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> 示例：计算标准误差
示例：计算标准误差
计算均值与比例的标准误差，以及回归斜率和截距的标准误差。
均值的标准误差
1.
定义矢量
D
。
<region id="ID0ECH1AB" top="192.00000000000003" left="19.200000000000003" height="111.24000000000001" width="105.79666666666667" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="44"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve">D</id> <matrix rows="36" cols="1"> <display size="4,1" /> <real>10.23</real> <real>10.12</real> <real>9.841</real> <real>10.28</real> <real>9.35</real> <real>10.03</real> <real>9.734</real> <real>9.325</real> <real>7.16</real> <real>12.01</real> <real>9.366</real> <real>9.537</real> <real>9.857</real> <real>9.946</real> <real>10.8</real> <real>9.912</real> <real>9.911</real> <real>13.22</real> <real>10.74</real> <real>9.962</real> <real>10.42</real> <real>10.21</real> <real>9.002</real> <real>9.799</real> <real>10.03</real> <real>9.227</real> <real>9.851</real> <real>11.78</real> <real>3.63</real> <real>10.57</real> <real>11.37</real> <real>10.18</real> <real>10.02</real> <real>13.93</real> <real>10.48</real> <real>6.377</real> </matrix> </define> </math> </region>
2.
计算均值的标准误差。
<region id="ID0EVJ1AB" top="288.00000000000006" left="19.200000000000003" height="25.6" width="106.07666666666668" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="46"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="*" xml:space="preserve">N</id> <apply> <id labels="FUNCTION" xml:space="preserve" label-is-contextual="true">length</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">D</id> </apply> </define> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
<region id="ID0EIK1AB" top="288.00000000000006" left="192.00000000000003" height="25.6" width="60.573333333333338" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="48"> <eval xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">N</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="ID0E6K1AB" top="326.40000000000003" left="19.200000000000003" height="50.065333527326587" width="202.86933330357073" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="50"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="*" xml:space="preserve">SE_mean</id> <apply> <nthRoot /> <placeholder /> <apply> <mult /> <apply> <div /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">N</id> <apply> <minus /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">N</id> <real>1</real> </apply> </apply> <apply> <id labels="FUNCTION" xml:space="preserve" label-is-contextual="true">var</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">D</id> </apply> </apply> </apply> </define> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
<region id="ID0ERL1AB" top="345.6" left="268.80000000000007" height="25.6" width="134.46666666666667" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="52"> <eval xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">SE_mean</id> <unitOverride> <placeholder /> </unitOverride> </eval> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
比例的标准误差
1.
定义矢量
Dp
。
<region id="ID0EVN1AB" top="499.20000000000005" left="19.200000000000003" height="129.84" width="86.006666666666661" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="54"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="*" xml:space="preserve">Dp</id> <matrix rows="100" cols="1"> <display size="5,1" /> <real>0</real> <real>1</real> <real>0</real> <real>0</real> <real>0</real> <real>0</real> <real>1</real> <real>0</real> <real>0</real> <real>1</real> <real>0</real> <real>0</real> <real>1</real> <real>0</real> <real>1</real> <real>0</real> <real>1</real> <real>0</real> <real>1</real> <real>0</real> <real>1</real> <real>0</real> <real>1</real> <real>1</real> <real>0</real> <real>1</real> <real>0</real> <real>0</real> <real>1</real> <real>0</real> <real>0</real> <real>1</real> <real>0</real> <real>1</real> <real>1</real> <real>0</real> <real>1</real> <real>0</real> <real>0</real> <real>0</real> <real>0</real> <real>1</real> <real>1</real> <real>1</real> <real>0</real> <real>0</real> <real>0</real> <real>0</real> <real>1</real> <real>0</real> <real>0</real> <real>0</real> <real>1</real> <real>0</real> <real>0</real> <real>0</real> <real>1</real> <real>0</real> <real>0</real> <real>1</real> <real>0</real> <real>0</real> <real>0</real> <real>0</real> <real>1</real> <real>0</real> <real>0</real> <real>0</real> <real>0</real> <real>0</real> <real>1</real> <real>1</real> <real>0</real> <real>0</real> <real>0</real> <real>0</real> <real>0</real> <real>1</real> <real>0</real> <real>0</real> <real>0</real> <real>0</real> <real>0</real> <real>0</real> <real>0</real> <real>0</real> <real>1</real> <real>0</real> <real>1</real> <real>0</real> <real>1</real> <real>1</real> <real>1</real> <real>0</real> <real>0</real> <real>0</real> <real>1</real> <real>0</real> <real>0</real> <real>0</real> </matrix> </define> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
2.
