7.3.1 Interval estimation: basic principles (1) 区间估计：基本原理（一）在线视频

大家好
Hello, everyone

欢迎回到轻松学统计课堂
Welcome back to the Easy Learning Statistics Class

接下来我们介绍区间估计的
Now, we will introduce the basic principles

基本原理
of interval estimation

上一节我们介绍了
In the last lecture

点估计的有关知识
we talked about point estimation

虽然点估计非常地简单 方便
Although point estimation is very simple, convenient

容易操作
and easy to operate

但是它也存在着
it also exists

非常明显的缺点
very obvious disadvantages

第一个缺点是
The first disadvantage is that

它不能够帮助我们反映
it doesn't help us reflect

这一次估计的误差有多大
how much the error of the estimate is

因为我们直接用的
Because we use directly

就是某一次观测的具体的值
the value of a particular observation

去代表了总体的参数值
to represent the parameter value of the population

那误差有多大是不能反映的
The magnitude of the error is not reflected

其实我们知道某一次抽样
Well, we know that between a sample

和总体参数之间
and the population parameter

必然会存在着或大或小的误差
there must be errors, big or small

第二个
Second

既然它不能反映估计的误差
since it doesn't reflect the error of the estimates

它也无法反映估计的精确程度
nor does it reflect the accuracy of the estimates

或者叫准确程度
or the degree of accuracy

我到底是很接近总体的真实值
Whether it is close to the real value of the population

还是离总体的真实值很远
or it is far from the true value of the population

这些我都没有办法来反映
cannot be reflected

所以点估计
So for point estimation

它在进行操作的时候
in practical operation

有两个非常明显的缺点
there are two very obvious disadvantages

那接下来我们就来介绍
Next, we’ll introduce

区间估计的基本原理
the basic principle of interval estimation

所谓区间估计
So called interval estimation

是利用样本统计量和抽样分布
uses sample statistics and sampling distribution

来估计总体参数的可能区间
to estimate the possible range of population parameters

这个方法是1934年的时候
This method is a theory of interval estimation

由统计学家奈曼所创立的
developed by statistician Naiman

一种区间估计的理论
in 1934

在正式开始介绍区间估计的
Before we begin the formal introduction to the basic principles

基本原理之前
of interval estimation

我想和大家先来做一个游戏
I want to play a game with you first

这个游戏我们日常生活里边
This game may also be encountered

可能也会遇到
in our daily life

比如我下面请大家来猜猜
For example, I'm going to ask you to guess

我今天来上课
When I come to the class today

包里带了多少钱
how much money do I have in my bag

那我想你得到这个命题以后
I think after you get this proposition

你就开始来想
you will begin to think

老师今天包里会带多少钱呢
how much money the teacher has in his bag today

有同学可能就会想
Some students my think

老师 508块
it is 508 yuan

或者同学给出299块
or some students may think it is 299 yuan

等等这样的一些数字
or other figures

都可能是我今天包里所带的钱
Any of these could be the money I have in my bag today

但是我想这个时候你自己心里
But I don't think you have much confidence

也没多少底
in yourself at the moment

你给出来的数字能猜中
Can your figure guess correctly

我今天包里带的钱吗
the money I have in my bag today

你直接给出一个数字
Just give me a number and guess how much money I have in my bag

来猜我包里有多少钱
To guess how much money I have in my bag

这就相当于是点估计
Is equivalent to point estimation

用一个值去估计另一个值
Use one value to estimate another

那有一些同学
Some students

可能是另外的形式
may use other forms

来猜我包里带了多少钱
To guess how much money I have in my bag

他可能是
he may do it

以一个区间的形式来给出的
by giving an interval

比如你可能给出一个区间
For example, you might give an interval

0到1万块
from 0 to 10,000 yuan

这个时候你可能信心十足地说
At this point you may say with confidence

老师 我有百分之百的把握
Teacher: I'm 100 percent sure that

今天你来上课
when you give the lecture, today

包里不会超过一万块钱
you have no more than 10,000 yuan in your bag

恭喜你 答对了
Congratulation, your answer is right

老师来上个课
When I give the lecture

真的不会带一万块钱在包里
I really don't carry 10,000 yuan in my bag

那这个时候我肯定就说
I'll say it then

你给的区间这么宽
the interval you gave is so wide

当然就可以
which, of course

把我包里带的多少钱给加（包括）进去
