328331em;"><span style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span style="height: 2.7em;"></span><span><span style="margin-right: 0.02778em;">D</span></span></span></span><span></span></span><span><span style="height: 0.15em;"><span></span></span></span></span></span></span><span style="margin-right: 0.277778em;"></span><span><span><span><span><span style="height: 0.69444em;"><span style="top: -3em;"><span style="height: 3em;"></span><span><span style="height: 0.88888em; vertical-align: -0.19444em;"></span><span><span></span></span><span></span></span></span></span><span></span></span><span><span style="height: 0.19444em;"><span></span></span></span></span></span></span></span><span><span style="height: 0.36687em; vertical-align: 0em;"></span><span>=</span><span style="margin-right: 0.277778em;"></span></span><span><span style="height: 0.64444em; vertical-align: 0em;"></span><span>0</span></span></span></span></span></td> 
   <td><span><span><span> ∥ D ‾ S D / n ∥ ≥ t α 2 ( n − 1 ) \|\frac{\overline D}{S_D/\sqrt{n}}\| \ge t_{\frac{\alpha}{2}}(n-1) </span><span><span><span style="height: 1.59296em; vertical-align: -0.549134em;"></span><span>∥</span><span><span></span><span><span><span><span style="height: 1.04383em;"><span style="top: -2.62587em;"><span style="height: 3em;"></span><span><span><span><span style="margin-right: 0.05764em;">S</span><span><span><span><span style="height: 0.3448em;"><span style="top: -2.35671em; margin-left: -0.05764em; margin-right: 0.0714286em;"><span style="height: 2.5em;"></span><span><span style="margin-right: 0.02778em;">D</span></span></span></span><span></span></span><span><span style="height: 0.143293em;"><span></span></span></span></span></span></span><span>/</span><span><span><span><span style="height: 0.805905em;"><span style="top: -3em;"><span style="height: 3em;"></span><span style="padding-left: 0.833em;"><span>n</span></span></span><span style="top: -2.76591em;"><span style="height: 3em;"></span><span style="min-width: 0.853em; height: 1.08em;">   </span></span></span><span></span></span><span><span style="height: 0.234095em;"><span></span></span></span></span></span></span></span></span><span style="top: -3.23em;"><span style="height: 3em;"></span><span style="border-bottom-width: 0.04em;"></span></span><span style="top: -3.394em;"><span style="height: 3em;"></span><span><span><span><span><span><span style="height: 0.92833em;"><span style="top: -3em;"><span style="height: 3em;"></span><span style="margin-right: 0.02778em;">D</span></span><span style="top: -3.83033em;"><span style="height: 3em;"></span><span style="border-bottom-width: 0.049em;"></span></span></span></span></span></span></span></span></span></span><span></span></span><span><span style="height: 0.549134em;"><span></span></span></span></span></span><span></span></span><span>∥</span><span style="margin-right: 0.277778em;"></span><span>≥</span><span style="margin-right: 0.277778em;"></span></span><span><span style="height: 1.1408em; vertical-align: -0.3908em;"></span><span><span>t</span><span><span><span><span style="height: 0.33408em;"><span style="top: -2.85em; margin-left: 0em; margin-right: 0.05em;"><span style="height: 3em;"></span><span><span><span><span></span><span><span><span><span style="height: 0.691543em;"><span style="top: -2.656em;"><span style="height: 3em;"></span><span><span><span>2</span></span></span></span><span style="top: -3.2255em;"><span style="height: 3em;"></span><span style="border-bottom-width: 0.049em;"></span></span><span style="top: -3.384em;"><span style="height: 3em;"></span><span><span><span style="margin-right: 0.0037em;">α</span></span></span></span></span><span></span></span><span><span style="height: 0.344em;"><span></span></span></span></span></span><span></span></span></span></span></span></span><span></span></span><span><span style="height: 0.3908em;"><span></span></span></span></span></span></span><span>(</span><span>n</span><span style="margin-right: 0.222222em;"></span><span>−</span><span style="margin-right: 0.222222em;"></span></span><span><span style="height: 1em; vertical-align: -0.25em;"></span><span>1</span><span>)</span></span></span></span></span></td> 
  </tr> 
  <tr> 
