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相关性猜测：用散点与线性回归估计ρ

相关性猜测：用散点与线性回归估计ρ
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2021-01-19 16:45
函数解析几何概率统计函数图像
该范例通过在随机生成的散点数据上，展示“猜测相关性系数ρ并验证”的互动过程，同时给出相关系数与回归直线的直观对应关系。核心数学概念包括：皮尔逊相关系数 PMCC（衡量线性相关强弱与方向）、线性回归/最小二乘拟合的直线（FitLineY）、以及在合适参数下的统计量解释（如方差与期望的常见背景概念）。作图与交互要点：利用 SelectedIndex 选择模型，Sequence 与 RandomUniform 生成样本与自变量集合，If 与 Zip 进行条件分支与数据配对，PMCC 计算相关系数ρ；再用 FitLineY 生成拟合直线，并通过 Polygon 与 Textfield 等对象显示/标注结果与提示信息（如猜测标签、正确/错误消息）；同时维护 xlistNormal、ylistNormal、xlistOther、ylistOther、xylist 与 rline、showcorrel 等显示对象，实现“散点+回归线+相关系数提示”的联动。适合中学后段至大学入门概率统计课程，用于相关性、回归思想与数据分析直观理解的课堂演示或上机练习。

GeoGebra 命令及参数

  1. SelectedIndex(models) → chosenmodel
  2. Sequence(Max[-4, Min[RandomNormal[0, 1], 4]], i, 1, 100) → xlistNormal
  3. RandomUniform(1, 3) → b
  4. RandomUniform(0.6, 1.5) → a
  5. RandomUniform((-sqrt((b + 4) / a)), sqrt((b + 4) / a), 100) → xlistOther, {1.7003313692087823, -1.1277502688964416, -1.4078792452588733, 1.3150959596831446, -0.020381214120730684, 1.6404255909687044, 1.074281299695697, 1.0498008487021717, -1.0627550195170277, -0.7534060318883677, -1.4764562716169511, -1.018908786712169, -0.8005900971521478, 0.6192988627952241, 1.0901308783262877, 1.9084540787821946, 0.9448841059650801, -1.5491789164550809, -1.4444500079819922, 0.12919110819643032, 1.8330270997035567, 1.5778974997993223, -0.2653047203180787, 1.9301898505503894, -1.3177715053332986, -1.7101592944944188, 1.9000943240200023, -1.7365807168324936, -0.34406672994609444, 1.4839370277252233, -1.8184222579683245, 0.45539916210450637, 1.2906079238353498, -0.20606093971558503, -1.659444940315778, -0.4609767523820443, -0.3265096233838727, -0.0651585713579168, 0.34719700184103774, 0.8892401836490491, 1.735200632763472, 1.2477686229659948, 1.2613627022567548, -0.2680196214179382, -0.2335013050632253, 0.4923454768926929, -0.20283920019997081, -0.4424142235270745, 1.3863952222126603, -0.6402037848894637, -0.19065385600626494, -0.22461193683024683, -0.08477931779156744, -1.50058964269489, -1.7316215344420485, -0.45341502298914604, 1.8020659143819382, -0.13966839665367337, -1.6047262767437025, 1.4554501567766442, 1.049189221303305, 1.5179787775945914, 0.22851482307218274, -0.5177805397564215, -1.0972734216267188, 1.5059566301744465, 1.8606390712853715, 0.6247416428711541, -0.3441529735573894, -0.3727403137179077, -0.27649149007175566, -1.2854324369032597, -0.027879960513713664, 1.1048201242788573, -1.5250223569924324, 0.2102316896005194, -0.9274448332524461, 0.598311295531958, 0.362893748897126, -1.6180266935823004, -1.688771307783909, 1.2519088633190194, 1.7846433048997674, -0.6318082096428963, 0.33283235704586134, 1.9224252131911943, 0.07303951759284177, -1.5065088483508118, -1.2086549795974193, -0.4884796012488888, 0.7824516894745781, -1.2252411026153256, -0.6048734379826703, 0.9336299739928646, -0.8206653662604222, 0.023618947536423907, -0.38462832127061675, 1.741603331636153, -0.537689389470976, 0.05141142380062402}
  6. If(chosenmodel ≟ 1, xlistNormal, xlistOther) → xlist
  7. RandomUniform(-1, 1) → rho
  8. Zip(Max[-4, Min[(rho * xx) + (sqrt(1 - rho^(2)) * RandomNormal[0, 1]), 4]], xx, xlist) → ylistNormal
  9. Zip(Max[-4, Min[f(xx) + RandomNormal[0, c], 4]], xx, xlist) → ylistOther
  10. If(chosenmodel ≟ 1, ylistNormal, ylistOther) → ylist
  11. Zip((xx, yy), xx, xlist, yy, ylist) → xylist
  12. PMCC(xylist) → r
  13. FitLineY(xylist) → rline
  14. Polygon(Corner[2], Corner[3], (0.75 * Corner[3]) + (0.25 * Corner[4]), (0.75 * Corner[2]) + (0.25 * Corner[1])) → panel, g, h, i, j
  15. If(abs(guess - r) > 0.2, 0, If[0.1 < abs(guess - r) ≤ 0.2, 1, If[abs(guess - r) ≤ 0.1, 2, 3]]) → guessstate
  16. Textfield(guessinput) → InputBox1

AI 绘图提示语参考

绘制一组随机散点图，通过调整参数生成正态分布和非正态分布的数据集，计算散点图的皮尔逊相关系数，并根据系数绘制回归线。用户需要根据图形估计相关系数，然后与计算出的系数进行比较，正确估计会显示正确提示。
以上提示语为根据原始绘图命令整理而来，仅供参考。