【统计学】第六章

Evernote Export

1.什么是统计量？为什么要引进统计量？统计量中为什么不含任何未知参数？

统计量是样本中的一个函数。由样本构造具体的统计量，实际上是对样本所含的总体信息按某种要求进行加工处理，把分散在样本中的信息集中到统计量的取值上，不同的统计推断问题要求构造不同的统计量。
统计量在统计学中具有极其重要的地位，它是统计推断的基础。统计量在统计学中的地位相当于随机变量在概率论中的地位。

2.判断下列样本函数中哪些是统计量？哪些不是统计量？

T1​=10(X1​+X2​+...+X10​)​

是统计量

T2​=min(X1​,X2​,...,X10​)

不是统计量

T3​=X10​−μ

不是统计量

T4​=σ(X10​−μ)​

不是统计量

3.什么是次序统计量

次序统计量称为参数估计和假设检验的一类重要的统计量。中位数、分位数、四分位数都是次序统计量。

4.什么是充分统计量

在实际统计推断中，是用统计量的值进行判断，而不是由样本观测值进行推断。这就意味着，经过加工后，有了统计量的值即可。在统计学中，假如一个统计量能把含在样本中有关总体的信息一点都不损失地提取出来，那对保证后边的统计推断质量具有重要意义。统计量加工过程中一点信息都不损失的统计量称为充分统计量。

5.什么是自由度

自由度是统计学中常用的一个概念，它具有解释为独立变量的个数，还可以解释为二次型的秩。

6.简述 x2分布、t分布、F分布及正态分布之间的关系

1.X2分布当X趋近于无穷大时，分布的极限分布是正态分布
2.t分布的密度函数是一偶函数
3.F分布中，两个自由度的位置不可互换。如果随机变量X服从t(n)分布，则X2服从F(1,n)的F分布。

7.什么是抽样分布？

为了判断总体的某些特征，需要从总体中按一定的抽样技术抽取若干个个体，将这一抽取过程称为抽样。
抽样分布在总体X的分布类型已知时，若对任意自然数n，都能导出统计量T=T(X1​,X2​,...Xn​)的分布的数学表达式，这种分布称为精确的抽样分布。

8.简述中心极限定理的意义

中心极限定理设从均值为μ、方差为σ2（有限）的任意一个总体中抽取样本量为n的样本，当n充分大时，样本均值X的抽样分布近似服从均值为μ、方差为nσ2​的正态分布。

练习题

调节一个装瓶机使其对每个瓶子的灌装量均值为μ盎司，通过观察这台装瓶机对每个瓶子的灌装量服从标准差 1.0盎司的正态分布。随机抽取由这台机器灌装的9个瓶子形成一个样本，并测定每个瓶子的灌装量。试确定样本均值偏离总体均值不超过0.3盎司的概率。