can include the money in bag

我说你给的区间宽的意思是
What I mean by saying your interval is wide is that

你的结果不准确
your result is not accurate

如果我要你提高准确度
If I want you to improve accuracy

那这个时候你可能
you may

就想把区间要缩短一点点
qant to shorten the interval a little bit

那你就可能给出另外一个区间
Then you might give me another interval

老师 我猜你今天包里带的钱
Teacher: I guess the money in your bag today

在50到500块之间
is between 50 and 500 yuan

对 这一次比0到1万块的区间
Compared with the interval from 0-10,000 yuan

要短很多 要准确得多了
this interval is much shorter and much more accurate

但是这一次你还敢说
But this time, do you dare to say

你有100%的把握吗
you are 100% sure

估计你可能信心
You may be confident, but

就没有那么充足了
You are less sure

你可能只敢说
You might just say

老师 我大概只有90%的把握
Teacher: I'm only about 90% sure

你今天包里带的钱
that the money in your bag today

在50到500块之间
is between 50 and 500 yuan

这是一个日常生活里边
This is a small game

可能经常会玩的小游戏
that may be played in everyday life

其实通过这个小游戏
Actually through this little game,

我们基本上面可以
we can basically

用它来学习区间估计的
learn the basic principles of

基本原理
interval estimation

我们看到给出一个区间的时候
We see that when we give an interval

我们需要给出一个区间的下限
we need to give a lower limit and

一个区间的上限
and an upper limit of the interval

以及和它对应的你的信心水平
and your level of corresponding confidence

你有100%的把握
100% sure

还是只有90%的把握
or only 90% sure

所以我们接下来
Next

利用样本的信息
when we use the information of the sample

去估计总体的可能区间的时候
to estimate the possible interval of the population

我们可能也涉及到
we may involve

这样几个内容
such contents

这个区间的上端点
As the upper end of this interval

这个区间的下端点
the lower end of the interval

以及和这个区间对应的
and your level of confidence

你的信心程度
corresponding to the interval

在刚才的游戏里边
in the game we played just now

你们可能觉得
you may think

有一点点不太容易
it is a little hard to

猜我包里钱的
guess how much money you have in your bag

这个游戏有一点点难度
The game is a little difficult

原因是你没有获得过
Because you have no information

我的行为习惯的有关信息
about my behavior

比如你通过某些信息了解到
Let's say you learn something

我一般的行为习惯
about my general behavior

我在出来上课的时候
the habit of how much money I carry in my bag

我包里带多少钱的习惯
when I go out to the class

或者我出去购物的时候
or when I go shopping

我一般包里带多少钱
how much money do I usually carry in my bag

如果你已经获得过
if you already have

相关的这些信息的话
the related information

那么我想你给出
I think it might be much easier

我今天包里带多少钱的
for you to give the interval of

这个区间
how much money I carry today

可能要容易得多
in my bag

这个就是
That's why

我们为什么要抽样的原因
we do sampling

因为我们通过抽样
Because by sampling

是可以获得样本的有关信息
you may get Information about the sample

而在它的基础上
and on its basis

接下来去估计总体的参数
you can estimate the parameters of the population

来构造这个区间
so as to construct the interval

所以经过这个游戏环节
So through this game link

我想我们大概可以把区间估计
I think we can sort out the information that

所要用到的信息捋一捋
will be used in interval estimation

第一 我们可能需要
First, we may need

有样本的有关信息
information about the sample

这个我们刚才前面已经铺垫过
We've already talked about this

估计值或者估计量
estimates or estimators

第二个
Second

我需要通过一定的方法
I need a certain method

把区间的下限和上限
to calculate out the lower limit and upper limit

Of the interval

第三个跟它对应的
Third, we may also need

可能还有一个信息
the corresponding information

就是信心程度
That is confidence degree

接下来我们通过一个例子
Next we’ll use an example

来描述区间估计的基本原理
to describe the basic principle of interval estimation

这个例子里边说的是
This example is about

Traveller公司
Traveller Company

是一家专门生产
which is a specialist manufacturer

旅行箱包的公司
of travel bags and suitcase

为了监控公司的产品质量
To monitor the quality of the company's products

那么这家公司每个月
this company takes

都要随机地抽取一个顾客样本
a random sample of customers every month

进行调查
to investigate

来了解顾客的满意分数
customer satisfaction scores

