7.4.2 Interval estimation of the mean: small sample case 均值的区间估计：小样本情形在线视频

接下来我们介绍小样本情形下
Next, we are going to introduce the interval estimation process of

总体均值的区间估计过程
the population mean in the small sample case

在小样本的情况下
In the case of small samples

总体均值的抽样分布
the sampling distribution of the population mean

依赖于总体的分布
depends on the distribution of the population

我们仅讨论总体
We're only talking about total

服从正态分布的情形
where the population obeys a normal distribution

根据前面抽样分布的定理
According to the sampling distribution theorems

我们知道总体
we know that when the population

是正态总体的情况下
is a normal population

如果标准差已知
If the standard deviation is known and

小样本出现了
the sample is small

X_bar还是会服从正态分布的
X-bar will still obey a normal distribution

但是如果总体是个正态总体
But if the population is a normal population

总体标准差未知的情况下
the population standard deviation is unknown

那如果样本容量小于30的话
and the sample size is less than 30

这个时候X_bar
X-bar at this time

就不再服从正态分布了
will no longer obey a normal distribution

由它所构造的t统计量
The t statistic constructed by it

那t统计量的一般形式
The general form of the t statistic

大家还会写吗
can you still write

我们前面在学抽样分布的时候
In learning the sampling distribution

学过t统计量
we learned about the t statistic

那t统计量在这里的表达形式是
The expression for the t statistic right here is

（公式如上）
(The formula is as above)

这个t统计量的形式
The form of this t-statistic

看起来非常类似于
looks very similar to

标准的正态分布
standard normal distribution

但是这里我要说明的是
But here's what I want to say is that

它并不是由
it is not transformed from

标准正态分布转化而来的
the standard normal distribution

它是由t统计量的构造
It's constructed by the t statistic

那t统计量的分子
The numerator of the t statistic

是一个标准的正态分布
is a standard normal distribution

它的分母是由卡方统计量
Its denominator is constructed by

除以它的自由度
dividing the chi-square statistic by its degrees of freedom

然后开根号来构造的
and then taking the square root

那么这个t统计量它服从的
This t statistic obeys

就是t分布
t distribution

并且它有一个自由度和它对应
and has a corresponding degree of freedom

这个自由度的话
The corresponding degree of freedom

是n-1和它对应的
is n-1

所以在小样本的情形下面
So in the small sample case

我们对总体均值的区间估计
for the interval estimation of the population mean

也要先了解它X一拔的抽样分布
we should first know its X-bar sampling distribution

根据前面的学习
According to the previous study

我们了解到t分布的图形特征
we know that the graphical characteristics of the t-distribution

和标准正态分布的图形特征
and those of the standard normal distribution

基本是类似的
are basically similar

比如它们都是对称的钟形分布
For example, they're both symmetrical bell distributions

它们都是关于Y轴对称的
they're both symmetric about the Y-axis

所以基于这样的一个基本知识
So based on this basic knowledge

我们前面的区间估计的原理
the principles of the previous interval estimation

可以再一次
can once again be

应用到t分布的图形上面
applied to the graph of the t-distribution

那以X_bar为中心所构建的区间
The intervals built around X_bar

仍然是在负无穷到正无穷
are still randomly fluctuating

这个区间上面随机地波动的
on the intervals from minus infinity to infinity

在它波动的过程里边
In the process of their fluctuation

有的区间能包含μ
some intervals can cover μ, and

有的区间不能包含μ
some can’t

我们依然希望
Still we hope

能包含μ的区间占大多数
a majority of intervals can cover μ

比如90% 比如95%
Let's say 90 %, 95 %

如果有了这个置信水平
If we have this confidence level

同样的我们可以通过查表
similarly, we can, by looking up the table

得到对应的临界值
find the corresponding critical values

只不过这一次
Just in this time

不再是查标准正态分布表
instead of looking up the standard normal distribution table

而是查t分布表而已
we look up the t distribution table

所以区间估计的
So the basic principles of interval estimation

基本原理还是有用的
are still useful

只不过抽样分布的形式
It's just that the sampling distribution form

从正态分布换成了t分布而已
has changed from a normal distribution to a t-distribution

在t分布里边
In the t distribution

对于给定的置信度
for a certain confidence degree

同样可以通过查表
by looking up the table

找到它对应的临界值
we can find its critical value

它的临界值的一般表达方法
The general expression of its critical value

用（公式如上）来表达
is as the formula above

利用临界值
By using the critical value

同样地可以把极限误差
the limit error

Ex_bar计算出来
Ex_bar, can also be calculated

那这个时候它就等于
At this time, it equals

（公式如上）
(The formula is as above)

因此在正态总体条件下
So in the case of a normal population

总体均值的区间估计
the interval estimate of the population mean

在标准差未知的小样本情形下
at unknown standard deviation and small samples

可以采用下面的方法进行
can adopt the following

下端点的算法
algorithm for the lower endpoint

（公式如上）
(The formula is as above)

上端点的算法仍然是（公式如上）
The algorithm for the upper endpoint is still (formula above)

