7.4.1 Interval estimation of the mean: large sample case 均值的区间估计：大样本情形在线视频

大家好
Hello, everyone

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

上一讲为大家介绍了
I introduced you in the last lecture

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

接下来为大家介绍
Next, I will introduce you

单个总体均值的区间估计
the interval estimation of single population mean

在这一讲里面
In this lecture

为大家安排了大样本情形
we will talk about the interval estimation methods of population mean

以及小样本情形下的
in both the large size

总体均值的区间估计方法
and small sample size

在我们正式
Before we formally

开始学习大样本情形下
start to learn

单个总体均值的
the interval estimation method for

区间估计方法之前
large sample size

我们先稍微回顾一下
let’s first recall a little about

上一讲关于区间估计的
the basic principles of interval estimation

基本原理的内容
learned in the previous lecture

我们通过上一讲的学习知道
We learned that in the last lecture

区间估计离不开
interval estimation is inseparable from

三个重要的信息
three pieces of important information

第一个重要的信息是
The first piece of important information is

样本的均值
sample mean

第二个重要的信息是
The second piece of important information is

样本均值的抽样分布
sampling distribution of the sample mean

第三个重要的信息则是
The third piece of important information is

我们可能接受的可靠程度
the acceptable degree of reliability

或者把握程度
or degree of assurance

好 接下来我们正式地来学习
Ok, let's get down to learn

大样本情形下的
the interval estimation of the population mean

总体均值的区间估计
in the case of large samples

大样本情形下
In the case of large samples

单个总体均值的区间估计
the interval estimation of single population mean

分两个内容
is composed of two contents

第一 总体标准差已知的情形
First, in the situation where the population standard device is known

那其实在我们上一讲
That’s actually

区间估计的基本原理
the basic principle of interval estimation we learned in the last lecture

这一讲里边
In that lecture

我们所使用的例子
the example we used

也就是Traveller公司的例子里边
was Traveller Company

根据以往的调查显示
in which, the previous survey showed

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

稳定在12分左右
was stabilized at about 12 points

那这个其实就是标准差已知的情形
That is the situation

标准差已知的情形
where the standard deviation is known

所以大样本情形下
Therefore, in the case of large sample

并且标准差已知
and the standard deviation is known

我们就直接用上一讲
let’s directly use the example

区间估计基本原理的例子
of basic principles of interval estimation in the last lecture

在置信水平为95%的时候
At a 95% confidence level

根据抽样分布
according to the sampling distribution

我们可以找到
we can find

计算极限误差的方法
the method of calculating the limit error

那么我们上一次计算的
In our previous calculation

结果是3.35
the result is 3.35

因此我们可以构建
so we can construct

总体均值的区间为
the interval of population mean as

（公式如上）
(The formula is as above)