计算比例的标准误差。
<region id="ID0EIQ1AB" top="652.80000000000007" left="19.200000000000003" height="25.6" width="111.92333333333335" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="56"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="*" xml:space="preserve">Np</id> <apply> <id labels="FUNCTION" xml:space="preserve" label-is-contextual="true">rows</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">Dp</id> </apply> </define> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
<region id="ID0E2Q1AB" top="652.80000000000007" left="211.20000000000005" height="25.6" width="77.073333333333338" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="58"> <eval xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">Np</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="ID0ETR1AB" top="691.2" left="19.200000000000003" height="71.600000000000009" width="80.093333333333334" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="60"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="*" xml:space="preserve">p</id> <apply> <div /> <apply> <summation /> <lambda> <boundVars> <placeholder /> </boundVars> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">Dp</id> </lambda> <upperBound> <placeholder /> </upperBound> </apply> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">Np</id> </apply> </define> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
<region id="ID0EGS1AB" top="691.2" left="211.20000000000005" height="25.6" width="69.136666666666684" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="62"> <eval xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">p</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="ID0E4S1AB" top="748.80000000000007" left="19.200000000000003" height="50.065333527326587" width="148.55933330357075" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="64"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="*" xml:space="preserve">SE_pr</id> <apply> <nthRoot /> <placeholder /> <apply> <div /> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">p</id> <parens> <apply> <minus /> <real>1</real> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">p</id> </apply> </parens> </apply> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">Np</id> </apply> </apply> </define> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
<region id="ID0EPT1AB" top="748.80000000000007" left="211.20000000000005" height="25.6" width="111.61333333333334" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="66"> <eval xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">SE_pr</id> <unitOverride> <placeholder /> </unitOverride> </eval> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
回归的标准误差
1.
定义矢量
Dr
。
<region id="ID0ETV1AB" top="902.40000000000009" left="19.200000000000003" height="111.24000000000001" width="157.08333333333334" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="68"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="*" xml:space="preserve">Dr</id> <matrix rows="100" cols="2"> <display size="4,2" /> <real>0.006</real> <real>1.967</real> <real>4.925</real> <real>4.752</real> <real>8.114</real> <real>5.871</real> <real>9.552</real> <real>8.52</real> <real>8.457</real> <real>9.737</real> <real>14.94</real> <real>11.6</real> <real>12.04</real> <real>15.66</real> <real>17.01</real> <real>15.83</real> <real>18.25</real> <real>17.29</real> <real>21.92</real> <real>21.6</real> <real>24.38</real> <real>25.78</real> <real>24.7</real> <real>25.31</real> <real>28.31</real> <real>28.9</real> <real>30.98</real> <real>30.06</real> <real>29.33</real> <real>33.2</real> <real>31.88</real> <real>34.39</real> <real>32.04</real> <real>34.38</real> <real>36.94</real> <real>39.19</real> <real>38.42</real> <real>40.72</real> <real>40.29</real> <real>42.72</real> <real>43</real> <real>44.68</real> <real>44.86</real> <real>43.76</real> <real>46.13</real> <real>47.59</real> <real>49.76</real> <real>47.84</real> <real>50.46</real> <real>52.5</real> <real>50.74</real> <real>51.71</real> <real>55.46</real> <real>55.13</real> <real>58.83</real> <real>55.77</real> <real>60.11</real> <real>57.96</real> <real>62.09</real> <real>59.78</real> <real>63.66</real> <real>62.4</real> <real>65.41</real> <real>66.61</real> <real>64.62</real> <real>69.17</real> <real>68.59</real> <real>69.13</real> <real>72.75</real> <real>71.75</real> <real>72.36</real> <real>75.23</real> <real>74.28</real> <real>77.91</real> <real>77.7</real> <real>75.98</real> <real>80.2</real> <real>79.5</real> <real>78.14</real> <real>81.86</real> <real>82.66</real> <real>85.22</real> <real>85.29</real> <real>87.21</real> <real>84.55</real> <real>86.57</real> <real>87.43</real> <real>87.7</real> <real>92.17</real> <real>92</real> <real>91.26</real> <real>91.01</real> <real>96.03</real> <real>94.05</real> <real>96.77</real> <real>95.57</real> <real>99.76</real> <real>99.72</real> <real>100.2</real> <real>102.5</real> <real>3</real> <real>4.437</real> <real>6.156</real> <real>7.886</real> <real>9.017</real> <real>11.48</real> <real>10.14</real> <real>11.25</real> <real>14.66</real> <real>12.88</real> <real>13.94</real> <real>24.25</real> <real>25.74</real> <real>18.96</real> <real>22.75</real> <real>27.42</real> <real>26.23</real> <real>30.16</real> <real>36.39</real> <real>33.87</real> <real>34.31</real> <real>27.87</real> <real>37.21</real> <real>31.59</real> <real>41.51</real> <real>42.62</real> <real>41.86</real> <real>50</real> <real>48.37</real> <real>55.33</real> <real>50.28</real> <real>62.26</real> <real>58.29</real> <real>58.04</real> <real>47.71</real> <real>48.7</real> <real>42.91</real> <real>71.5</real> <real>46.89</real> <real>45.09</real> <real>68.64</real> <real>66.35</real> <real>62.18</real> <real>66.02</real> <real>53.72</real> <real>81.21</real> <real>87.03</real> <real>91.05</real> <real>65.4</real> <real>58.23</real> <real>92.25</real> <real>85.09</real> <real>58.76</real> <real>90.65</real> <real>62.66</real> <real>70.49</real> <real>110.5</real> <real>97.78</real> <real>89.57</real> <real>91.31</real> <real>93.56</real> <real>106</real> <real>102.6</real> <real>66.37</real> <real>73.44</real> <real>124.1</real> <real>118.3</real> <real>95.47</real> <real>108.6</real> <real>137.9</real> <real>85.36</real> <real>83.35</real> <real>143.5</real> <real>78.03</real> <real>81.14</real> <real>87.87</real> <real>144.7</real> <real>135.6</real> <real>82.15</real> <real>137.9</real> <real>100.4</real> <real>97.68</real> <real>113</real> <real>116.5</real> <real>154.4</real> <real>132.8</real> <real>157.6</real> <real>102.6</real> <real>126.4</real> <real>104.2</real> <real>98.96</real> <real>146.2</real> <real>145.4</real> <real>125</real> <real>173</real> <real>125</real> <real>184.9</real> <real>172.1</real> <real>136.4</real> <real>124.6</real> </matrix> </define> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
2.
