8.5.1 Analysis of examples of individual population proportion and variance test 例题分析 单个总体比例及方差检验在线视频

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

今天我们给大家讲的就是
Today we will give you a lecture on

一个总体比例的假设检验
hypothesis testing on a overall ratio

和一个总体方差的假设检验
and hypothesis test on a population variance

我们先来看
Let’s first focus on

一个总体比例的假设检验
the hypothesis testing on a overall ratio

那么在这里
Here

我们要给大家回顾
we shall tell everyone to recall

我们第六章当中
some conclusions once mentioned

曾经提到过的一些结论
in Chapter VI

在大样本的前提下
Under the premise of large sample

按照中心极限定理
according to the conclusion we are told

告诉我们的结论
by the central limit theorem

那么样本比例
the sample proportion

我们是近似服从正态分布的
approximately obeys the normal distribution

在第六章的讲述当中
As related in Chapter VI

我们说
we say

关于总体比例的统计推断
regarding the statistical inference of overall ratio

我们对大样本的要求
we have relatively strict

比较严格
requirement on large sample

在这种情况下
In such cases

我们的大样本的要求
the requirement on large sample

一般是（公式如上）
typically is (the formula as above)

并且（公式如上）
and (the formula as above)

我们把这样的样本
We deem samples like this

视为大样本
as large samples

在大样本的情况下
In the case of large sample

我们的样本比例
the sample proportion

是近似服从正态分布的
approximately obeys the normal distribution

服从什么样的正态分布呢
What kind of normal distribution then

我们是近似服从均值为大P
It approximately obeys such a normal distribution

大P就是总体的比例
where the mean is capital P

方差为（公式如上）
namely the population proportion

这样的一个正态分布
and where the variance is (as in the formula above)

那这里
So here

我们当然很容易可以构造
we can of course easily construct

在一个总体比例的
a test statistic of overall ratio

假设检验下
Under the hypothesis testing

大样本情形下
in the circumstance of large sample

我们的检验统计量
the test statistic

那么这里
here

我们就可以用p-P
can be used as p-P

当然这里我们假定
Of course we postulate here

原假设总是成立的
the null hypothesis always holds

所以这个大P
So the capital P

就是按原假设
is hypothesized to be greater than p0

假设大P等于P0
by the null hypothesis

所以我们的检验统计量
So the test statistic

就是（公式如上）
is (as in the formula above)

就是该有大P的地方
Namely, where there should be capital P

我们都使用大P0
we use capital p0 instead

因为我们总是认为
since we always believe

原假设是成立的
the original hypothesis holds

而原假设就是P等于P0
while the original hypothesis is P is equal to p0

这里我们希望大家
Here we hope everyone

对总体比例的区间估计
have a review

做一个回顾
of the interval estimation on overall ratio

在区间估计当中
In interval estimation

总体比例的区间估计式是
the formula for interval estimation on overall ratio is

（公式如上）
(the formula above)

那么这里
Here

大家一定要把
everyone must

我们假设检验当中
make a distinction between

使用的检验统计量
the test statistic used

和在区间估计当中
in hypothesis testing

使用的区间估计的这个式子
and the formula for interval estimation used

把它区别开来
in interval estimation

注意它们的不同
and notice the differences between them

当时我们讲
Once we explained

区间估计的时候
why there is no such (formula as above)

我们的区间估计式当中
during interval estimation

为什么没有出现（公式如上）

而是出现了（公式如上）
but (the formula as above)

这主要是因为
This is mainly because

我们做区间估计的目的
the objective of making an interval estimation

是要估计总体的比例
is to estimate the population proportion

而总体的比例是不知道的
while the population proportion is unknown

所以在大样本的情况下
Hence in the case of the large sample

我们用样本方差（公式如上）
we substitute the sample variance (as in the formula above)

去替换掉了
for

总体方差（公式如上）
the population variance (as in the formula above)

从而完成了
thereby going through

区间估计的整个过程
the whole process of interval estimation

但是有些同学
But some students

在假设检验当中
are prone to making a mistake

容易犯的一个错误
in hypothesis testing

就是在下面的
namely somewhere

（公式如上）的地方
(in the formula as above)

那有很多同学
Many students

容易写成（公式如上）
are prone to writing as (the formula as above)

那么如果有同学
If someone

犯了这个错误
makes such a mistake

我想还是大家对于
I guess he

假设检验的基本原理不够清楚
is not clear enough about the rationale of hypothesis testing

因为我们的假设检验
Since hypothesis tests

总是在原假设成立的前提下
always consider the problem

去考虑问题
under the premise that the null hypothesis holds

那么我们有原假设
So we have the null hypothesis

那么原假设就是P等于P0
that P is equal to p0

不论双侧还是单侧
be it two-sided or one-sided case

因为单侧
For one-sided test

它可能有大于等于P0
it is likely greater than or equal to p0

小于等于P0
or smaller than or equal to P0

但是我们都取
In either case we examine

等于的情况来考察
in the equal-to case

这样的话
This way

我们在根号下
everyone must write P0(1 – P0)