根据以往的调查
According to the past surveys

满意分数的标准差
the standard deviation of the satisfaction score

稳定在12分左右
is stable at about 12 points

这里先停一下
Let's stop here first

大家注意到
You notice that

这里出现了一个标准差为12分
there is a standard deviation of 12 points

那这里边我想先考一下大家
I want to test you first

那从轻松学统计的课堂
we have studied statistics

课堂学习到现在
in this class up to now

我们知道标准差
We know the standard deviation

有三个不同的形式
have three different forms

第一 总体标准差
First, population standard deviation

第二 样本标准差
Second, sample standard deviation

第三 抽样标准差
Third, sampling standard deviation

那么这里
Then, it is said here that

写了一个根据以往的调查
according to the previous investigation

满意分数的标准差稳定
the standard deviation of the satisfaction score is stable

在12分左右
at about 12 points

这个12
For this 12

我如果要用一个相应的符号
if I am to use a corresponding notation

把它表达出来
to express it

我应该选哪一个呢
which should I select

我是选σ=12
σ=12

还是选S=12
S=12

还是选σ除以根号n等于12呢
or σ over the square root of n is equal to 12

这个信息大家一定要
You should make clear about

先把它弄清楚
this information

那么这里边我们告诉大家
Here, we tell you

一个基本的原则
a basic principle

就是如果出现了
That is, if there is description

根据以往的调查的字眼
based on the previous survey

我们所显示的信息
then the related information

所代表的都是总体的标准差
represents the standard deviation of the population

也就是说这个12分
That is to say, for the 12 points

我们可以用σ=12分
we may use σ =12 points

来表达相关的内容了
to express the relevant content

也就是说这一次总体的
This is to say for this population

标准差是已知的
the standard deviation is given

最近一次49名顾客的
Then, it is said according to the recent sampling of

抽样显示
49 customers

那这里
Here

我们才有最新一次的抽样出现
the most recent sampling occurs

所以刚才前面的12分
Therefore, the previous 12 points

一定不是样本标准差
Must not be the sample standard deviation

那最新一次抽样
In the most recent sampling

这个49名
49 customers

也就是我们的样本容量
are the sample size

那我们可以
Then, we may

用n=49来表示了
use n=49 to express

满意分数的样本均值为82分
The sample mean of the satisfaction score is 82 points

那这个显示的
So

就是X_bar等于82了
X_bar is equal to 82

有了这些信息
With such information

接下来要我们来建立
we can set up

总体满意分数的区间
the interval of the satisfaction scores of the population

也就是要给它一个区间估计
That is to make an interval estimation

根据咱们刚才游戏环节的分析
According to our analysis of the game

我们需要准备一个信心程度
we need to prepare a level of confidence

第二 样本的信息我们已经有了
Second, we already have the information

第三个 想办法把区间的
Third, we need to figure out

上限和下限给它计算出来
the upper and lower limits of the interval

区间的上限和下限
For the upper and lower limits of the interval

怎么样来计算
How to calculate them

这里边
There is

有一个非常技巧性的环节
a very skilled link

首先我们可以想像
First we can imagine

如果对于区间的长度
if, for the length of the interval

没有任何要求的话
there's no requirement

那这样的区间
then, for such interval that

以样本信息为基础的区间
is based on sample information

可以有无穷多个
there could be an infinite number of them

因为我可以是0到1万
I can take an interval from 0 to 10,000

这不是有1万块的区间嘛
so this is just that interval of 0- 10,000 yuan

我也可以是50到500
I can set an interval from 50 to 500

那这个区间的话
Then, this interval

它的长度有450块钱
has a length of 450 yuan

又或者我是从0到500
Or I may set the interval to be from 0 to 500

那它又换了一个区间
This is another interval

因此如果区间的长度
Therefore, if the interval length

没有一个约束的话
is not restricted

我们可以构造的区间
we may construction

有若干多个
thousands of intervals

所以首先第一步
Therefore, in the first step

我们可能对于区间的长度
we may set a requirement

会有一个要求
on the length of interval

比如我们在刚才的
For example, in the

这个游戏环节里边
game link we played just now

我可能会附加一个条件
I might attach a condition

因为我知道你给出的区间的
Because I know that there's an infinite number of possibilities

可能性有无穷多种
you may give for the interval

所以我就会可能附加一个条件