计算穿过上述数据集的最佳拟合线的斜率和截距。
<region id="ID0EFY1AB" top="1017.6000000000001" left="19.200000000000003" height="27.752000000000002" width="69.24666666666667" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="70"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="*" xml:space="preserve">X</id> <apply> <matcol /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">Dr</id> <real>0</real> </apply> </define> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
<region id="ID0EYY1AB" top="1017.6000000000001" left="192.00000000000003" height="27.752000000000002" width="68.613333333333344" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="72"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="*" xml:space="preserve">Y</id> <apply> <matcol /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">Dr</id> <real>1</real> </apply> </define> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
<region id="ID0EOZ1AB" top="1036.8000000000002" left="19.200000000000003" height="25.6" width="118.24333333333334" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="74"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="*" xml:space="preserve">m</id> <apply> <id labels="FUNCTION" xml:space="preserve" label-is-contextual="true">slope</id> <sequence> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">X</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">Y</id> </sequence> </apply> </define> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
<region id="ID0EA11AB" top="1036.8000000000002" left="192.00000000000003" height="25.6" width="139.07" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="76"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="*" xml:space="preserve">b</id> <apply> <id labels="FUNCTION" xml:space="preserve" label-is-contextual="true">intercept</id> <sequence> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">X</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">Y</id> </sequence> </apply> </define> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
3.
计算斜率的标准误差。
<region id="ID0ET21AB" top="1113.6000000000001" left="19.200000000000003" height="25.6" width="108.87666666666667" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="78"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="*" xml:space="preserve">Nr</id> <apply> <id labels="FUNCTION" xml:space="preserve" label-is-contextual="true">rows</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">Dr</id> </apply> </define> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
<region id="ID0EB31AB" top="1171.2000000000003" left="19.200000000000003" height="74.465333527326592" width="286.21528235077858" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="80"> <define xmlns="http://schemas.mathsoft.com/math50"> <id labels="*" xml:space="preserve">SE_slope</id> <apply> <nthRoot /> <placeholder /> <apply> <div /> <apply> <summation /> <lambda> <boundVars> <id xml:space="preserve">i</id> </boundVars> <apply> <pow /> <parens> <apply> <minus /> <apply> <indexer /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">Y</id> <apply> <minus /> <id xml:space="preserve">i</id> <real>1</real> </apply> </apply> <parens> <apply> <plus /> <apply> <mult /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">m</id> <apply> <indexer /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">X</id> <apply> <minus /> <id xml:space="preserve">i</id> <real>1</real> </apply> </apply> </apply> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">b</id> </apply> </parens> </apply> </parens> <real>2</real> </apply> </lambda> <lowerBound> <real>1</real> </lowerBound> <upperBound> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">Nr</id> </upperBound> </apply> <apply> <mult /> <apply> <mult /> <parens> <apply> <minus /> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">Nr</id> <real>2</real> </apply> </parens> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">Nr</id> </apply> <apply> <id labels="FUNCTION" xml:space="preserve" label-is-contextual="true">var</id> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">X</id> </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="ID0EP31AB" top="1171.2000000000003" left="307.20000000000005" height="25.6" width="130.57666666666668" xmlns="http://schemas.mathsoft.com/worksheet50"> <math resultRef="82"> <eval xmlns="http://schemas.mathsoft.com/math50"> <id labels="VARIABLE" xml:space="preserve" label-is-contextual="true">SE_slope</id> <unitOverride> <placeholder /> </unitOverride> </eval> <resultFormat> <matrix size="12,12" offset="0,0" show-indices="false" expand-nested-arrays="false" /> </resultFormat> </math> </region>
4.
计算截距的标准误差。
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