範例：常態均數的 T-測試

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範例：常態均數的 T-測試

檢驗兩個常態總體具有相等均數的假設。

1. 定義要比較的數據集。

按一下以複製此運算式

按一下以複製此運算式

2. 收集樣本統計。

每個數據集的樣本數	<region id="ID0E3BACB" top="403.20000000000005" left="38.400000000000006" width="137.98000000000002" height="25.6" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="91"> <define xmlns="http://schemas.mathsoft.com/math50"> <id xml:space="preserve" labels="">n1</id> <apply> <id xml:space="preserve" labels="FUNCTION" label-is-contextual="true">length</id> <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">data1</id> </apply> </define> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region> <region id="ID0EFCACB" top="441.60000000000008" left="38.400000000000006" width="137.98000000000002" height="25.6" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="95"> <define xmlns="http://schemas.mathsoft.com/math50"> <id xml:space="preserve" labels="">n2</id> <apply> <id xml:space="preserve" labels="FUNCTION" label-is-contextual="true">length</id> <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">data2</id> </apply> </define> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>	<region id="ID0E1CACB" top="403.20000000000005" left="384.00000000000006" width="65.516666666666666" height="25.6" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="93"> <eval xmlns="http://schemas.mathsoft.com/math50"> <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">n1</id> <unitOverride> <placeholder /> </unitOverride> </eval> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region> <region id="ID0EIDACB" top="441.60000000000008" left="384.00000000000006" width="65.516666666666666" height="25.6" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="97"> <eval xmlns="http://schemas.mathsoft.com/math50"> <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">n2</id> <unitOverride> <placeholder /> </unitOverride> </eval> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
取樣均數	<region id="ID0ENEACB" top="480.00000000000006" left="38.400000000000006" width="136.66000000000003" height="25.6" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="99"> <define xmlns="http://schemas.mathsoft.com/math50"> <id xml:space="preserve" labels="">m1</id> <apply> <id xml:space="preserve" labels="FUNCTION" label-is-contextual="true">mean</id> <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">data1</id> </apply> </define> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region> <region id="ID0E2EACB" top="518.40000000000009" left="38.400000000000006" width="136.66000000000003" height="25.6" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="103"> <define xmlns="http://schemas.mathsoft.com/math50"> <id xml:space="preserve" labels="">m2</id> <apply> <id xml:space="preserve" labels="FUNCTION" label-is-contextual="true">mean</id> <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">data2</id> </apply> </define> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>	<region id="ID0EQFACB" top="480.00000000000006" left="384.00000000000006" width="82.250000000000014" height="25.6" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="101"> <eval xmlns="http://schemas.mathsoft.com/math50"> <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">m1</id> <unitOverride> <placeholder /> </unitOverride> </eval> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region> <region id="ID0E5FACB" top="518.40000000000009" left="384.00000000000006" width="82.250000000000014" height="25.6" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="105"> <eval xmlns="http://schemas.mathsoft.com/math50"> <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">m2</id> <unitOverride> <placeholder /> </unitOverride> </eval> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
取樣標準差	<region id="ID0EDHACB" top="614.40000000000009" left="38.400000000000006" width="197.0459999702374" height="50.065333527326587" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="111"> <define xmlns="http://schemas.mathsoft.com/math50"> <id xml:space="preserve" labels="">s1</id> <apply> <mult /> <apply> <id xml:space="preserve" labels="FUNCTION" label-is-contextual="true">stdev</id> <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">data1</id> </apply> <apply> <nthRoot /> <placeholder /> <apply> <div /> <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">n1</id> <apply> <minus /> <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">n1</id> <real>1</real> </apply> </apply> </apply> </apply> </define> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region> <region id="ID0ERHACB" top="672.00000000000011" left="38.400000000000006" width="197.0459999702374" height="50.065333527326587" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="115"> <define xmlns="http://schemas.mathsoft.com/math50"> <id xml:space="preserve" labels="">s2</id> <apply> <mult /> <apply> <id xml:space="preserve" labels="FUNCTION" label-is-contextual="true">stdev</id> <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">data2</id> </apply> <apply> <nthRoot /> <placeholder /> <apply> <div /> <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">n2</id> <apply> <minus /> <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">n2</id> <real>1</real> </apply> </apply> </apply> </apply> </define> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>	<region id="ID0EGIACB" top="614.40000000000009" left="384.00000000000006" width="90.876666666666679" height="25.6" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="113"> <eval xmlns="http://schemas.mathsoft.com/math50"> <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">s1</id> <unitOverride> <placeholder /> </unitOverride> </eval> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region> <region id="ID0EUIACB" top="672.00000000000011" left="384.00000000000006" width="90.876666666666679" height="25.6" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="117"> <eval xmlns="http://schemas.mathsoft.com/math50"> <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">s2</id> <unitOverride> <placeholder /> </unitOverride> </eval> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
合併兩個均數時的自由度	<region id="ID0EZJACB" top="556.80000000000007" left="38.400000000000006" width="105.12666666666667" height="25.6" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="107"> <define xmlns="http://schemas.mathsoft.com/math50"> <id xml:space="preserve" labels="*">ν</id> <apply> <minus /> <apply> <plus /> <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">n1</id> <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">n2</id> </apply> <real>2</real> </apply> </define> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>	<region id="ID0EOKACB" top="556.80000000000007" left="384.00000000000006" width="56.493333333333339" height="25.6" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="109"> <eval xmlns="http://schemas.mathsoft.com/math50"> <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">ν</id> <unitOverride> <placeholder /> </unitOverride> </eval> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
數據集中差分的標準誤差	<region id="ID0ERLACB" top="729.60000000000014" left="38.400000000000006" width="313.71933330357075" height="52.217333527326588" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="119"> <define xmlns="http://schemas.mathsoft.com/math50"> <id xml:space="preserve" labels="*">s</id> <apply> <nthRoot /> <placeholder /> <apply> <mult /> <apply> <div /> <apply> <plus /> <apply> <mult /> <parens> <apply> <minus /> <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">n1</id> <real>1</real> </apply> </parens> <apply> <pow /> <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">s1</id> <real>2</real> </apply> </apply> <apply> <mult /> <parens> <apply> <minus /> <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">n2</id> <real>1</real> </apply> </parens> <apply> <pow /> <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">s2</id> <real>2</real> </apply> </apply> </apply> <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">ν</id> </apply> <parens> <apply> <plus /> <apply> <div /> <real>1</real> <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">n1</id> </apply> <apply> <div /> <real>1</real> <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">n2</id> </apply> </apply> </parens> </apply> </apply> </define> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>	<region id="ID0EFMACB" top="729.60000000000014" left="384.00000000000006" width="82.896666666666675" height="25.6" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="121"> <eval xmlns="http://schemas.mathsoft.com/math50"> <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">s</id> <unitOverride> <placeholder /> </unitOverride> </eval> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>

