理論関連事項

統計学の基本事項,確率分布の詳細,各種データ解析法の理論的背景について.

母集団の平均が既知のとき,標準正規分布に従う母集団から n 個の標本を取り出したときに得られる以下のような統計量V,すなわち,n 個のデータの2乗の和は自由度n のカイ二乗分布に従う.

\begin{eqnarray*}V=\sum_{i=1}^{n}(\frac{x_i-\mu}{\sigma})^2\tag{1}\end{eqnarray*}

また,平均 μ および分散 σ2 が未知のの正規分布に従う母集団から n 個の標本を取り出したとき,取り出した標本のそれぞれの値をZ変換した2乗の和である以下の統計量W は自由度 n-1 のカイ二乗分布に従う.

\begin{eqnarray*}W=\sum_{i=1}^{n}(\frac{x_i-\overline{x}}{\sigma})^2\tag{2}\end{eqnarray*}

以上の計算で得られた V または W が自由度 n または n-1 のカイ二乗分布に従うことを利用し95%または99%信頼区間を予測するための不等式を立てる.例えば,平均および分散 σ2 が未知のの正規分布に従う母集団から8個の標本を取り出したとき,母分散の値を95%の信頼区間で予測するには以下のようにする.左右の数値は以下のカイ二乗分布表より取得する.

\begin{eqnarray*}1.69\leq W=\sum_{i=1}^{n}(\frac{x_i-\overline{x}}{\sigma})^2\leq 16.01\tag{3}\end{eqnarray*}
自由度0.9950.9750.9500.9000.1000.050.0250.005
10.000039270.00098210.0039320.015792.7063.8415.0247.879
20.010030.050640.10260.21074.6055.9917.37810.6
30.071720.21580.35180.58446.2517.8159.34812.84
40.2070.48440.71071.0647.7799.48811.1414.86
50.41170.83121.1451.619.23611.0712.8316.75
60.67571.2371.6352.20410.6412.5914.4518.55
70.98931.692.1672.83312.0214.0716.0120.28
81.3442.182.7333.4913.3615.5117.5321.95
91.7352.73.3254.16814.6816.9219.0223.59
102.1563.2473.944.86515.9918.3120.4825.19
112.6033.8164.5755.57817.2819.6821.9226.76
123.0744.4045.2266.30418.5521.0323.3428.3
133.5655.0095.8927.04219.8122.3624.7429.82
144.0755.6296.5717.7921.0623.6826.1231.32
154.6016.2627.2618.54722.312527.4932.8
165.1426.9087.9629.31223.5426.328.8534.27
175.6977.5648.67210.0924.7727.5930.1935.72
186.2658.2319.3910.8625.9928.8731.5337.16
196.8448.90710.1211.6527.230.1432.8538.58
207.4349.59110.8512.4428.4131.4134.1740
218.03410.2811.5913.2429.6232.6735.4841.4
228.64310.9812.3414.0430.8133.9236.7842.8
239.2611.6913.0914.8532.0135.1738.0844.18
249.88612.413.8515.6633.236.4239.3645.56
2510.5213.1214.6116.4734.3837.6540.6546.93
2611.1613.8415.3817.2935.5638.8941.9248.29
2711.8114.5716.1518.1136.7440.1143.1949.64
2812.4615.3116.9318.9437.9241.3444.4650.99
2913.1216.0517.7119.7739.0942.5645.7252.34
3013.7916.7918.4920.640.2643.7746.9853.67
3114.4617.5419.2821.4341.4244.9948.2355
3215.1318.2920.0722.2742.5846.1949.4856.33
3315.8219.0520.8723.1143.7547.450.7357.65
3416.519.8121.6623.9544.948.651.9758.96
3517.1920.5722.4724.846.0649.853.260.27
3617.8921.3423.2725.6447.215154.4461.58
3718.5922.1124.0726.4948.3652.1955.6762.88
3819.2922.8824.8827.3449.5153.3856.964.18
392023.6525.728.250.6654.5758.1265.48
4020.7124.4326.5129.0551.8155.7659.3466.77
4121.4225.2127.3329.9152.9556.9460.5668.05
4222.142628.1430.7754.0958.1261.7869.34
4322.8626.7928.9631.6355.2359.362.9970.62
4423.5827.5729.7932.4956.3760.4864.271.89
4524.3128.3730.6133.3557.5161.6665.4173.17
4625.0429.1631.4434.2258.6462.8366.6274.44
4725.7729.9632.2735.0859.776467.8275.7
4826.5130.7533.135.9560.9165.1769.0276.97
4927.2531.5533.9336.8262.0466.3470.2278.23
5027.9932.3634.7637.6963.1767.571.4279.49
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