7.5.1 Interval estimation of the mean: small sample case 区间估计：总体比例和方差在线视频

大家好
Hello, everyone

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

上一讲给大家介绍了
In the last lecture, we introduced you

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

接下来为大家介绍
Now, we are going to introduce you

单个总体比例
interval estimation of

和方差的区间估计
the single population proportion and variance

首先我们来看
First let’s see

总体比例的区间估计
the interval estimation of population proportion

对总体比例P的区间估计
The interval estimation of population proportion P

在原理上
in principle,

和总体均值的区间估计
is the same with the interval estimation

是一样的
of population mean

在第三章的时候
In Chapter 3

我们介绍过总体比例这个指标
we talked about the indicator of the population proportion

它是用来表达
It is used to express

是非标志的平均数的
the mean of mark of yes or no

当我们的标志表现
When our mark performance

只能够用是和非来作答的时候
can only be answered with yes or no

我们可以通过特殊的计数处理
we can, by special counting processing,

赋值0和1给它
assign 0 and 1 to it

在这种情况下
In this case,

我们依然可以计算
we can still compute

它所对应的平均水平
its corresponding average level

也就是我们现在所提到的
or the population proportion

总体比例
we are mentioning

因此从这个角度上来讲
So from this angle

总体比例P本质上是一个均值
the population proportion P is essentially a mean

因此对于它的区间估计的原理
So it's interval estimation principles

和总体均值的估计的原理
are the same as those

是相同的
of the population mean estimation

回忆我们上一节讲的内容
Recall the content of our previous lecture

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

我们是要利用样本均值的
we use the sample distribution

抽样分布来完成整个过程的
of sample mean to complete the process

那如果我们现在
Now, if we want to do

要对总体比例P进行区间估计
the interval estimation of the total proportion P

同样要利用样本比例小写p的
we will also use the sampling distribution of

抽样分布来进行估计了
sample proportion p (lowercased)

那么样本比例
Then, what does the sampling distribution like

它所服从的抽样分布
that is obeyed by

又是什么样子的呢
sample proportion

我们仅仅讨论以下这种情形
We only discuss the form

样本比例所服从的
of sampling distribution obeyed by

抽样分布的形式
the sample proportion in the following case

如果n大于30
N is greater than 30,

也就是我们说的大样本情形
which is what we call the large sample case

另外如果np大于等于5
in addition, np is greater than or equal to 5

还有n(1-p)大于等于5
And n(1-p) is greater than or equal to 5

那这里我有一个问题
So I have a question here

想先问一下大家
I want to ask you first

大家能告诉我
Can you tell me

n乘以p是代表什么吗
what does n times p mean

大家会不会
Are you thinking

还在想这个是不是均值呢
it is a mean

还是什么意思呢
or something else

其实回到p的算法
When we go back to the algorithm for p

很快我们可以
we’ll soon know

了解np是什么意思
what np means

因为我们在计算p的时候
Because when we compute p

用的是n{\fs10}1{\r}除以n来计算的
we divide n{\fs10}1{\r} by n

那个n{\fs10}1{\r}代表的
That n{\fs10}1{\r} represents

就是标志表现为
the number of units when the

是所代表的单位数了
mark performance is yes

所以n乘以p当然就还原到
So n times p definitely goes back to

标志表现为是所对应的单位数
the corresponding units when the mark performance is yes

那n乘以1-p
So n times 1-p

当然就是它的对立面了
is of course its opposite

就是标志表现为否
the number of units of the population when the

所对应的总体单位数
when the mark performance is no

那它们组合起来的意思
So what they mean together

就是大样本情形
is the big sample case

另外我的每一种标志
and for each of my mark

所对应的单位数都不少于5个
the corresponding number of units is no less than 5

也就是标志表现为是
That is, no matter whether the mark performance is yes

和标志表现为否
or no

所对应的单位数都不少于5个
the corresponding numbers of units are no less than 5

那么就满足了乘数
So that satisfies the premise for the discussion of

也就是样本比例抽样分布
the multiplier which is the sampling distribution of sample proportion

