7.3.2 Interval estimation: basic principles (2) 区间估计：基本原理（二）在线视频

好 所以现在我们已经有了
Now, we have the information

样本均值是82分
that the sample mean is 82 points

那以它为中心对称的区间
and the interval symmetric with it as the center

如果我能够把区间的长度
Then, how to figure out

计算出来的话
the length of the interval

那就左边和右边各加减一个
Add or subtract one half of the length

长度的一半
on the left and the right

我们就可以想办法
So that we can have a method

把它计算出来了
to calculate it

当然接下来就是
And then, of course

一个非常关键的问题
a very critical issue is to

就是帮助我们来计算
help us calculate

这个左边和右边加减的
the figure to be

这个数字
subtracted or added on the left and right

我们通常把这个数字
We usually call this figure

称之为抽样误差
as sampling error

那抽样误差它反映的
The sampling error reflects

是一个无偏估计
the absolute value of the difference

和它所对应的总体参数之差的绝对值
Between an unbiased estimate and its corresponding population parameter

如果用符号来表示的话
If you express it by notation

我们通常会用（公式如上）
ee usually use the (formula above)

因为它在左边也行
As it can be put on the left

在右边也行
or on the right

把这个式子绝对值符号去掉
So let's remove the absolute value sign

把它两边移项整理
and do transposition of the items on both sides

非常快地
Very quickly

我们就得到变形后的式子
we get the deformed formula

（公式如上）
(As above)

那也就说
That is to say

从这个式子里边来看的话
from this formula

我们基本上面看到了
we basically see

跟商品质量标识类似的
an expression similar to

一个表达
commodity quality label

出现了X_bar加减某个数字的
An expression for X_bar plus or minus a figure

一个表达式
appears

那当然我们看到商品的
Of course when we see the quality label

质量标识的时候
of a commodity

我们并不会认为
We don't think

这个塑料容器的重量
the weight of this plastic container

是90.5减5
is 90.5 minus 5

也就是85.5克 或者是95.5克
or 85.5 grams or 95.5 grams

我们并不会这么认为
We don't think so

我们看到这样的信息以后
After we see this information

我们会认为
we would think that

这个塑料容器的自身重量
the weight of the plastic container itself

会落在85.5到95.5
falls in an interval

这个区间之内
between 85.5 and 95.5

所以这个变形后的
So this deformed

（公式如上）
formula (as above)

就是我们接下来要表达的
is the expression use to express

这个区间的上下的
the upper and lower ends

两个端点的表达式了
of the interval

这个式子里边X_bar已经有了
We already have X_bar in this formula, and

唯一还待求解的
The only one left to be solved

就是（符号如上）
is (notation as above)

那接下来我们就要想办法
So next, we will try

把（符号如上）给它计算出来
to figure out the (notation as above)

现在（符号如上）有多大
How big is the (notation as above)

