示例：正态均值的 T 检验

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示例：正态均值的 T 检验

对两个正态总体具有相同均值的假设进行检验。

1. 定义要比较的数据集。

2. 收集样本统计信息。

每个数据集的样本数	<region id="ID0EGILBB" 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="ID0EPILBB" 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="ID0EEJLBB" 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="ID0ESJLBB" 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>
样本均值	<region id="ID0EXKLBB" 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="ID0EFLLBB" 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="ID0E1LLBB" 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="ID0EIMLBB" 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>
样本标准差	<region id="ID0ENNLBB" 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="ID0E2NLBB" 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="ID0EQOLBB" 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="ID0E5OLBB" 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>
合并两个均值时的自由度	<region id="ID0EDQLBB" 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="ID0EYQLBB" 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>
数据集中差异的标准差	<region id="ID0E2RLBB" 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="ID0EPSLBB" 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. 定义显著性水平。

4. 计算检验统计。

5. 声明虚假设和备择假设。

H0: m1 ≤ m2

H1: m1 > m2

6. 计算 p 值并检验假设。在本示例中，当原假设为真 (不拒绝 H0) 时，所有布尔表达式的计算值均为 1。

检验统计大于观察值的概率为 0.946 (假设原假设为真)。p 值与显著性水平之间的比较结果显示：没有证据表明备择假设为真。

7. 计算临界区的极限并检验假设。

接受原假设。没有证据表明 m1 大于 m2。

8. 绘制学生分布图像 (蓝色)、临界区边界图像 (绿色) 及检验统计图像 (红色)。

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