七、檢驗你的問題

總體(population)和樣本(sample)。樣本的選取原則是樣本選擇儘可能和總體的特徵匹配。測量樣本和總體特徵近視程度的量數叫做抽樣誤差(sampling error)。

零假設(null hypothesis) 和研究假設(research hypothesis)。零假設是研究起點也是測量實際的研究結果的基準。零假設是變數間無關的陳述，而研究假設是變數間有關係的明確陳述。例如：

          零假設： 9年紀學生的ABC記憶考試的平均成績和12年級學生的平均成績沒有差異。

         研究假設： 9年紀學生的ABC記憶考試的平均成績不同於12年級學生的平均成績。

零假設對應總體，研究假設對應樣本。總體不能直接進行檢驗， 只能依據樣本的研究假設的檢驗結論來做出間接推論。

無方向研究假設(nondirectional research hypothesis)反映群體間的差異，但是差異的方向是不確定的。有方向研究假設(directional research hypothesis)反映群體間的差異，而且差異的方向是確定的。

八、概率和概率的重要性

學習概率是理解正態曲線的基礎，也是理解推論統計的基礎。

正態曲線表示均值、中位數和眾數相等的數值分佈。正態曲線沒有偏度。正態曲線有一個很好的波峰（只有一個），而且波峰正好處於中間。正態曲線沿著中心將曲線對摺，兩邊會完全重疊。正態曲線的雙尾是漸進的(asymptotic)

標準值（standard scores）

z值（z score）

 

z值表示偏離均值的標準差的個數，如果資料分佈是正態分佈，曲線的不同面積可用用標準差或者z值得不同數值表示（即z值和麵積(概率)存在對應關係，可以查表）。z值的零界點1.65包含了曲線覆蓋面積的45%，與另一邊的50%加起來就是95%，也就是在x軸的這一點上留下了5%。5%是統計學家採用的標準，即觀察到的結果的概率小於5%，我們可以斷定這是不可能的，除非有非概率事件發生。

python z值對應正態分佈曲線面積的查表函式。

def get_zarea(zvalue):
    z_area = [
        [0.0000,0.0040,0.0080,0.0120,0.0160,0.0199,0.0239,0.0279,0.0319,0.0359],
        [0.0398,0.0438,0.0478,0.0517,0.0557,0.0596,0.0636,0.0675,0.0714,0.0753],
        [0.0793,0.0832,0.0871,0.0910,0.0948,0.0987,0.1026,0.1064,0.1103,0.1141],
        [0.1179,0.1217,0.1255,0.1293,0.1331,0.1368,0.1406,0.1443,0.1480,0.1517],
        [0.1554,0.1591,0.1628,0.1664,0.1700,0.1736,0.1772,0.1808,0.1844,0.1879],
        [0.1915,0.1950,0.1985,0.2019,0.2054,0.2088,0.2123,0.2157,0.2190,0.2224],
        [0.2257,0.2291,0.2324,0.2357,0.2389,0.2422,0.2454,0.2486,0.2517,0.2549],
        [0.2580,0.2611,0.2642,0.2673,0.2704,0.2734,0.2764,0.2794,0.2823,0.2852],
        [0.2881,0.2910,0.2939,0.2967,0.2995,0.3023,0.3051,0.3078,0.3106,0.3133],
        [0.3159,0.3186,0.3212,0.3238,0.3264,0.3289,0.3315,0.3340,0.3365,0.3389],
        [0.3413,0.3438,0.3461,0.3485,0.3508,0.3531,0.3554,0.3577,0.3599,0.3621],
        [0.3643,0.3665,0.3686,0.3708,0.3729,0.3749,0.3770,0.3790,0.3810,0.3830],
        [0.3849,0.3869,0.3888,0.3907,0.3925,0.3944,0.3962,0.3980,0.3997,0.4015],
        [0.4032,0.4049,0.4066,0.4082,0.4099,0.4115,0.4131,0.4147,0.4162,0.4177],
        [0.4192,0.4207,0.4222,0.4236,0.4251,0.4265,0.4279,0.4292,0.4306,0.4319],
        [0.4332,0.4345,0.4357,0.4370,0.4382,0.4394,0.4406,0.4418,0.4429,0.4441],
        [0.4452,0.4463,0.4474,0.4484,0.4495,0.4505,0.4515,0.4525,0.4535,0.4545],
        [0.4554,0.4564,0.4573,0.4582,0.4591,0.4599,0.4608,0.4616,0.4625,0.4633],
        [0.4641,0.4649,0.4656,0.4664,0.4671,0.4678,0.4686,0.4693,0.4699,0.4706],
        [0.4713,0.4719,0.4726,0.4732,0.4738,0.4744,0.4750,0.4756,0.4761,0.4767],
        [0.4772,0.4778,0.4783,0.4788,0.4793,0.4798,0.4803,0.4808,0.4812,0.4817],
        [0.4821,0.4826,0.4830,0.4834,0.4838,0.4842,0.4846,0.4850,0.4854,0.4857],
        [0.4861,0.4864,0.4868,0.4871,0.4875,0.4878,0.4881,0.4884,0.4887,0.4890],
        [0.4893,0.4896,0.4898,0.4901,0.4904,0.4906,0.4909,0.4911,0.4913,0.4916],
        [0.4918,0.4920,0.4922,0.4925,0.4927,0.4929,0.4931,0.4932,0.4934,0.4936],
        [0.4938,0.4940,0.4941,0.4943,0.4945,0.4946,0.4948,0.4949,0.4951,0.4952],
        [0.4953,0.4955,0.4956,0.4957,0.4959,0.4960,0.4961,0.4962,0.4963,0.4964],
        [0.4965,0.4966,0.4967,0.4968,0.4969,0.4970,0.4971,0.4972,0.4973,0.4974],
        [0.4974,0.4975,0.4976,0.4977,0.4977,0.4978,0.4979,0.4979,0.4980,0.4981],
        [0.4981,0.4982,0.4982,0.4983,0.4984,0.4984,0.4985,0.4985,0.4986,0.4986],
        [0.4987,0.4987,0.4987,0.4988,0.4988,0.4989,0.4989,0.4989,0.4990,0.4990]
    ]
    zvalue = abs(zvalue)
    if zvalue>3.09:
        return 0.4990
    zvalue = int(zvalue*100)
    i = zvalue / 10
    j = zvalue % 10
    area = z_area[i][j]
    return area

print get_zarea(0.02)
print get_zarea(1.65)
print get_zarea(3.09)

輸出

0.008
0.4505
0.499