如果这三个条件同时满足了
If all three of these conditions are met,

那么样本比例近似地
so the sample proportion approximately

会服从正态分布
obeys a normal distribution

它所服从的正态分布两个参数
The two parameters of the normal distribution it obeys

当然对应还是均值
of course are mean

和它的方差
and its variance

这两个参数
The algorithm for the two parameters

它的算法依然类似于
are still similar to that of

均值的参数的算法
mean parameter

比如样本比例它的数学期望会等于多少呢
Let's say what will be the mathematical expectation

会等于多少呢
of sample proportion

当然要等于总体的比例了
Of course, it will be equal to the population proportion

对不对
Right

我们是利用样本比例
We're using sample proportions

去估计总体的比例
to estimate the proportion of the population

所以样本比例
So the sample proportion,

我们前面也提到它的数学期望
we also talked about its mathematical expectation,

是恰好等于总体的比例的
is just equal to the proportion of the population

那另外样本比例它所对应的
Then, what's the variance of the normal distribution

正态分布的方差是等于多少呢
of the sample proportion

通过抽样分布的基本原理我们知道
We know through the basic principles of the sampling distribution

样本平均数的方差
that the variance of the sample mean

它是等于除以根号n
is equal to σ over the square root of n

那样本比例本质上是一个均值
As the sample proportion is essentially an average

因此它的方差
its variance

它的标准差
its standard deviation

也会等于σ除以根号n
will also be equal to σ over the square root of n

只不过它的σ的算法比较特殊
It's just that the algorithm for σ is special

大家还记得
Do you remember

比例的方差的算法吗
the algorithm for the proportion variance

我想有的同学
I think some of you

可能已经想起来了
may have remembered

对 比例的方差的算法是
Yes, the algorithm for proportion variance is

（公式如上）
(The formula is as above)

那标准差对应的就等于
then, what corresponds to standard deviation is equal to

（公式如上）
(The formula is as above)

注意这个地方的P都是大写的
Notice the Ps here are all in capital letters

都是总体所对应的方差了
They are all variances corresponding to the population

那如果我们能够把样本比例
If we can write the form of

所服从的分布的形式
the distribution obeyed by

能够写出来
sample proportion

那这种情况下面
In this case,

我们就可以利用前面的
we can use the previous

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

来计算总体比例的置信区间了
to compute the confidence interval for the population proportion

利用样本比例
By using the sampling distribution

所服从的抽样分布
obeyed by sample proportion,

我们就可以进一步来计算
we can further compute

它所对应的极限误差
its corresponding limiting error

我们这里用E
Here, we use E

小写的p组合来代表
and lowercased p combination to represent

样本比例所对应的极限误差
the limiting error corresponding to the sample proportion

Ep来代表
That’s Ep

由于它的抽样分布
Because its sampling distribution

仍然是一个正态分布
is still a normal distribution

那么我们所用的临界值
so our critical value

依然用（公式如上）来表达
is still expressed by (the formula as above)

那这个时候乘以（字符如上）
At this time, when multiplied by (character above)

（字符如上）代表的是样本比例的
(The character above) represents the standard deviation of

标准差
sample proportion

那样本比例的标准差
The standard deviation of sample proportion

刚才我们已经分析过
as we analyzed already,

它还是等于（公式如上）
is still equal to (the formula as above)

而此时的σ是等于
and then σ is equal to

（公式如上）
(the formula as above)

那所以在样本比例的
So in the process of interval estimate

区间估计过程里边
of sample proportion,

没有新的知识点
there is no new knowledge point

只是把我们原来的结论
We just promote the application

在此处进行了推广应用
of our original conclusion here

在我们刚才的式子里面
In the formula, we just wrote,

大P是等待着估计的总体参数
uppercased P is the population parameter waiting to be estimated

它的值一般都是不知道的
Its value is generally unknown

它不知道也就意味着
Being unknown means

总体的方差没有办法求解
there's no way to solve the variance of the population

所以我们一般会用样本的方差
So we usually use the variance of the sample

（公式如上）
(The formula as above)

来替代总体的方差
To replace the variance of the population

（公式如上）
(The formula as above)

对它做一个点估计
make a point estimate of it

如果进行了点估计
If point estimate is made

那么这个时候的极限误差的
then the uppercased P in the formula for

计算公式里边的大P
Limiting error

相应地就都要换成小p
need to be changed into lowercased p

因此极限误差的计算公式
So in the formula for calculating the limiting error

Ep就等于（公式如上）
Ep is equal to (the formula as above)