我没有计算出来
I have not calculated it

比如我假定
Let’s say I assume

它是这么长的一个距离
it is a distance with this length

那以这么长的一个距离
Then, in the distance with this length

以X_bar为中心
taking X-bar as center

来构造一个区间
let’s construct an interval

它就会变成这么长
It's going to be this long

那根据我们前面的学习
According to what we learned earlier

X_bar是一个随机变量
bar is a random variable

它是一个随机变量
As it's a random variable

因此它一定有一个抽样分布
it must have a sampling distribution

还记得X_bar的抽样分布
Do you remember what the sampling distribution

是什么吗
of bar is

回忆回忆我们上一章抽样
Recall what we learned in the last chapter about sampling

和抽样分布的知识
and sampling distributions

我们知道在我们这个题目里边
We know that in this question

已经给定了大样本
s large sample is given

并且已经给定了
and the standard deviation of the population

总体的标准差已知
is also known

根据这些信息
According to this information

我们可以判断X_bar
we can judge X_bar

一定是服从正态分布的
must obey normal distribution

那如果有了X_bar
If we have X-bar, and

它是服从正态分布的
it obeys normal distribution

那总体标准差是等于12的
so the standard deviation of the population is equal to 12

样本容量是等于49的
the sample size is equal to 49

那根据中心极限定理
According to the central limit theorem

我们知道这个时候
we know at this time

样本均值服从的是
sample mean obeys

一个正态分布 它的均值
a normal distribution, and its mean

是等于总体的均值μ
is equal to the population mean μ

它的标准差是等于抽样标准差
Its standard deviation is equal to the sampling standard deviation

除以根号n
σ squared over the square root of n

把数字代进去
Substitute the number into it

12除以根号49等于1.71
12 divided by the square root of 49 is 1.71

的一个这样的正态分布
we get such a normal distribution

如果我们判断X_bar
If we judge that X-bar

是一个正态分布的话
is a normal distribution

那么接下来
Then

我们就可以看这张图片
we can look at this picture

那么在这张图片上面
On this picture

我们看到下面有一个横轴
we see that there's a horizontal axis down here

上面有一个对称的
It has a symmetrical

钟形的曲线
bell-shaped curve

那这个对称的钟形的曲线
So this symmetrical bell curve

就是X_bar所服从的正态分布了
is the normal distribution that X_bar obeys

当然正中间对应的
and of course, the center corresponds to

就是μ总体的均值
μ mean of the population

X_bar刚才我们说了
X_bar, as we just said

它是一个随机变量
is a random variable that

那它是围绕着μ左右波动的
moves around the μ

也就是说它可以
which means it can

在整个横轴上面
oscillating from left to right

从左到右都在波动的
on the whole horizontal axis

刚才我们假定Ex_bar
We just assumed Ex_bar

是等于这么长的一个距离
is equal to this long distance

那两倍的（符号如上）
The double (notation as above)

也就是总区间的长度
is the length of the total interval

大概就会有这么长
It would be about that long

这么长的一个区间
an interval with this length

随着X_bar的随机性
With the randomness of X bar

也会在我们整个数轴上
it will also oscillate from left to right

从左到右不停地波动
on the whole horizontal axis

那我们自己也可以在纸上面
Then we can draw and see

画一画 比划比划
on the paper ourselves

一比划我们就会发现
Then we will discover

这个区间在左右移动的过程里边
the interval, when moving left and right

并不总是能够
cannot always cover

把μ涵盖在里边的
the μ

在我们的图片上面
On our picture

假定这一段是
assume this section is

我们现在的（符号如上）
our present (notation as above)

我们可以发现
we can find that

当X_bar为中心的
when the interval with X-bar as center

固定长度的区间
and having a fixed length

在比较左边的位置的时候
is at the far left position

我们发现这个时候这个区间
we find that this interval at this time

并不能够把μ包含在里边
cannot cover the μ

当这个区间往右边挪
When the interval moves to the right

挪一点点
little by little

挪到什么位置
to which position

挪到它的右端点
to its right endpoint

恰好覆盖到的时候
When it just covers the μ

这个位置是
This position is

我们第一次从左到右
when we move it from left to right

能够覆盖到μ的区间
the interval capable of covering the μ

那这个位置我们给它定下来
we fix the position

假如这个位置的话
If this position

我们是μ左边（符号如上）的位置
is at the left of μ (notation as above)

那这个区间可以包含μ
and the interval can cover μ

这个区间继续往右边移
we continue to move the interval to the right

我们发现往右边移动的过程里边
and we see that as we move it to the right

这些区间都能够包含到μ
all these intervals can include μ

继续往右边移
Let's keep moving to the right

移到这个区间的左端点
When we move it to the left endpoint of the interval

恰好和μ重合的时候
and it just coincides with μ

我们把这个点记录下来
Let's record this point

它在μ的右边的（字符如上）的位置
It is at the left of μ (notation as above)

