【统计学】第六章

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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。

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