那总体比例的1-α的置信区间
The 1-α confidence interval of the population proportion

就可以通过下面
can be constructed by

这个式子来构造
the following formula

它的区间的下端点等于
The lower endpoint of its interval is equal to

（公式如上）
(The formula as above)

它的区间的上端点等于
The upper endpoint of its interval is equal to

（公式如上）
(The formula as above)

那在这个式子里边（公式如上）
So in this formula (as above),

仍然是标准正态分布
it is still the corresponding critical value

右侧尾部面积为（公式如上）时
of standard normal distribution

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

这个我们前面已经分析过
We have analyzed this before and

不在啰嗦
will talk no more

好 接下来我们来看一个例子
Ok, let's look at an example

仍然使用ABSPC.xls中的数据
The data in abspc.xls will be still used

24寸是指拉杆箱的长加宽加高
24 inches mean that the sum of the length, width and height of the trolley case

三个数之和不大于135厘米
is not greater than 135 cm

这是飞机规定的
This is the size

标准登机拉杆箱的尺寸
of the standard trolley cases stipulated for planes

必须进行托运
that must be consigned

下页表格里边列出了
The table in the following page lists

大妞妈妈抽查的36只拉杆箱
the sum of the dimensions of length, width and height of

长宽高的尺寸之和
36 trolley cases sampled by the girl’s mother

试在95%的置信水平下
Under the 95% confidence level, try to estimate

估计总体中符合飞机规定的
the interval of the proportion of the trolley cases

标准登机拉杆箱尺寸的拉杆箱
meeting the standard trolley case size for plane boarding

所占比例的区间
in the population

那这就是一个总体比例
So this is an example of interval estimation

进行区间估计的例子
of population proportion

在这个表格里面
In this table,

我们根据ABSPC.xls
we use the data

里边的数据
in abspc.xls

对长宽高这三个尺寸
To perform summing of the three dimensions:

进行了求和
length, width and height

那么在这个表格里边
In this table,

我们可以观察一下
we can observe

超过了135厘米的拉杆箱
the trolley cases exceeding 135 cm

我已经用红色的字体
which have been marked

把它标出来了
in red

我们可以数一下
Let’s count them:

有1 2 3 4 5
1, 2, 3, 4 ,5

6 ,7, 8, 9 ,10

总共是有10个拉杆箱的长宽高
There are a total of 10 trolley cases

尺寸之和是超过了135厘米
whose length, width and height dimensions add up to more than 135 cm

也就是不符合飞机规定的
That is, they do not meet the size of

标准的登机拉杆箱的尺寸了
standard trolley cases for plane boarding

那我们就可以利用这些信息
We can use that information

来帮助我们去构造
to help us construct

刚才题目里边要的区间
the interval required in the problem

那首先依题意已知n=36
First, we know from the problem that n=36

是一个大样本的情形
which is a large sample case

那n乘以p也就是说
So n times p,

符合飞机规定的拉杆箱有26只
gives that 26 trolley cases are meeting the regulation for plane boarding

n（1-p）
Then n（1-p）

也就是我们说的
gives that there are 10 trolley cases

不符合飞机规定的
not meeting the regulation

标准尺寸有10只
For plane boarding

这两个数字都比5只要来得更大
both of the numbers are greater than 5

就符合我们
which conforms to the premise

刚才讨论的样本比例
For the normal distribution obeyed by

服从正态分布的前提条件
sample proportion we discussed just now

好 所以这三个条件出现了
Ok, as all three conditions are here

我们就可以判断样本比例
we can judge that the sample proportion

这一次一定会近似地
This time will be sure to approximately

服从正态分布的
obey a normal distribution

那1-α是等于95%
1-α is equal to 95%

由于样本比例服从正态分布
As the sample proportion is normally distributed

那我们就可以查临界值
so we can look up the critical value

（公式如上）等于1.96
which (formula above) is equal to 1.96

我们把这些信息先给它准备好
Let's get this information ready first

接下来第一步我们来计算
Next, we’ll compute

样本比例帮助我们获得
the sample proportion to help us

点估计值
get the point estimate

小p是等于（公式如上）
The lowercased p is equal (formula above)

等于26除以36
to 26 divided by 36

计算的结果是72.22%
The result is 72.22 %

第二步
Second step,

我们接下来计算极限误差
Let’s compute the limiting error

Ep等于（公式如上）
Ep is equal to (formula as above)