在这里（公式如上）
Here (the formula above)

是在自由度为n-1的t分布里边
the corresponding critical value

右侧尾部面积为（公式如上）时
is given for the t distribution with a n-1 degree of freedom and

所对应的临界值了
when the tail area of the right side is (the above formula)

因为它和标准正态分布的
Because it has the same features

特点是一样的
as the standard normal distribution

也是左右对称的
It's also left-right symmetric

因此中间有1-α
So we have 1-α in the middle, and

左右两边就各分到有
the left and right sides each

（公式如上）的面积
have an area (as shown in the formula above)

我们来看一个例子
Let's look at an example

在ABSPC.xls的数据里边
In the abspc.xls data

大妞妈妈同时记录了
the girl's mother recorded at the same time

所抽查到的拉杆箱的承重量
the carrying weight of the trolley cases sampled

其中有28个拉杆箱
of which, 28 cases

提供了该项指标值
provided this indicator value

具体的数字在下面的表格里面
The numbers are in the table below

其中表中的大写的N
where the capital N

表示没有这一项数据
represents no data of this item

接下来要我们根据上述资料
Next, we are going to, based on the above information,

建立置信度为90%的
establish an interval of

总体均值的区间
the population mean with 90% confidence

假定承重量指标
It is assumed that the carrying indictor

总体服从正态分布
population obeys a normal distribution

那有了这些信息
Having such information

我们也同样地可以先分析
we can also first make an analysis

假定总体是个正态总体
Let's say the population is a normal population

n=28 是个小样本
n =28 is a small sample

并且没有信息直接告诉我们
and there's no information to tell us directly about

总体的标准差
the standard deviation of the population

所以我们刚才提到的
So there comes in the t-distribution

t分布的情形就出现了
we’ve just mentioned

正态总体 小样本
Normal population, small sample

并且标准差未知
and unknown standard deviation

那这个时候我们就可以判断
Now, we can judge that

X_bar是服从
X-bar obeys

自由度为27的t分布
a t distribution with 27 degrees of freedom

当然另外还告诉了
And, of course, I told you

我们置信度是90%
the confidence is 90%

那根据1-α等于90%
According to 1-α equaling 90%

等一下我们要查的表
the table that we're going to look up is

就不再是标准正态分布表
no longer the standard normal distribution table

而是t分布表了
but a t-distribution table

接下来我们的步骤
The following steps

和前面的例子基本类似
will be basically similar to those in the previous example

第一 计算点估计值
First, calculate the point estimate

第二 计算极限误差
Second, calculate the limiting error

解 依题意
Solution：according to the question

总体服从正态分布 n=28
The population obeys a normal distribution, n=28

此时总体方差未知
The population variance is unknown

可以用自由度为27的t分布
You can use a t distribution of 27 degrees of freedom

进行总体均值的区间估计
to carry out the interval estimation of the population mean

首先第一步计算样本平均数
The first step is to calculate the sample average

（公式如上）
(The formula is as above)

将数据代入进来
Substitute the data into the formula

计算的结果是35.36公斤
The result is 35.36 kg

这是平均的承重量
This is the average weight

那为了计算极限误差
In order to calculate the limiting error

我们需要有一个准备
we need to have a preparation

就是样本标准差
That's the sample standard deviation

这个我们在前面也分析过
We’ve analyzed previously

样本标准差的算法
The algorithm of sample standard deviation

刚才也提醒过
I reminded you just now that

就是一定是无偏性处理以后的
It must be the formula after unbiasedness treatment

一定是（公式如上）
must be the (formula above)

把信息代入进来
Substitute the information into the formula

计算的结果是17.9
The result is 17.9

有了这个信息
With this information

接下来计算极限误差
we’ll calculate the limiting error

（公式如上）
(The formula is above)

那这个地方的话
Here

有一个（公式如上）出现了
there is a (formula as above)