好 我们接下来把这个区间
OK, let’s continue to move

继续往右移
the interval rightward

我们发现
We find

当我们继续往右移的时候
as we continue to move it to the right

这个区间又不能包含μ了
this interval becomes unable to cover μ

也就是说由于X_bar
So that means because X_bar

是一个随机变量
is a random variable

其实以它为中心
The interval symmetrical

所构造的对称的区间
about it

也是一个随机的区间
is also a random interval

那这个随机的区间
This random interval

在移动的过程里边
in its movement process

有的能包含μ
may cover μ

有的不能包含μ
Or may not cover μ

当然 从研究者的角度来讲
From the researcher's point of view, of course

我们希望能包含μ的区间
we hop the intervals capable of covering μ

尽可能地多
are as many as possible

这样我的可靠程度才比较高
so that the degree of reliability will be high

我的把握程度才比较高
For the high degree of reliability

高到一个什么样的情况呢
Up to how high

高到一个什么样的情况
Up to how high

才是我们可以接受的呢
will it be acceptable to us

比如我们刚才猜钱的游戏里边
Like in the money guessing game we just played

我可以说我有100%的把握程度
I could say I'm 100 percent sure

我也可以说我有90%的把握程度
or I could say I'm 90 percent sure

但是如果你说你的把握程度
But if you say you are only

只有60%
60% sure

这个可能稍微偏低了一点
this is probably a little bit too low

所以对于我们来讲的话
So for us

我们希望这个区间
we want there are

能包含μ的区间尽可能地多
possibly more intervals that can cover μ

尽可能地多 多到什么程度呢
Possibly more, to what extent

这个尽可能地多
How does the possibly more intervals

它又是怎么样来刻画的呢
Be portrayed

显然在我们图片上面来看的话
Obviously in the picture above

这个区间它是落在μ的左边（符号如上）
such intervals are on the left of μ (notation as above)

和μ的右边（字符如上）的位置
and on the right of μ (notation as above)

在正态分布下面
Under normal distribution

这个区间对应
such intervals must correspond to

一定会有一个概率
a probability

跟它相对应
The corresponding

那这个概率通常情况下
probability is generally

我们把它描述为
described as

刚才我们也表达过
We said just now that

我们希望我们能包含μ的区间
we hope to have possibly more intervals

尽可能地多
capable of covering μ

也就是希望1-α的值
That's we want the value of 1-α

尽可能地大才比较好
to be possibly large

所以对我们来讲的话
Therefore

我们可能希望1-α达到90%
we might want 1-α to reach 90 %

或者达到95%
or reach 95%

这样的一个比较大的一个概率
If the probability is large

那我们估计的把握程度
the degree of sureness of estimation

就会比较高一些
will be higher

已知X_bar是个正态分布
We know that X_bar is a normal distribution

如果我们又已知
If we also know

中间对称区间的面积
the area of the symmetric interval in the center

1-α是95%的话
1-α is 95%

那么接下来我们就可能通过
then, we may use

正态分布的有关知识
the knowledge related to normal distribution

把Ex_bar求解出来
to solve Ex_bar

我们来看一下
Let’s see

那由于X_bar是服从均值为μ
Since X_bar is a normal distribution where the mean is constant

方差为1.71的平方的正态分布
and the variance is 1.71 squared

那么就有下面的关系式成立
the following relationship holds

下面这个关系式要表达的是
What the following relation is to express is

当X（字符如上）和μ的误差
When the error between X (notation above) and μ

小于等于（字符如上）
less than or equal to (notations as above)

用1-α来表示
it is expressed by 1-α

通常把称之为置信水平
1-α is usually referred to as the confidence level

或者可靠程度都可以
or reliability

它反映的是估计结果的可信程度
It reflects how credible

可信程度
the estimation result is

那这是在一般的正态分布
This is in the case of general normal distribution

下面的概率表达式
The following probability expression

我们可以稍做变化
can be changed a little

把它变成标准的正态分布
to make it for standard normal distribution

在下面这个式子里边
In the following formula

我们可以通过对刚才
we may

这个概率式子
divide the left and right

左右两边同时除以（符号如上）
of the probability formula by (notations as above)

就可以把刚才的
to transform the

一般的正态分布
general normal distribution

转换成标准的正态分布
into standard normal distribution

那也就是说
That is to say

有下面的这个式子成立
the following formula holds

（公式如上）
(The formula is as above)

这是一个标准的正态分布
This is a general expression

它的区间和对应的概率
For the intervals and corresponding probability

一般的表达式
of standard normal distribution

如果事先给定了一个置信度
if a confidence is given in advance

那么我们就可以根据
we may, according to

标准正态分布表
the standard normal distribution table

找到它对应的临界值
to find its critical value

（公式如上）
(Formula as above)

进而我们就可以计算
and then we can calculate

抽样误差（字符如上）
sampling error (notations as above)

大家还会查标准正态分布表
Can you look at the standard normal distribution table

得到它对应的临界值吗
to find its corresponding critical value

那这个内容
This content

我们前面已经学习过
was taught before

希望大家能够利用
Hope you can use

标准正态分布表
the standard normal distribution table to

找到相应的临界值
find the corresponding critical value

那如果我们能够找到
If we can find

（公式如上）
(Formula as above)