那p我们已经计算好了
the p has been computed, which

是72.22%
is 72.22%

接下来我们就只要代到
Then, we just substitute it

下面这个式子里边就可以
into the following formula

（公式如上）等于14.63%
The result (formula as above) is 14.63%

也就意味着我们的极限误差
That means our limiting error

达到了14.63%
reached 14.63%

相对来说
Relatively speaking,

还是一个比较大的误差
it's still a big error

好 有了这两个信息
Ok, with these two pieces of information

我们就可以构造
we can construct

总体比例的95%的置信区间
a 95% confidence interval for the population proportion

下限是72.22%-14.63%
The lower limit is 72.22% - 14.63%

等于57.59%
and equals 57.59%

上限是72.22%+14.63%
The upper limit is 72.22%+14.63%

等于86.85%
and equals to 86.85%

所以从整个
So the total

求解的步骤上面来看
steps of solution

和刚才前面的均值的估计
is not different from the previous

并没有不同
mean estimation

只不过在讨论样本比例的
only when we're talking about the sampling distribution

抽样分布的时候
of sample proportion,

它的附加条件更多一些而已
it has a few more additional conditions

这个地方大家要稍微注意一下
This is a place needing a little bit of attention

接下来我们把总体均值
Next, we are going to make a simple summary

和总体比例的区间估计的步骤
of the steps of interval estimate of the population mean

简单做一个总结
and population proportion

通过刚才前面的例题
Through the previous examples,

我们可以总结这几个步骤
we can summarize these steps

第一步是要点估计值
The first step is to do point estimate,

那我们要计算样本平均数
for which we need to compute the sample mean

或者是样本比例
or sample proportion

第二步
The second step is

我们是想获得抽样极限误差
to get the limiting error of the sampling

那通常在
Generally,

获得抽样极限误差之前
before getting the limiting error of sampling

我们会有一个过渡地计算
we need a transition computation

就是样本标准差
for the sample standard deviation

那当然这个地方
Of course, here

我还得提醒大家
I still want to remind you that

如果是样本均值情形下
when we compute the sample standard deviation

计算样本标准差的时候
in the case of sample mean,

一定记得做无偏性的调整
be sure to do unbiased adjustment

第三步
The third step is to

就可以利用抽样分布
use the sampling distribution

来计算抽样极限误差
to compute the limiting error of the sampling

第四步
The fourth step is to

根据点估计值和抽样极限误差
based on the point estimate and the sampling limit error

我们就可以构造总体均值
to construct the population mean

或者是总体比例的置信区间
or the confidence interval of the population proportion

那这四大步骤
These four steps, to

对于总体均值
the population mean

和总体比例来讲的话
and population proportion

都是通的
are applicable

这就是一个小小的总结
So much for the little summary

接下来我们要介绍
Next, we are going to introduce

总体方差的区间估计
interval estimation of population variance

我想通过前面的学习
I think, through the previous study

如果现在要我们来对
if we are going to do

单个总体方差
the interval estimate

去进行区间估计的话
for single population variance

我想大家基本上
you have had a basic understanding of

已经了解到它的步骤
its steps

还是要获得点估计值
We still need to get the estimate, and

以及获得区间的上限和下限
the upper limit and lower limit of the interval

这是一般的方法
That's the general approach

所以对于我们来讲
For us,

对于方差去进行区间估计
for the variance interval estimation,

步骤上和均值以及比例的
the steps are similar to those

过程有点类似
for mean and proportion

但是由于方差本身的特殊性
But because of the particularity of variance itself,

我们在构造它的区间的时候
when constructing its interval

还是和均值以及比例的估计
compared with the mean and proportion estimation

稍稍有一些不同
there is still a little difference

那我们考虑在正态总体里边
We need to consider the corresponding statistics

方差所对应的统计量
in the variance of a normal population

（公式如上）
(Formula as above)