   <td><span><span><span> H 0 : μ D ≤ 0 , H 1 : μ D &gt; 0 H_0:\mu_{D}\le0,H_1:\mu_D&gt;0 </span><span><span><span style="height: 0.83333em; vertical-align: -0.15em;"></span><span><span style="margin-right: 0.08125em;">H</span><span><span><span><span style="height: 0.301108em;"><span style="top: -2.55em; margin-left: -0.08125em; margin-right: 0.05em;"><span style="height: 2.7em;"></span><span><span>0</span></span></span></span><span></span></span><span><span style="height: 0.15em;"><span></span></span></span></span></span></span><span style="margin-right: 0.277778em;"></span><span>:</span><span style="margin-right: 0.277778em;"></span></span><span><span style="height: 0.83041em; vertical-align: -0.19444em;"></span><span><span>μ</span><span><span><span><span style="height: 0.328331em;"><span style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span style="height: 2.7em;"></span><span><span><span style="margin-right: 0.02778em;">D</span></span></span></span></span><span></span></span><span><span style="height: 0.15em;"><span></span></span></span></span></span></span><span style="margin-right: 0.277778em;"></span><span>≤</span><span style="margin-right: 0.277778em;"></span></span><span><span style="height: 0.87777em; vertical-align: -0.19444em;"></span><span>0</span><span>,</span><span style="margin-right: 0.166667em;"></span><span><span style="margin-right: 0.08125em;">H</span><span><span><span><span style="height: 0.301108em;"><span style="top: -2.55em; margin-left: -0.08125em; margin-right: 0.05em;"><span style="height: 2.7em;"></span><span><span>1</span></span></span></span><span></span></span><span><span style="height: 0.15em;"><span></span></span></span></span></span></span><span style="margin-right: 0.277778em;"></span><span>:</span><span style="margin-right: 0.277778em;"></span></span><span><span style="height: 0.73354em; vertical-align: -0.19444em;"></span><span><span>μ</span><span><span><span><span style="height: 0.328331em;"><span style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span style="height: 2.7em;"></span><span><span style="margin-right: 0.02778em;">D</span></span></span></span><span></span></span><span><span style="height: 0.15em;"><span></span></span></span></span></span></span><span style="margin-right: 0.277778em;"></span><span>&gt;</span><span style="margin-right: 0.277778em;"></span></span><span><span style="height: 0.64444em; vertical-align: 0em;"></span><span>0</span></span></span></span></span></td> 
   <td><span><span><span> D ‾ S D / n ≥ t α ( n − 1 ) \frac{\overline D}{S_D/\sqrt{n}} \ge t_{\alpha}(n-1) </span><span><span><span style="height: 1.59296em; vertical-align: -0.549134em;"></span><span><span></span><span><span><span><span style="height: 1.04383em;"><span style="top: -2.62587em;"><span style="height: 3em;"></span><span><span><span><span style="margin-right: 0.05764em;">S</span><span><span><span><span style="height: 0.3448em;"><span style="top: -2.35671em; margin-left: -0.05764em; margin-right: 0.0714286em;"><span style="height: 2.5em;"></span><span><span style="margin-right: 0.02778em;">D</span></span></span></span><span></span></span><span><span style="height: 0.143293em;"><span></span></span></span></span></span></span><span>/</span><span><span><span><span style="height: 0.805905em;"><span style="top: -3em;"><span style="height: 3em;"></span><span style="padding-left: 0.833em;"><span>n</span></span></span><span style="top: -2.76591em;"><span style="height: 3em;"></span><span style="min-width: 0.853em; height: 1.08em;">   </span></span></span><span></span></span><span><span style="height: 0.234095em;"><span></span></span></span></span></span></span></span></span><span style="top: -3.23em;"><span style="height: 3em;"></span><span style="border-bottom-width: 0.04em;"></span></span><span style="top: -3.394em;"><span style="height: 3em;"></span><span><span><span><span><span><span style="height: 0.92833em;"><span style="top: -3em;"><span style="height: 3em;"></span><span style="margin-right: 0.02778em;">D</span></span><span style="top: -3.83033em;"><span style="height: 3em;"></span><span style="border-bottom-width: 0.049em;"></span></span></span></span></span></span></span></span></span></span><span></span></span><span><span style="height: 0.549134em;"><span></span></span></span></span></span><span></span></span><span style="margin-right: 0.277778em;"></span><span>≥</span><span style="margin-right: 0.277778em;"></span></span><span><span style="height: 1em; vertical-align: -0.25em;"></span><span><span>t</span><span><span><span><span style="height: 0.151392em;"><span style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span style="height: 2.7em;"></span><span><span><span style="margin-right: 0.0037em;">α</span></span></span></span></span><span></span></span><span><span style="height: 0.15em;"><span></span></span></span></span></span></span><span>(</span><span>n</span><span style="margin-right: 0.222222em;"></span><span>−</span><span style="margin-right: 0.222222em;"></span></span><span><span style="height: 1em; vertical-align: -0.25em;"></span><span>1</span><span>)</span></span></span></span></span></td> 
  </tr> 
  <tr> 