那（字符如上）非常快的
that (notation as above) can be quickly

就可以用（公式如上）
Be expressed by the

来表达了
(Formula as above)

那这样的话
In this way

我们就找到了一种方法
we find a method

帮助我们
to help us

把那一段误差计算出来
calculate the error at that section

那在计算这一段误差的时候
In calculating the error of this section

我们利用了
we use

一个非常重要的信息
very important information

就是X_bar的抽样分布
It's the sampling distribution of X_bar

那接下来我们继续
So let's move on

来计算Ex_bar
to calculate Ex bar

根据上面例题里边给定的信息
according to the information given in the example above

如果1-α=95%
If 1-α=95%,

查标准正态分布表
Look up the standard normal distribution table and

可以得到（公式如上）
we can get (formula as above)

是等于1.96的
is equal to 1.96

那把数字代入进去
Then, substitute the number

（公式如上）
(Formula as above)

那（符号如上）刚才
Then (notation as above)

我们也已经计算它是等于1.71
We've calculated that it's equal to 1.71

那1.96乘以1.71的话等于3.35
So 1.96 times 1.71 is equal to 3.35

也就是说
That is to say

我把刚才我手势表达的
I've calculated the little distance

这一小段距离给计算出来了
that I just indicated with my hand

它就等于3.35
That's equal to 3.35

在我给定的这些条件下面
Under these conditions that I've given you

有了这一小段距离
when we have this little distance

那区间的下限和区间的上限
the lower end and the upper end of the interval

就可以算出来了
can be calculated

区间的上限
The upper limit of the interval

当然就等于82加上3.35
is of course equal to 82 plus 3.35

区间的下限就等于82减去3.35
The lower end of the interval is equal to 82 minus 3.35

这样的话置信度为95%的区间
Then, a confidence range of 95%

就可以建立起来
can be established

它是从78.65到85.35分
It's from 78.65 to 85.35 points

这就根据
Then

我们刚才所给定的信息
according to the given information and

利用区间估计的基本原理
by using the basic principle of interval estimation

计算出来了
we can calculate

它的一个置信区间
its confidence interval

那这个置信区间
This confidence interval

唯一和刚才我们的信息对应
only correspond to the information given just now

第一个信息样本均值
First, sample mean

第二个信息1-α
Second, 1-α

抽样误差的意义
The meaning of the sampling error

在这个例子里边
in this example

还可以这样来表述
can also be expressed this way

以样本均值为中心的
For an interval of minus or plus 3.35

正负3.35的区间
with sample mean as center

包含总体均值的概率是95%
the probability of including the population mean is 95%

或者说样本均值产生的
Or the sampling error generated

抽样误差是3.35
by sample mean is 3.35

或更小的概率是0.95
or the smaller probability is 0.95

常用的置信度除了95%
The usual confidence levels are 95%

还有90% 95.45%
90%, 95.45% and

和99.73%
99.73%

它们对应的临界值
Their corresponding critical values

这几个大家可以记住
which are commonly used and

这是常用的
you may remember

分别是1.645 2和3
are 1.645, 2 and 3, respectively

可以分别反映各自的估计区间
They can reflect the degrees of assurance

所对应的把握程度
of respective estimation intervals

最后我们稍微总结一下
Finally, let's just conclude a little bit

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

通过刚才的讲述
Through the above description

我们看到
we can see that

如果要构造总体均值的
to construct the confidence interval

置信区间
of population mean,

有三个重要的信息要帮助我们
ee need three pieces of important information

第一个重要的信息 样本均值
The first important piece of information is the sample mean

一定要有抽样作为基础
ewhich must be based on sampling

第二个
The second

一定要已知样本均值的
the sampling distribution of the sample

抽样分布
mean must be known

那这个我们在前一章已经学过
We already learned it in the previous chapter

第三个
The third is

是和这个区间对应的置信水平
the confidence level corresponding to such interval

一定要给定
which must be given

如果不给定我们的误差有多少
If the error is not given

也是无法计算的
calculation cannot be made

这就是区间估计的基本原理
So much for the basic principles of interval estimation

谢谢大家
Thank you