So I may attach a condition that

正负五十块都算你对
50 yuan more or less is right

那我说正负五十块都算你对
When I say 50 yuan more or less is right

实际上我就给了一个条件给你
I gave you a condition

就是区间的长度是100块
That is the length of the interval is 100 yuan

比我少50块我也算你对
If it is 50 yuan more, you are right

比我多50块我也算你对
If it is 50 yuan less, you are also right

这种方法在我们日常生活里边
This method is very common

如果你留意过
in our daily life

非常地常见
If you notice

在一些产品质量标志里边
In some product quality signs

非常地常见
it is very common

比如我们如果是去买一些
If we go to buy some

袋装食品的时候
bagged foods

它的袋装上面
on its package bag

它的袋子上面可能标注3公斤加减3%
It may be labelled that the weight is 3 kilograms plus or minus 3%

3公斤加减3%
3 kilograms plus or minus 3%

那就告诉我们
tells us that

这一袋产品的重量
the weight of this bag of product

在3公斤加减0.09公斤的
is in the range of

范围之内波动的
3 kilograms plus or minus 0.09 kilogram

又或者
Or

我们去买一些容器的时候
when we go to buy some containers

这个容器它可能标明它的自重
this container might indicate its dead weight

它说它的自重是90.5加减5克
It says its weight is 90.5 plus or minus 5 grams

这个也给出了一个区间
This also gives an interval

这个区间的长度的话
The length of this interval

就是10克了
is 10 grams

所以这种方法非常地常见
So it's very common method

就是我们可能
We may

只能够寻找一个特殊的区间
only look for a particular interval

那这个区间的长度是固定的
the length of which is fixed

当然对我们来讲
Of course, to us

即使退一步
even a step back

区间的长度是固定的
the length of the interval is fixed

如果以82分为基础
If we take 82 points as base

来构建一个固定长度的区间
to construct a fixed length interval

其实还是有若干多种可能
there are still many possibilities

我只要保证它的区间长度
As long as the interval length is guaranteed

比如我举一个例子
Let me give you an example

这个区间长度是10分
The interval length is 10 points

我只要以82分为基础
and I only need to take 82 points as basis

我的总长度是10分
The total length is 10 points

那么我的区间
Then, there can still be

还是可以有任意多个
a number of possibilities

因为我的左右的端点
Because my left and right endpoints

可以做调整
can be adjusted

我可以是80分到90分
It can be between 80 to 90 points

那这个是以82分为基础的
It is an interval that is based on

一个区间吧
82 points

我左边减2分 右边加8分
I subtract 2 points from the left side and add 8 points to the right side

这不是一个区间嘛
And this is also an interval

我也可以是75分到85分
It can also between 75 points to 85 points

这个区间长度也是10分
This interval length is also 10 points

我也是以82分为基础
And it is also based on 82 points

我左边减 右边加
I subtract from the left and add to the right

只不过左边和右边减的
Only that subtracted from the left is not equal to that

和加的不相等而已
added to the right

因此即使退一步
Therefore, even one step back

区间长度固定下来
When the interval length is fixed

以样本信息为基础的区间
and based on sample information

仍然还是有若干多个
there are still a number of possibilities

所以接下来
Therefore, the interval

我们所要构建的区间
we will construct next

一定是一个特殊的区间
must be a special interval

刚才前面举到的例子
In the previous example

产品的质量标识
the product quality label

比如3公斤加减3%
such as 3 kilograms minus or plus 3%

比如90.5加减5克
and 90.5 minus or plus 5 kilograms

这样的产品标识
Such product label

给我们接下来构造区间
gives us a good idea

提供了一个非常好的启发
for constructing the interval

就是最好的
That is the best

或者是说最常用的区间
Or most commonly used interval

它是以某个数为基础对称的区间
should be the symmetric interval

为基础对称的区间
based on a certain figure

那正负5就表明左右两边是对称的了
So plus or minus 5 means that the left and right sides

是对称的了
are symmetric

也只有这样的区间
Only such interval

才能够通过我们接下来的方法
by our following method

把它以样本信息为基础
and based on sample information

给构造出来
can be constructed

所以在我们构造总体均值的
So in the process of constructing the interval

区间的过程里边
of the population mean

一定是有这样的一个环节
there must be such a link

找到的一定是
To find

一个非常特殊的区间
a special interval that

是以样本信息为基础的
is based on sample information

对称的区间
is Symmetrical interval