3. 定義顯著程度。

4. 計算測試統計。

5. 指定虛無假設與對立假設。

H0: m1 ≤ m2

H1: m1 > m2

6. 計算 p 值並測試假設。在此範例中，當虛無假設為真 (不拒絕 H0) 時，所有布林運算式的計算結果為 1。

<region id="ID0EITACB" top="1248.0000000000002" left="38.400000000000006" width="146.95000000000002" height="25.6" xmlns="http://schemas.mathsoft.com/worksheet50">
  <math resultRef="129">
    <define xmlns="http://schemas.mathsoft.com/math50">
      <id xml:space="preserve" labels="*">P</id>
      <apply>
        <mult />
        <real>2</real>
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          <apply>
            <minus />
            <real>1</real>
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              <id xml:space="preserve" labels="FUNCTION" label-is-contextual="true">pt</id>
              <sequence>
                <apply>
                  <absval />
                  <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">t</id>
                </apply>
                <id xml:space="preserve" labels="VARIABLE" label-is-contextual="true">ν</id>
              </sequence>
            </apply>
          </apply>
        </parens>
      </apply>
    </define>
    <resultFormat>
      <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" />
    </resultFormat>
  </math>
</region>

假設虛無假設為真，則檢驗統計有 0.946 的機率會大於所觀察到的結果。p 值與顯著程度的比較，無法證實對立假設為真。

7. 計算臨界區域的限制，並測試假設。

接受虛無假設。無證據顯示 m1 大於 m2。

8. 繪製 T-測試 (藍色)、臨界區域的邊界 (綠色) 及測試統計 (紅色)。

按一下以複製此運算式

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            <gridFrequency>9</gridFrequency>
            <gridLabels display="true" />
            <gridLines />
            <tickMarks display="true" />
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          <axisLabel />
          <markers>
            <id xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">crit</id>
            <id xml:space="preserve" xmlns="http://schemas.mathsoft.com/math50">t</id>
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            <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
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            <tickMarks display="true" />
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          <axisLabel />
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            <general precision="3" show-trailing-zeros="false" radix="dec" zero-threshold="15" imaginary-value="i" exponential-threshold="3" />
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相關主題

T-測試

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