等于78.65到85.35分
the interval is equal to

这样的一个区间
78.65 to 85.35 points

注意这个地方稍微说明一下就是
Note: let me make a little explanation here

一般总体的这个区间
In the intervals of general population

两端都可以是闭口的
both ends can be closed

都可以是闭区间
they can be closed intervals

因为我们知道对于
Because we know that

连续型变量来讲
for continuous variables

它在某个特定值
at a certain value

对应的概率是等于0的
the corresponding probability is equal to 0

因此我们所指的95%的这个区间
so the 95% interval we are referring to

它是可以包含
can include

这个端点在里面的
the endpoints

通常情况下
Generally

我们把这个区间
we call this interval

称之为置信区间
as confidence interval

它对应的置信水平是95%
It corresponds to a confidence level of 95%

这就是大样本
This is, under conditions of large sample

并且σ已知的情形下面
and known σ

我们如何来
how we

估计总体均值的方法
estimate the population mean

这个通过我们上一讲的例子
This can be obtained

就可以得到
through the example cited in the last lecture

那总体均值的置信区间的含义
In respect of the confidence interval for the population mean

我们也稍微描述一下
let me give a little description

它是指以样本均值为中心
It means that, of all intervals constructed by

所构造的所有区间里边
taking sample mean as center

有95%的区间
95% intervals

可以包含总体参数的真实值
can contain the truth- value of the population parameter

而有5%的区间
while 5% intervals

是包含不到总体参数的真值的
do not contain the truth value of the population parameter

所以这个95%和5%
So the concepts of 95% and 5%

这两个概率是基于所有的区间
are based on all intervals

来讨论它的可能性的
in discussion the possibility

而不是指我们某一次抽样
It does not mean that the intervals constructed

所构造的区间
based on a certain sample

能够包含总体真值的概率是95%
have a 95% probability of covering the population truth value

或者它不能包含
or 5% probability

总体真值的概率是5%
of not covering the population truth value

不是指某一次
It does not mean a certain sampling

要注意的是
What should be noticed is that

我们某一次抽样结束了
after a certain sampling

以它为信息所构造的区间
the intervals constructed based on its information

要不100%能包含总体参数值
either 100% can contain total parameter values

要不100%不能包含总体参数值
or 100% cannot contain total parameter values

因为这个时候区间
Because the intervals here

是固定下来的
are fixed

而我们所谓的95%和5%
And what we call 95% and 5% are discussed

是基于区间是个随机的区间
based on the fact

来讨论的
the intervals are random intervals

因为X_bar
Because -bar

它本身是个随机变量
is a random variable itself

所以这个地方
Therefore, here

大家一定要注意它的表述
you have to pay attention to the presentation

那根据刚才的描述我们知道
We know from the description that

要构造一个置信区间
constructing a confidence interval

包含两个部分
contains two parts

第一 点估计值
First, point estimate

第二 描述估计准确度的正负值
Second, the positive and negative values describing the estimation accuracy

通常情况下
Generally

也把这个正负值
the positive and negative values

称之为极限误差
are also called limiting error

它是反映样本估计值和
It reflects the maximum error range between

和总体参数值之间的
the sample estimates and

最大的误差范围
the population parameter values

也就是我们刚才说的
That is, the figures

正负3% 正负5克
plus or minus 3%, plus or minus 5 grams

这样的一个数字
we just said

就是我们的最大的误差范围
are our maximum error range

超出这个范围
Errors beyond this range

我就不能再接受了
are unacceptable to us

那我们稍微地描述一下
Then, we’ll talk a little about

在大样本已知
the formula for interval estimation

总体标准差的情况下
under the case of large samples with

区间估计的式子
known population standard deviation

它可以用（公式如上）
The formula (as above) can be used

来计算区间的下限
to calculate the lower limit of the interval

它可以通过
and can through【It can pass】

（公式如上）
the formula (as above)

计算区间的上限
calculate the lower limit of the interval

在这个式子里边
In this formula

1-α是置信系数
1- is confidence coefficient

（公式如上）
(The formula is as above)

是标准正态分布里边
in standard normal distribution

右侧尾部中所提供的面积
the area provided in the right tail

为（公式如上）的时候
when (the formula is as above)

所对应的临界值
is the corresponding critical value

这是关于这些信息的描述
This is a description of the information

第二个情况
The second case is

大样本并且未知的时候
the sample is large and σ is unknown

其实在大多数的情况下
Actually, in most cases

总体的参数值
the parameter value of the population

都是等待着去估计的
is waiting for estimation

所以它的标准差
So its standard deviation

通常可能也是未知的
may generally be unknown

而根据我们前面所学过
According to the theorem of the sampling distribution

抽样分布的定理
that we’ve learned before

在大样本的情况下
in the large sample case

还是可以用样本的标准差s
the standard deviation of our sample, s

注意 这个s是指已经对总体的
Note: this s is the one that has undergone unbiasedness treatment

标准差做过无偏性处理的s
to the overall standard deviation

通常我们用s作为
Usually we use s as

总体标准差的点估计值
the point estimate of the population standard deviation

这种情况下
In this case

中心极限定理会生效
the central limit theorem will work

那么样本平均数
So the sample mean

仍然还是会服从正态分布
will still obey normal distribution

那如果它还服从正态分布的话
If it still obeys normal distribution

我们在区间估计的
the interval random fluctuation

基本原理里边
we described

所描述的那个随机
in the basic principles of

那个区间的随机波动
interval estimation

就还有效果
will still work

并且它在随机波动的时候
Moreover, it is still under

还是在正态分布的情况下面
the normal distribution that

来波动的
the random fluctuation occurs

因此这个时候我们还是可以用
So at this time, we can still use

σ已知时候的方法
the process similar to the method

相似的过程
used when σ is known

来估计总体均值所对应的区间
to estimate the interval corresponding to the population mean

只不过此时原来的σ
It's just that the original

要被现在的s所替换
should be replaced by the current s

其他的环节并不会发生改变
and the other links will not be changed

计算下限的方法还是
The method for calculating the lower limit is still

（公式如上）
(The formula as above)