%23%23%23%23%201.%E4%BB%80%E4%B9%88%E6%98%AF%E7%BB%9F%E8%AE%A1%E9%87%8F%EF%BC%9F%E4%B8%BA%E4%BB%80%E4%B9%88%E8%A6%81%E5%BC%95%E8%BF%9B%E7%BB%9F%E8%AE%A1%E9%87%8F%EF%BC%9F%E7%BB%9F%E8%AE%A1%E9%87%8F%E4%B8%AD%E4%B8%BA%E4%BB%80%E4%B9%88%E4%B8%8D%E5%90%AB%E4%BB%BB%E4%BD%95%E6%9C%AA%E7%9F%A5%E5%8F%82%E6%95%B0%EF%BC%9F%0A%3E%E7%BB%9F%E8%AE%A1%E9%87%8F%E6%98%AF%E6%A0%B7%E6%9C%AC%E4%B8%AD%E7%9A%84%E4%B8%80%E4%B8%AA%E5%87%BD%E6%95%B0%E3%80%82%E7%94%B1%E6%A0%B7%E6%9C%AC%E6%9E%84%E9%80%A0%E5%85%B7%E4%BD%93%E7%9A%84%E7%BB%9F%E8%AE%A1%E9%87%8F%EF%BC%8C%E5%AE%9E%E9%99%85%E4%B8%8A%E6%98%AF%E5%AF%B9%E6%A0%B7%E6%9C%AC%E6%89%80%E5%90%AB%E7%9A%84%E6%80%BB%E4%BD%93%E4%BF%A1%E6%81%AF%E6%8C%89%E6%9F%90%E7%A7%8D%E8%A6%81%E6%B1%82%E8%BF%9B%E8%A1%8C%E5%8A%A0%E5%B7%A5%E5%A4%84%E7%90%86%EF%BC%8C%E6%8A%8A%E5%88%86%E6%95%A3%E5%9C%A8%E6%A0%B7%E6%9C%AC%E4%B8%AD%E7%9A%84%E4%BF%A1%E6%81%AF%E9%9B%86%E4%B8%AD%E5%88%B0%E7%BB%9F%E8%AE%A1%E9%87%8F%E7%9A%84%E5%8F%96%E5%80%BC%E4%B8%8A%EF%BC%8C%E4%B8%8D%E5%90%8C%E7%9A%84%E7%BB%9F%E8%AE%A1%E6%8E%A8%E6%96%AD%E9%97%AE%E9%A2%98%E8%A6%81%E6%B1%82%E6%9E%84%E9%80%A0%E4%B8%8D%E5%90%8C%E7%9A%84%E7%BB%9F%E8%AE%A1%E9%87%8F%E3%80%82%0A%3E%E7%BB%9F%E8%AE%A1%E9%87%8F%E5%9C%A8%E7%BB%9F%E8%AE%A1%E5%AD%A6%E4%B8%AD%E5%85%B7%E6%9C%89%E6%9E%81%E5%85%B6%E9%87%8D%E8%A6%81%E7%9A%84%E5%9C%B0%E4%BD%8D%EF%BC%8C%E5%AE%83%E6%98%AF%E7%BB%9F%E8%AE%A1%E6%8E%A8%E6%96%AD%E7%9A%84%E5%9F%BA%E7%A1%80%E3%80%82%E7%BB%9F%E8%AE%A1%E9%87%8F%E5%9C%A8%E7%BB%9F%E8%AE%A1%E5%AD%A6%E4%B8%AD%E7%9A%84%E5%9C%B0%E4%BD%8D%E7%9B%B8%E5%BD%93%E4%BA%8E%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8F%E5%9C%A8%E6%A6%82%E7%8E%87%E8%AE%BA%E4%B8%AD%E7%9A%84%E5%9C%B0%E4%BD%8D%E3%80%82%0A%0A%23%23%23%23%202.