   <td><span><span><span> H 0 : μ D ≥ 0 , H 1 : μ D &lt; 0 H_0:\mu_{D}\ge0,H_1:\mu_D&lt;0 </span><span><span><span style="height: 0.83333em; vertical-align: -0.15em;"></span><span><span style="margin-right: 0.08125em;">H</span><span><span><span><span style="height: 0.301108em;"><span style="top: -2.55em; margin-left: -0.08125em; margin-right: 0.05em;"><span style="height: 2.7em;"></span><span><span>0</span></span></span></span><span></span></span><span><span style="height: 0.15em;"><span></span></span></span></span></span></span><span style="margin-right: 0.277778em;"></span><span>:</span><span style="margin-right: 0.277778em;"></span></span><span><span style="height: 0.83041em; vertical-align: -0.19444em;"></span><span><span>μ</span><span><span><span><span style="height: 0.328331em;"><span style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span style="height: 2.7em;"></span><span><span><span style="margin-right: 0.02778em;">D</span></span></span></span></span><span></span></span><span><span style="height: 0.15em;"><span></span></span></span></span></span></span><span style="margin-right: 0.277778em;"></span><span>≥</span><span style="margin-right: 0.277778em;"></span></span><span><span style="height: 0.87777em; vertical-align: -0.19444em;"></span><span>0</span><span>,</span><span style="margin-right: 0.166667em;"></span><span><span style="margin-right: 0.08125em;">H</span><span><span><span><span style="height: 0.301108em;"><span style="top: -2.55em; margin-left: -0.08125em; margin-right: 0.05em;"><span style="height: 2.7em;"></span><span><span>1</span></span></span></span><span></span></span><span><span style="height: 0.15em;"><span></span></span></span></span></span></span><span style="margin-right: 0.277778em;"></span><span>:</span><span style="margin-right: 0.277778em;"></span></span><span><span style="height: 0.73354em; vertical-align: -0.19444em;"></span><span><span>μ</span><span><span><span><span style="height: 0.328331em;"><span style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span style="height: 2.7em;"></span><span><span style="margin-right: 0.02778em;">D</span></span></span></span><span></span></span><span><span style="height: 0.15em;"><span></span></span></span></span></span></span><span style="margin-right: 0.277778em;"></span><span>&lt;</span><span style="margin-right: 0.277778em;"></span></span><span><span style="height: 0.64444em; vertical-align: 0em;"></span><span>0</span></span></span></span></span></td> 
   <td><span><span><span> D ‾ S D / n ≤ − t α ( n − 1 ) \frac{\overline D}{S_D/\sqrt{n}} \le -t_{\alpha}(n-1) </span><span><span><span style="height: 1.59296em; vertical-align: -0.549134em;"></span><span><span></span><span><span><span><span style="height: 1.04383em;"><span style="top: -2.62587em;"><span style="height: 3em;"></span><span><span><span><span style="margin-right: 0.05764em;">S</span><span><span><span><span style="height: 0.3448em;"><span style="top: -2.35671em; margin-left: -0.05764em; margin-right: 0.0714286em;"><span style="height: 2.5em;"></span><span><span style="margin-right: 0.02778em;">D</span></span></span></span><span></span></span><span><span style="height: 0.143293em;"><span></span></span></span></span></span></span><span>/</span><span><span><span><span style="height: 0.805905em;"><span style="top: -3em;"><span style="height: 3em;"></span><span style="padding-left: 0.833em;"><span>n</span></span></span><span style="top: -2.76591em;"><span style="height: 3em;"></span><span style="min-width: 0.853em; height: 1.08em;">   </span></span></span><span></span></span><span><span style="height: 0.234095em;"><span></span></span></span></span></span></span></span></span><span style="top: -3.23em;"><span style="height: 3em;"></span><span style="border-bottom-width: 0.04em;"></span></span><span style="top: -3.394em;"><span style="height: 3em;"></span><span><span><span><span><span><span style="height: 0.92833em;"><span style="top: -3em;"><span style="height: 3em;"></span><span style="margin-right: 0.02778em;">D</span></span><span style="top: -3.83033em;"><span style="height: 3em;"></span><span style="border-bottom-width: 0.049em;"></span></span></span></span></span></span></span></span></span></span><span></span></span><span><span style="height: 0.549134em;"><span></span></span></span></span></span><span></span></span><span style="margin-right: 0.277778em;"></span><span>≤</span><span style="margin-right: 0.277778em;"></span></span><span><span style="height: 1em; vertical-align: -0.25em;"></span><span>−</span><span><span>t</span><span><span><span><span style="height: 0.151392em;"><span style="top: -2.55em; margin-left: 0em; margin-right: 0.05em;"><span style="height: 2.7em;"></span><span><span><span style="margin-right: 0.0037em;">α</span></span></span></span></span><span></span></span><span><span style="height: 0.15em;"><span></span></span></span></span></span></span><span>(</span><span>n</span><span style="margin-right: 0.222222em;"></span><span>−</span><span style="margin-right: 0.222222em;"></span></span><span><span style="height: 1em; vertical-align: -0.25em;"></span><span>1</span><span>)</span></span></span></span></span></td> 
  </tr> 
 </tbody> 
</table>

![|  |  ||--|--||  |  |][_ _--_--_ _]

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