计算区间上限的方法
The method for calculating the upper limit

也仍然还是
is still

（公式如上）
(The formula as above)

那（公式如上）还是来自于
That (formula as above) is still

标准正态分布右侧尾部面积
from the corresponding critical value

为（公式如上）时候
of the right tail area of standard normal distribution

所对应的临界值了
when (formula as above)

这是关于原理的解释
This is an explanation of the principle

那接下来我们看一个例子
Next, let's look at an example

我们袭着开篇的视频
We will continue with the video at the beginning

假定大妞妈妈想进一步了解
Suppose the girl’s mother wants to further know

某网购商城旅行箱包的质量
about the quality of travel suitcases in an online shopping mall

她随机抽取了36个24寸
She picked randomly 36 24-inch

由ABS加PC材质的拉杆箱
trolley cases made of ABS and PC

获得有关的信息
The relevant information obtained

我把它储存在一个文件
was stored in a file

叫做ABSPC.xls
named ABSPC.xls

在我们下面的表格里面
In our table below

单独列出了箱体长度的数据
the data of box length are listed separately

那请根据这些数据
Please, based on these data

来构造该网购商城
construct the confidence interval of 95.45%

24寸ABS加PC材质的拉杆箱
of the average length of the trolley cases

箱体平均长度的95.45%的
made of ABS and PC

置信区间
in this online shopping mall

那下面表格里边有36个
The following table contains the relevant data

拉杆箱的长度的有关数据
about the length of 36 trolley cases

获得这些信息之后
After getting the information

如果要我们构造一个置信区间
if we are asked to construct a confidence interval

我们可以分析一下
we can make an analysis

第一 我们知道36个是个大样本
First we know that 36 cases mean it is a large sample

所以中心极限定理
So the central limit theorem

它是可以生效的
can be effective

如果没有告诉我们
If we are not told that

总体服从的是正态分布的话
the population follows a normal distribution

那还有一个我们已知的信息
we have a piece of known information that

就是1-α是等于95.45%
1-α is equal to 95.45%

那根据这个信息
Then, based on this information

根据36个大样本
and the information that the large sample of 35 cases

它所服从的是一个正态分布
obeys a normal distribution

那么这两个信息的话
Based on these two pieces of information

就可以帮助我们查找到
we can look up and find

临界值（公式如上）
the critical value (formula above)

接下来如果我们要构造
If we want to construct

这个置信区间
the confidence interval

根据我们前面的介绍
according to our previous introduction

我们要准备两个信息
we need to prepare two pieces of information

第一 点估计值
The first is point estimate

也就是这个平均长度所对应的
or the average sample length

样本平均长度了
corresponding to the average length

第二个就是极限误差
The second is limiting error

所以我们把整个步骤分解出来
Now, let's break down the whole step

那首先第一步
First

我们计算样本的平均长度
we calculate the average length of the sample

（公式如上）
(The formula as above)

把表中的数据代入进去
Substitute the data from the table into the formula

59加上61加上63
59 plus 61 plus 63

一直加到64
add up to 64

总共有36只箱子
There are altogether 36 cases

除以36
Divide by 36

计算出来这些箱子的平均的
the average length of these cases is calculated

长度是62.64厘米
to be 62.64 cm

那有了这个就点估计值
Having this point estimate

已经准备好了
we are ready for the second step

第二步我们计算极限误差
Calculating the limiting error

根据我们前面的分析
According to our previous analysis

大样本的情况在X_bar
in the large sample case, X-bar

仍然会服从正态分布
still obey a normal distribution

那么（字符如上）仍然可以用
Then (notation as above) can still be calculated

（公式如上）来计算
By (the formula as above)

只不过此时σ未知
But at this point σ is unknown and

我们需要用样本标准差s
So we need to use the sample standard deviation s

来代替它
to replace it

那s题目里边
In the question, S

并没有直接告诉我们
is not directly told to us

所以我们接下来一步就计算s
So, next we need to calculate s

样本标准差s等于
The sample standard deviation s is equal to

（公式如上）
(The formula is as above)

这个地方大家一定要注意
This is a place we should pay attention to

分母是n-1而不是n
The denominator is n-1 instead of n

这就是我前面提到的
This is what I mentioned previously

已经做过无偏性的调整了
unbiased adjustment has been made

把数字代进去
Substitute the number and

计算的结果等于3.17
the result is 3.17

那有了s 有了n
Having s and n

我们的极限误差
we can calculate

就可以计算出来了
the limiting error

把数字代入进来
Substitute the number into the formula

（公式如上）
(The formula is as above)