%E5%88%A4%E6%96%AD%E4%B8%8B%E5%88%97%E6%A0%B7%E6%9C%AC%E5%87%BD%E6%95%B0%E4%B8%AD%E5%93%AA%E4%BA%9B%E6%98%AF%E7%BB%9F%E8%AE%A1%E9%87%8F%EF%BC%9F%E5%93%AA%E4%BA%9B%E4%B8%8D%E6%98%AF%E7%BB%9F%E8%AE%A1%E9%87%8F%EF%BC%9F%0A%24%24T_1%3D%5Cfrac%7B(X_1%2BX_2%2B...%2BX_%7B10%7D)%7D%7B10%7D%24%24%20%0A%E6%98%AF%E7%BB%9F%E8%AE%A1%E9%87%8F%0A%24%24T_2%3Dmin(X_1%2CX_2%2C...%2CX_%7B10%7D)%24%24%0A%E4%B8%8D%E6%98%AF%E7%BB%9F%E8%AE%A1%E9%87%8F%0A%24%24T_3%3DX_%7B10%7D-%5Cmu%24%24%0A%E4%B8%8D%E6%98%AF%E7%BB%9F%E8%AE%A1%E9%87%8F%0A%24%24T_4%3D%5Cfrac%7B(X_%7B10%7D-%5Cmu)%7D%7B%5Csigma%7D%24%24%0A%E4%B8%8D%E6%98%AF%E7%BB%9F%E8%AE%A1%E9%87%8F%0A%23%23%23%23%203.%E4%BB%80%E4%B9%88%E6%98%AF%E6%AC%A1%E5%BA%8F%E7%BB%9F%E8%AE%A1%E9%87%8F%0A%3E%E6%AC%A1%E5%BA%8F%E7%BB%9F%E8%AE%A1%E9%87%8F%E7%A7%B0%E4%B8%BA%E5%8F%82%E6%95%B0%E4%BC%B0%E8%AE%A1%E5%92%8C%E5%81%87%E8%AE%BE%E6%A3%80%E9%AA%8C%E7%9A%84%E4%B8%80%E7%B1%BB%E9%87%8D%E8%A6%81%E7%9A%84%E7%BB%9F%E8%AE%A1%E9%87%8F%E3%80%82%E4%B8%AD%E4%BD%8D%E6%95%B0%E3%80%81%E5%88%86%E4%BD%8D%E6%95%B0%E3%80%81%E5%9B%9B%E5%88%86%E4%BD%8D%E6%95%B0%E9%83%BD%E6%98%AF%E6%AC%A1%E5%BA%8F%E7%BB%9F%E8%AE%A1%E9%87%8F%E3%80%82%0A%0A%23%23%23%23%204.%E4%BB%80%E4%B9%88%E6%98%AF%E5%85%85%E5%88%86%E7%BB%9F%E8%AE%A1%E9%87%8F%0A%3E%E5%9C%A8%E5%AE%9E%E9%99%85%E7%BB%9F%E8%AE%A1%E6%8E%A8%E6%96%AD%E4%B8%AD%EF%BC%8C%E6%98%AF%E7%94%A8%E7%BB%9F%E8%AE%A1%E9%87%8F%E7%9A%84%E5%80%BC%E8%BF%9B%E8%A1%8C%E5%88%A4%E6%96%AD%EF%BC%8C%E8%80%8C%E4%B8%8D%E6%98%AF%E7%94%B1%E6%A0%B7%E6%9C%AC%E8%A7%82%E6%B5%8B%E5%80%BC%E8%BF%9B%E8%A1%8C%E6%8E%A8%E6%96%AD%E3%80%82%E8%BF%99%E5%B0%B1%E6%84%8F%E5%91%B3%E7%9D%80%EF%BC%8C%E7%BB%8F%E8%BF%87%E5%8A%A0%E5%B7%A5%E5%90%8E%EF%BC%8C%E6%9C%89%E4%BA%86%E7%BB%9F%E8%AE%A1%E9%87%8F%E7%9A%84%E5%80%BC%E5%8D%B3%E5%8F%AF%E3%80%82%E5%9C%A8%E7%BB%9F%E8%AE%A1%E5%AD%A6%E4%B8%AD%EF%BC%8C%E5%81%87%E5%A6%82%E4%B8%80%E4%B8%AA%E7%BB%9F%E8%AE%A1%E9%87%8F%E8%83%BD%E6%8A%8A%E5%90%AB%E5%9C%A8%E6%A0%