这样的一个估计量
to consider such an estimate

那么这个估计量
This estimate

它是不依赖于任何未知参数
does not depend on any unknown parameters

都会服从
It will obey

自由度为n-1的卡方分布的
a chi-square distribution with n-1 degrees of freedom

这个我想前面
This was introduced

在抽样分布理论里边
in the sampling distribution

已经有介绍
theory

对于给定的置信度1-α
For a given degree of confidence of 1-α

我们注意到在卡方分布图里面
we notice that in the chi-square distribution

那卡方分布它是由
which is composed of

标准正态变量的平方和来构成的
the sum of the squares of the standard normal variables

因此所有卡方分布的值
So all the values of the chi-square distributions

都是大于0的
are greater than 0

卡方分布的图片
Pictures of chi square distribution

一般也是一个钟形分布
are generally of a bell-shaped distribution

不过它是一个偏态的钟形分布
But it's a skewed bell distribution

并且它的图形
and its graphs

都分布在第一象限里边
are all in the first quadrant

卡方分布的临界值的表达方法
The critical value of the chi-square distribution

一般情况下
is generally

是用下面的式子来表达的
expressed in terms of the following formula

（公式如上）
(Formula as above)

也就是说
That is,

当我们的卡方统计量的值
when the value of our chi-square statistic

大于（等于）临界值
is greater than or equal to the threshold

（公式如上）的时候
(Formula as above)

我们对应的右侧尾部面积
the corresponding right tail area

是等于（公式如上）
is equal to (formula as above)

那左侧的临界值的表达方式是
For the expression of the left side critical value

当我们的卡方统计量的值
When the value of our chi-square statistic

小于（等于）某个临界值
is less than (or equal to) a certain critical value

这个临界值的表达方式
the expression of such critical value

一般是（公式如上）
is generally (formula as above)

当我们这个卡方统计量的值
When the value of our chi-square statistic

小于（等于）这个临界值的时候
is less than (or equal to) this critical value

那么它左侧所对应的
its left side corresponding

尾部面积是（公式如上）
til area is (formula as above)

所以我们在卡方分布里边
So, in the chi-square distribution,

中间留出来的部分还是
the part in the middle is still 1-α

尾部两侧各分到（公式如上）
The tail of each side gets (formula above)

但是由于卡方分布的图形
But because the chi-square pattern

它不是对称的
is not symmetric

因此中间所对应的区域
so the corresponding zone in the middle

并不像前面的均值
is not a symmetrical interval as it is

和比例估计的时候
in the previous mean

是一个对称的区间
and proportion estimation

这个大家要注意一下
Please pay attention to it

而且它的临界值的表达方式
and the way its critical value is expressed

和前面的也略有差异
is a little different from the previous one

那么有了上面的这个铺垫
With the above information

我们就可以得到下面的式子
we can get the following formula

（公式如上）
(Formula as above)

当它落在这个区间里边的时候
When it falls in this interval,

它对应的概率等于1-α
its corresponding probability is 1-α

那有了这个式子
With this formula

我们就可以
we can

将它进行变形处理
Do deforming on it

得到总体方差的
to get the 1-α confidence interval

置信度为的置信区间
of the population variance

那么区间的下端点是
The lower endpoint of the interval is

（公式如上）
(Formula as above)

区间的上端点是
The upper endpoint of the interval is

（公式如上）
(Formula as above)

那（公式如上）平方的值
The value of (formula as above) squared

也可以通过查卡方分布表
can also be found by

来得到了
looking up the chi-square distribution table

如果已经得到方差的置信区间
If the confidence interval of variance has been obtained

将上限和下限相应地开根号
find the root of the upper limit and the lower limit correspondingly