它的查表的方法
Its table lookup method

和标准正态分布基本类似
is the same as for the standard normal distribution

好 接下来我来教大家查一下
Now, let me show you how to lookup

t分布表的临界值
the critical value of the t-distribution table

请大家拿出教材来
Please take out your textbooks

那我们拿到教材以后的话
Now, let's

翻到教材后面的目录
Go to the table of contents at the back of the textbook

目录里边你可以找到
You can find in the table of contents

有一个t分布双侧临界值表
A two-sided critical value table for t-distribution

那我们在进行区间估计的时候
So when we do interval estimation

这张表是最方便查临界值的
This table is very convenient for us to look up the critical values

那么在这个双侧临界值表里面
After getting this two-sided

拿到以后
critical value table

我们首先要观察一下图片
Let's look at the picture first

表格上面有一个图片
there is a picture on the table

那么这个图片里边显示
It is shown in the picture

两侧阴影部分面积加起来有α
the shaded areas on both sides add up to α

中间是1-α
In the middle is 1-α

那么在我们这个题目里边
In our question

中间1-α是90%
the middle 1-α accounts for 90%

那当然两侧的面积就是10%
So, of course, the area of both sides accounts for 10%

那这个10%我们到哪里去找呢
Where can we find the 10%

在我们表格里面的
In the table

第一行里边不是有一些数字
there are some numbers in the first line

有0.1 有0.05 有0.02
such as 0.1, 0.05, 0.02

有0.01这样的一些数字
0.01

那这一次我们对应的是0.1
So this time we're dealing with 0.1

因为是90%剩余的部分
Because it is the remaining part after 90% is reduced

所以我们现在就找到第一列
Now, find the first line

是我们要查的这一列
and look up in the first line

那根据前面的分析
According to the previous analysis

我们的自由度是27
the degrees of freedom are 27

所以在表格里面
So, in the table

往下找
Search downward

找到自由度为27交叉的位置
Find the position where it crosses with the 27 degrees of freedom

1.703就是我们这一个例题
1.703 is the critical value in t-distribution table

t分布表的临界值了
for this example

把这个数字查好以后
After finding the number

接下来我们就可以计算
we can calculate

极限误差的值
the value of the limiting error

那极限误差
For the limiting error

（公式如上）
(The formula is as above)

注意 这个地方是根号n
Notice this is the square root of n

不是根号n-1
not the square root of n-1

所以大家一定要注意一下
So you have to pay attention to

无偏性的调整
unbiased adjustment

是调整在s本身的算法里面
The adjustment is in the algorithm of S itself

不是任意你看到有n的地方
but not that wherever you see n

都要把它变成n-1
you have to turn it into n-1

好 回到我们刚才的算式里边
Now, go back to the previous algorithm

（公式如上）
(The formula is as above)

计算的结果是5.76公斤
The result 1s 5.76 kg

这样的一个数字
With this number

那有了点估计值
we have the point estimate

有了极限误差
the limiting error

90%的置信区间就可以算出来
Now, we can figure out the 90% confidence interval

下限是29.60公斤
The lower limit is 29.60 kg, and

上限是41.12公斤
The upper limit is 41.12kg

那这样的话
Then

我们就可以帮助大妞妈妈
we can help the girl’s mother

把24寸ABS加PC材质的旅行箱的
to calculate the average weight

平均承重量给它计算出来了
of the 24-inch ABS and PC suitcase

这就是在小样本的情形下
So that's the method for estimating

如何来估计总体均值的
the confidence interval for the population mean

置信区间的方法
in the case of small samples

它和大样本情形唯一的不同
The only difference from in the case of large samples

就是这一次我们利用了t分布
is that we use t-distribution

而原来我们利用的是正态分布
it was normal distribution we previously used

来帮助我们查表
to help us in table look-up

好 接下来我们对总体均值的
Then, we are going to make a little summary of

区间估计过程
the process of interval estimation of

稍微做一个总结
population mean

根据我们前面的例子
According to our previous example

我们可以总结单个总体均值
we can summarize the steps of estimating

区间估计的步骤
the mean interval of a single population

第一步通常是计算样本平均数
The first step is usually to calculate the average of the sample

帮助我们获得点估计值
to help us get a point estimate

可以是简单算法
It could be a simple algorithm or

也可以是加权的算法
a weighted algorithm

第二步
The second step is

通常是要计算样本标准差
usually to calculate the sample standard deviation

来帮助我们计算极限误差
to help us calculate the limiting error

样本标准差
Algorithms for sample standard deviation

和样本平均数的算法
and sample mean

是配套的
are matching

如果样本平均数用了简单算法
If the sample average calculation uses a simple algorithm

那样本标准差也是简单算法
the calculation of sample standard deviation will also use a simple algorithm

如果样本平均数用了加权算法
If the sample average calculation uses a weighted algorithm

那么样本标准差也是加权算法
the calculation of sample standard deviation will also use a weighted algorithm

另外 还要提醒大家的是
Another thing to remind you of is

一定要做无偏性的调整
to be sure to make unbiased adjustments

第三步是计算极限误差
The third step is to calculate the limiting error

在大样本的时候
In the case of large samples

极限误差的算法
the limiting error algorithm

依赖于正态分布
depends on the normal distribution

因为X_bar它的抽样分布
Because X_bar’s sampling distribution

是个正态分布
is a normal distribution

那你可以是
You can use

（公式如上）
(The formula is as above)

在小样本的情形下面
In the case of small samples

我们计算极限误差
calculation of limiting error

是依赖于自由度为n-1的t分布
depends on the t-distribution with n-1 degrees of freedom

这个时候是查t分布表
At this time, look up the t-distribution table

得到临界值（公式如上）
to get the critical value (formula above)

所以（公式如上）
Therefore, (the formula is as above)

第四步则是构造
The fourth step is to construct

置信水平为1-α的区间
the interval with a confidence level of 1-α

下端点（公式如上）
with the lower endpoint of (formula above)

上端点（公式如上）
and the upper endpoint of (formula above)

这样的话我们就基本上完成了总体均值的
So we're done with the process of

完成了总体均值的
the interval estimation

区间估计的过程
of the population mean

好 这一讲就是这样
So much for this lecture

谢谢大家
Thank you