B7%E6%9C%AC%E4%B8%AD%E6%9C%89%E5%85%B3%E6%80%BB%E4%BD%93%E7%9A%84%E4%BF%A1%E6%81%AF%E4%B8%80%E7%82%B9%E9%83%BD%E4%B8%8D%E6%8D%9F%E5%A4%B1%E5%9C%B0%E6%8F%90%E5%8F%96%E5%87%BA%E6%9D%A5%EF%BC%8C%E9%82%A3%E5%AF%B9%E4%BF%9D%E8%AF%81%E5%90%8E%E8%BE%B9%E7%9A%84%E7%BB%9F%E8%AE%A1%E6%8E%A8%E6%96%AD%E8%B4%A8%E9%87%8F%E5%85%B7%E6%9C%89%E9%87%8D%E8%A6%81%E6%84%8F%E4%B9%89%E3%80%82%E7%BB%9F%E8%AE%A1%E9%87%8F%E5%8A%A0%E5%B7%A5%E8%BF%87%E7%A8%8B%E4%B8%AD%E4%B8%80%E7%82%B9%E4%BF%A1%E6%81%AF%E9%83%BD%E4%B8%8D%E6%8D%9F%E5%A4%B1%E7%9A%84%E7%BB%9F%E8%AE%A1%E9%87%8F%E7%A7%B0%E4%B8%BA%E5%85%85%E5%88%86%E7%BB%9F%E8%AE%A1%E9%87%8F%E3%80%82%0A%0A%23%23%23%23%205.%E4%BB%80%E4%B9%88%E6%98%AF%E8%87%AA%E7%94%B1%E5%BA%A6%0A%3E%E8%87%AA%E7%94%B1%E5%BA%A6%E6%98%AF%E7%BB%9F%E8%AE%A1%E5%AD%A6%E4%B8%AD%E5%B8%B8%E7%94%A8%E7%9A%84%E4%B8%80%E4%B8%AA%E6%A6%82%E5%BF%B5%EF%BC%8C%E5%AE%83%E5%85%B7%E6%9C%89%E8%A7%A3%E9%87%8A%E4%B8%BA%E7%8B%AC%E7%AB%8B%E5%8F%98%E9%87%8F%E7%9A%84%E4%B8%AA%E6%95%B0%EF%BC%8C%E8%BF%98%E5%8F%AF%E4%BB%A5%E8%A7%A3%E9%87%8A%E4%B8%BA%E4%BA%8C%E6%AC%A1%E5%9E%8B%E7%9A%84%E7%A7%A9%E3%80%82%0A%0A%23%23%23%23%206.%E7%AE%80%E8%BF%B0%20x2%E5%88%86%E5%B8%83%E3%80%81t%E5%88%86%E5%B8%83%E3%80%81F%E5%88%86%E5%B8%83%E5%8F%8A%E6%AD%A3%E6%80%81%E5%88%86%E5%B8%83%E4%B9%8B%E9%97%B4%E7%9A%84%E5%85%B3%E7%B3%BB%0A%3E1.%24X%5E2%24%E5%88%86%E5%B8%83%E5%BD%93%24X%24%E8%B6%8B%E8%BF%91%E4%BA%8E%E6%97%A0%E7%A9%B7%E5%A4%A7%E6%97%B6%EF%BC%8C%E5%88%86%E5%B8%83%E7%9A%84%E6%9E%81%E9%99%90%E5%88%86%E5%B8%83%E6%98%AF%E6%AD%A3%E6%80%81%E5%88%86%E5%B8%83%0A%3E2.t%E5%88%86%E5%B8%83%E7%9A%84%E5%AF%86%E5%BA%A6%E5%87%BD%E6%95%B0%E6%98%AF%E4%B8%80%E5%81%B6%E5%87%BD%E6%95%B0%0A%3E3.F%E5%88%86%E5%B8%83%E4%B8%AD%EF%BC%8C%E4%B8%A4%E4%B8%AA%E8%87%AA%E7%94%B1%E5%BA%A6%E7%9A%84%E4%BD%8D%E7%BD%AE%E4%B8%8D%E5%8F%AF%E4%BA%92%E6%8D%A2%E3%80%82%E5%A6%82%E6%9E%9C%E9%9A%8F%E6%9C%BA%E5%8F%98%E9%87%8FX%E6%9C%8D%E4%BB%8E%24t(n)%24%E5%88%86%E5%B8%83%EF%BC%8C%E5%88%99%24X%5E2%24%E6%9C%8D%E4%BB%8EF(1%2Cn)%E7%9A%84F%E5%88%86%E5%B8%83%E3%80%82%0A%0A%23%23%23%23%207.