数字代进去
Substitute the number into the formula

（公式如上）
Substitute the number into the formula

也就是说它是以1.06的
So it fluctuates with

正负值来波动的
the minus and plus values of 1.06

接下来我们就可以
Then, we can

把区间的上端点
calculate the upper endpoint and

和下端点计算出来
the lower endpoint

下端点等于62.64减去1.06
The lower endpoint is equal to 62.64 minus 1.06

上端点等于62.64加上1.06
The upper endpoint is equal to 62.64 plus 1.06

当然最后要过渡到
And of course, eventually you will

把它的值计算出来
calculate its value

61.18到62.70厘米
To calculate

这样的一个区间
the interval

给它计算出来
between 61.18 to 62.70 cm

那这样的话
Then

我们就可以通过前面的原理
we can break down the steps

把步骤分解
according to the previous principle

把我们已知的信息
Substitute the known information

往这个里边来套 来代入
into it

这样的话我们就可以得到
In this way we can get

总体平均长度的95.45%的
the 95.45% confidence interval of

置信区间
the population average length

所以说实话
So to be honest

如果只是要按步骤去执行
if you conduct the calculation of confidence interval

置信区间的计算的话
as per the steps

我觉得这个就是依葫芦画瓢
I think this is a follow-suit

把我们的信息往里边套就行了
Just substitute our information into it

这个过程并不难
This process is not difficult

我觉得关键的一点
The key point is still

还是回到上一讲的
the basic principles of interval estimation

区间估计的基本原理
we introduced in the previous lecture

就是你一定要理解了
You have to understand

这个地方为什么是（公式如上）
why it is this (formula as above) to be used here

而不是别的
not any other

这个（公式如上）它对应的
This (formula above) corresponds to

是标准正态分布表是为什么
the standard normal distribution table. Why

那这个就是由X_bar的
Because it is from the X-bar

抽样分布来的
sampling distribution

所以这一点我是建议大家
So this is something that I suggest

一定要把它理解透
that you should understand well

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

好 接下来我们来看一下
Now, let’s look at

几个注意事项
a few attentions

第一个 估计的可靠程度
First, the reliability of estimation

也就是用置信系数1-α来描述
which is described by the confidence coefficient, 1-

一般在抽样之前就可以确定
It can usually be determined before sampling

根据所有可能样本
according to the probability of the intervals

所建立的区间
constructed based on the sample

能包含总体参数的概率
That can include the population parameter

就是我们所说的1-α
That’s what we called 1-

这个我们前面也提过
We mentioned previously that

一定是基于
The probability

所有可能的区间来讨论
Must be discussed

它包含的概率的
based on all possible intervals

第二个要注意的是
The second attention is

估计的准确程度
the degree of accuracy of the estimation

是通过置信区间的长度
which is expressed by

来表示的
the length of the confidence interval

那置信区间的长度有多少呢
What is the length of the confidence interval

大家会计算吗
Can you calculate

好 我想有的同学可能
Ok, I think some of you might

已经算出来了
have calculated

置信区间的长度
the length of the confidence interval

那不就是等于两倍的
That's equal to twice

极限误差嘛
the limit error

所以用（符号如上）来描述就可以
So just describe it with (notation above)

并且置信区间的长度越长
and the longer the length of the confidence interval

准确度就会越差
the accuracy is worse

那我们知道
We know that

在置信度一定的情况下
with a certain degree of confidence

置信区间的长度
the length of the confidence interval

和样本容量的大小
and the size of the sample

是呈反方向变动的
change in an opposite direction

因此如果要提高估计的准确度
So if you want to improve the accuracy of your estimates

我们可以通过
we can do this

扩大样本容量的方法来实现
by expanding the sample size

第三个要注意的问题是
The third attention is

可靠程度和准确程度
degree of reliability and degree of accuracy

此消彼长
which offset each other and

它们是不可能同时兼得的
and cannot be improved at the same time

因此我们一定要在准确程度
Therefore, we must

和可靠程度之间
find a relatively balance

寻找到一个相对来说
between the degree of accuracy

比较均衡的总和
and the degree of reliability

这是在总体均值估计的时候
These are the three issues needing our attention

我们要注意的三个小问题
in the population mean estimation