就可以得到标准差的置信度
and you will get the confidence interval

为的置信区间
With the 1-α confidence degree of standard deviation

那这个方法我们不再重复
we're not going to repeat this method

因为前面我们在讲方差和标准差
as we already introduced previously

这一组概念的时候
when we talked about the concepts of variance and

也有介绍到
standard deviation

好 接下来我们来看一个例子
Ok, let’s look at an example

仍然延用
We’ll continue to use

这一讲例一里面的数据
the data in the first example in this lecture

这一次我们来帮大妞妈妈
This time, we’ll help the girl's mother

了解24寸拉杆箱
to find the stability of the sum of length, width and height

长宽高尺寸之和的稳定性
Of 24-inch trolley cases

也就是要我们来构造
That is we are going to construct

24寸拉杆箱长宽高
the corresponding confidence interval of

尺寸之和的方差
the variance of the sum of length, width and height of

所对应的置信区间
24-inch trolley cases

假定这一次1-α等于95%
Let's say that this time, 1-α is equal to 95%

那么依题意
According to the problem meaning

我们已知n=36
we have known n=36

1-α=95%
1-α=95%

有了这两个信息
With this information

我们就可以通过卡方分布表
we can look up the corresponding critical value

来查到相应的临界值
in the chi-square distribution table

那接下来
Next,

我也教大家来查一下表
I'll show you how to look it up

请大家把教材准备好
Please get your textbooks ready

那大家拿到了教材以后
When you get the textbook

请翻到卡方分布临界值表
please turn to the critical vales table of chi-square distribution

这个附录
in the annex

那么在拿到翻到这个附录以后
After turning to this annex

我们仍然的是要先观察
we need to first observe

这个表格上面的图片
The picture in the table

在这个图片里边显示的
In this picture,

阴影部分的面积
the area of the shaded part

它这里用了α来表示
is expressed with α

它对应的数字
Its corresponding numbers

就在我们表格的第一行里面
are in the first line of the table

在我们这个例题里边
In our example

我们告诉了1-α是等于95%的
we told you that 1-α is equal to 95%

所以我们首先可以
So we can first

把右侧的临界值查出来
find the right side critical value

右侧的临界值所对应的
The area of the shaded part corresponding to

阴影部分的面积
the right side critical value

就应该是（公式如上）
should be (formula as above)

也就是0.025
0.025

所以我们需要
So we need to

找到0.025这一列
find the 0.025 column

那对应的自由度
The corresponding degrees of freedom

我们也知道是n-1
is known to be n=1

那么就是35
That is 35

我们在0.025这一列里边
We find, in the 0.025 column,

找到自由度为35的地方
the place where the degrees of freedom is 35

那么我们可以得到
Then, we can know

这个数字是53.203
the number is 53.203

同样的方法把左侧的临界值
Find out the left side critical value

给它找出来
by the same method

那左侧的临界值对应的
The area of the shaded part of

阴影部分的面积
the left side critical value

这一次就大了
is large this time

它应该等于1-2.5%
It should be equal to 1-2.5%

也就是0.975
That is 0.975

所以我们找到0.975这一列
So, we find the 0.975 column and

仍然找到自由度为35所对应的
find out the number corresponding to the 35 degrees of freedom

数字是20.569
is 20.569

所以这就是查卡方分布表临界值的方法
That's the method to look up the critical values

临界值的方法
in the chi-square distribution table

好 把这些信息准备好以后
OK, after getting ready the information,

接下来我们就可以来构造
we can construct

总体方差所对应的置信区间了
the confidence interval for the variance of the population

首先同样的要准备点估计值
First of all, again, you have to get the point estimate ready

也就是样本方差s平方
That is sample variance s squared

同样地
Similarly,

我们这里在算s平方的时候
in computing the s squared

用的是无偏估计量s平方
it is the unbiased s squared that should be used

好 所以是（公式如上）
So, it is (formula as above)

把例一中的数字代进来
Substitute the number in the first example

我们可以计算得到
We can know by computing that

s平方等于39.98
S squared is equal to 39.98

第二步
In the second step,

我们就直接计算区间的下限
we’ll directly compute the lower limit of the interval

那区间的下限是等于
The lower limit of the interval is equal to

（公式如上）
(Formula as above)

区间的上限是等于
The upper limit of the interval is equal to

（公式如上）
(Formula as above)

在我们已经计算好区间的下限
After the lower limit and upper limit of the interval

和区间的上限的情况下
have been computed,

那么方差的95%的置信区间其实就已经产生了
the 95% confidence interval for the variance

其实就已经产生了
is actually generated,

是从26.30到68.03这个区间
It goes from 26.30 to 68.03

如果将上限和下限开根号的话
If we take the root of the upper limit and the lower limit

那么我们就得到标准差的95%的
we’ll have a 95% confidence interval

置信区间为5.13到8.25
of the standard deviation

这个范围
which is between 5.13 and 8.25

这是总体方差的
This is the process of constructing the confidence interval

置信区间的构造的过程
of the variance of a population

我们看到
We can see that

它和均值以及比例的过程
the process of mean and proportion

还是稍有差异
is slightly different

这个原因就由于卡方分布
The reason for this is that the chi-square distribution

不是一个对称的分布所造成的
is not a symmetrical distribution

好 这就是关于总体比例
Ok, this is the content about

以及总体方差的
the interval estimate of

区间估计的内容
the population variance

到这里就全部地介绍完了
That's all

谢谢大家
Thank you