%E4%BB%80%E4%B9%88%E6%98%AF%E6%8A%BD%E6%A0%B7%E5%88%86%E5%B8%83%EF%BC%9F%0A%3E%E4%B8%BA%E4%BA%86%E5%88%A4%E6%96%AD%E6%80%BB%E4%BD%93%E7%9A%84%E6%9F%90%E4%BA%9B%E7%89%B9%E5%BE%81%EF%BC%8C%E9%9C%80%E8%A6%81%E4%BB%8E%E6%80%BB%E4%BD%93%E4%B8%AD%E6%8C%89%E4%B8%80%E5%AE%9A%E7%9A%84%E6%8A%BD%E6%A0%B7%E6%8A%80%E6%9C%AF%E6%8A%BD%E5%8F%96%E8%8B%A5%E5%B9%B2%E4%B8%AA%E4%B8%AA%E4%BD%93%EF%BC%8C%E5%B0%86%E8%BF%99%E4%B8%80%E6%8A%BD%E5%8F%96%E8%BF%87%E7%A8%8B%E7%A7%B0%E4%B8%BA%E6%8A%BD%E6%A0%B7%E3%80%82%0A%3E%E6%8A%BD%E6%A0%B7%E5%88%86%E5%B8%83%E5%9C%A8%E6%80%BB%E4%BD%93%24X%24%E7%9A%84%E5%88%86%E5%B8%83%E7%B1%BB%E5%9E%8B%E5%B7%B2%E7%9F%A5%E6%97%B6%EF%BC%8C%E8%8B%A5%E5%AF%B9%E4%BB%BB%E6%84%8F%E8%87%AA%E7%84%B6%E6%95%B0n%EF%BC%8C%E9%83%BD%E8%83%BD%E5%AF%BC%E5%87%BA%E7%BB%9F%E8%AE%A1%E9%87%8F%24T%3DT(X_1%2CX_2%2C...X_n)%24%E7%9A%84%E5%88%86%E5%B8%83%E7%9A%84%E6%95%B0%E5%AD%A6%E8%A1%A8%E8%BE%BE%E5%BC%8F%EF%BC%8C%E8%BF%99%E7%A7%8D%E5%88%86%E5%B8%83%E7%A7%B0%E4%B8%BA%E7%B2%BE%E7%A1%AE%E7%9A%84%E6%8A%BD%E6%A0%B7%E5%88%86%E5%B8%83%E3%80%82%0A%0A%23%23%23%23%208.%E7%AE%80%E8%BF%B0%E4%B8%AD%E5%BF%83%E6%9E%81%E9%99%90%E5%AE%9A%E7%90%86%E7%9A%84%E6%84%8F%E4%B9%89%0A%3E**%E4%B8%AD%E5%BF%83%E6%9E%81%E9%99%90%E5%AE%9A%E7%90%86**%E8%AE%BE%E4%BB%8E%E5%9D%87%E5%80%BC%E4%B8%BA%24%5Cmu%24%E3%80%81%E6%96%B9%E5%B7%AE%E4%B8%BA%24%5Csigma%5E2%24%EF%BC%88%E6%9C%89%E9%99%90%EF%BC%89%E7%9A%84%E4%BB%BB%E6%84%8F%E4%B8%80%E4%B8%AA%E6%80%BB%E4%BD%93%E4%B8%AD%E6%8A%BD%E5%8F%96%E6%A0%B7%E6%9C%AC%E9%87%8F%E4%B8%BAn%E7%9A%84%E6%A0%B7%E6%9C%AC%EF%BC%8C%E5%BD%93n%E5%85%85%E5%88%86%E5%A4%A7%E6%97%B6%EF%BC%8C%E6%A0%B7%E6%9C%AC%E5%9D%87%E5%80%BCX%E7%9A%84%E6%8A%BD%E6%A0%B7%E5%88%86%E5%B8%83%E8%BF%91%E4%BC%BC%E6%9C%8D%E4%BB%8E%E5%9D%87%E5%80%BC%E4%B8%BA%24%5Cmu%24%E3%80%81%E6%96%B9%E5%B7%AE%E4%B8%BA%24%5Cfrac%7B%5Csigma%5E2%7D%7Bn%7D%24%E7%9A%84%E6%AD%A3%E6%80%81%E5%88%86%E5%B8%83%E3%80%82%0A%0A%23%23%23%20%E7%BB%83%E4%B9%A0%E9%A2%98%0A%23%23%23%23%20%E8%B0%83%E8%8A%82%E4%B8%80%E4%B8%AA%E8%A3%85%E7%93%B6%E6%9C%BA%E4%BD%BF%E5%85%B6%E5%AF%B9%E6%AF%8F%E4%B8%AA%E7%93%B6%E5%AD%90%E7%9A%84%E7%81%8C%E8%A3%85%E9%87%8F%E5%9D%87%E5%80%BC%E4%B8%BA%CE%BC%E7%9B%8E%E5%8F%B8%EF%BC%8C%E9%80%9A%E8%BF%87%E8%A7%82%E5%AF%9F%E8%BF%99%E5%8F%B0%E8%A3%85%E7%93%B6%E6%9C%BA%E5%AF%B9%E6%AF%8F%E4%B8%AA%E7%93%B6%E5%AD%90%E7%9A%84%E7%81%8C%E8%A3%85%E9%87%8F%E6%9C%8D%E4%BB%8E%E6%A0%87%E5%87%86%E5%B7%AE%201.0%E7%9B%8E%E5%8F%B8%E7%9A%84%E6%AD%A3%E6%80%81%E5%88%86%E5%B8%83%E3%80%82%E9%9A%8F%E6%9C%BA%E6%8A%BD%E5%8F%96%E7%94%B1%E8%BF%99%E5%8F%B0%E6%9C%BA%E5%99%A8%E7%81%8C%E8%A3%85%E7%9A%849%E4%B8%AA%E7%93%B6%E5%AD%90%E5%BD%A2%E6%88%90%E4%B8%80%E4%B8%AA%E6%A0%B7%E6%9C%AC%EF%BC%8C%E5%B9%B6%E6%B5%8B%E5%AE%9A%E6%AF%8F%E4%B8%AA%E7%93%B6%E5%AD%90%E7%9A%84%E7%81%8C%E8%A3%85%E9%87%8F%E3%80%82%E8%AF%95%E7%A1%AE%E5%AE%9A%E6%A0%B7%E6%9C%AC%E5%9D%87%E5%80%BC%E5%81%8F%E7%A6%BB%E6%80%BB%E4%BD%93%E5%9D%87%E5%80%BC%E4%B8%8D%E8%B6%85%E8%BF%870.3%E7%9B%8E%E5%8F%B8%E7%9A%84%E6%A6%82%E7%8E%87%E3%80%82%0A%E6%80%BB%E4%BD%93%E6%9C%8D%E4%BB%8E%E6%AD%A3%E6%80%81%E5%88%86%E5%B8%83%EF%BC%8C%E6%A0%B7%E6%9C%AC%E5%9D%87%E5%80%BC%E4%B9%9F%E6%9C%8D%E4%BB%8E%E6%AD%A3%E6%80%81%E5%88%86%E5%B8%83%EF%BC%8C%E4%B8%94%E5%9D%87%E5%80%BC%E7%9B%B8%E5%90%8C%EF%BC%8C%E6%96%B9%E5%B7%AE%E7%AD%89%E4%BA%8E%E6%80%BB%E4%BD%93%E6%96%B9%E5%B7%AE%E9%99%A4%E4%BB%A5%E6%A0%B7%E6%9C%AC%E9%87%8F%EF%BC%8C%24P%3D0.9%24%0A

 

来源：https://www.cnblogs.com/pandaboy1123/p/10690017.html