大家一定要写P0(1-P0)
under the radical sign

因为这就是假设
since this is the hypothesis

总体的比例就等于P0
that the population proportion is equal to p0

如果写成p(1-p)
Were it written as p(1-p)

那么我们说这里
we say it

就不符合我们假设检验的
does not conform to the basic logic

一个基本的逻辑了
of hypothesis testing

所以我们是在
So it is

原假设成立的情况下
in the case that the null hypothesis holds

去考察我们的小概率事件
that we examine whether

是否发生
the small probability event happens

如果小概率事件发生
If the small probability event should happen

我们就拒绝原假设
we simply reject the null hypothesis

这是我们整个的一个逻辑
This is the holistic logical frame

所以这个地方
So here

需要大家注意
everyone needs to pay attention

这里我们总结一下
Now we sum up

关于一个总体比例的假设检验
the hypothesis testing on a population proportion

我们使用的检验统计量
The test statistic we use

就是z检验统计量
is the z-test statistic

当然前提是大样本
Of course the premise is large sample

那么这个检验统计量
Then this test statistic

它的形式
has the form

我们刚才已经给大家
we have just presented

展示出来了
to everyone

那么接下来
Following this

我们来看具体的例题
we shall examine concrete examples

大家看一下这个例子
See the example below

一种以休闲和娱乐
A magazine themed

为主题的杂志
by leisure and entertainment

声称其读者群中
claims 80% of its readers group

有80%为女性
are female

为验证这一说法是否属实
To test whether this statement is veracious

某研究部门抽取了
some research department draws

由200人组成的一个随机样本
a random sample composed of 200 people

发现有146个女性
finding there are 146 females

经常阅读该杂志
who often read this magazine

那么这个时候
At this moment

我们取显著性水平
we select the significance level

α等于005
α=005

检验该杂志读者群中
to test whether the proportion of females is 80%

女性的比例是否为80%
among the reader's group of this magazine

那么根据我们刚才的分析
According to our analysis just now

这明显是一个双侧检验的问题
this is a two-sided test problem

就是他关心的是
Namely, what it concerns is

这个读者当中
whether the proportion of females is 80%

女性的比例是不是80%
among the readers

所以我们的原假设
It follows that our null hypothesis

就是大P等于80%
is capital P is equal to 80%

备择假设呢
while the alternative hypothesis

就是大P不等于80%
is capital P is unequal to 80%

我们的显著性水平
The significance level

α等于005
α=0.05

n等于200
and n=200

那么我们可以大家计算小p
Then we can figure out lowercased p

那么计算出小p之后
After figuring out lowercased p

大家可以看看
everyone can see

np和n（1-p）
whether both np and n(1–p)

是不是都是大于等于5的
are greater than or equal to 5

那么这就是一个大样本
This is a large sample

接下来
Next

我们要计算检验统计量的
we shall calculate the specific value

具体的取值
of the test statistic

我们把数据代进去
by substituting the data

那么刚才的样本比例
The sample proportion just now

我们通过题目的信息
can be figured out

可以计算我们的样本比例
via the information in the problem

就是073
to be 0.73

接下来我们用073减去大P0
Next we subtract capital P0 from 0.73

就是我们认为
We think

大P现在等于大P0
capital P is now equal to capital P0

就是08
namely 08

所以（公式如上）
Hence (the formula above)

那么这个地方我刚才提到了
This is where students are most prone to mistake

这是最容易出错的地方
as I have mentioned just now

千万不要写成了
Make sure not to write it as

073×1-073
0.73×(1-0.73)

那么应该写080×1-080
but as 0.80×(1-0.80)

因为我们始终认为
Since we always believe

原假设是成立的
the null hypothesis holds

总体的比例就是08
the population proportion is 08

那么我们计算
Now we calculate

z检验统计量的取值
the value of z-test statistic

那么看到z检验统计量的取值
Upon seeing the value of z-test statistic

是负的2475
is -2475

而我们的显著性水平
the significance level

是α等于005
α=005

那么临界值是正负196
and the critical value is ±196

所以我们最终的决策
we definitely make the final decision

当然是要拒绝原假设
to reject the null hypothesis

因为我们检验统计量的值
Since the value of the test statistic

出现在了拒绝域
falls in the rejection region

也就是说
this means

该杂志的说法并不属实
the magazine’s statement is not true

那么这个就是
Above is

关于总体比例的假设检验
an example we have given to everyone

我们给大家举的例子
about the hypothesis testing on population proportion

下面我们再来看
Next we examine

一个总体方差的假设检验
the hypothesis testing on a population variance

那么关于一个
Regarding the hypothesis testing on a

总体方差的假设检验
population variance

我们的前提条件
our precondition

就是假设总体服从正态分布
is the assumption that the population obeys the normal distribution

或者近似服从正态分布
or approximately obeys the normal distribution

那么在这个前提下
Under this premise,

我们使用一个服从卡方分布的
we use a test statistic

检验统计量
obeying the χ-distribution

就是（公式如上）
as (in the formula above)

就是样本方差（公式如上）
Namely, the sample variance (as in the formula above)

它就服从自由度为
obeys the χ2-distribution

n-1的卡方分布
with the degree of freedom of n – 1

这个分布
This distribution

我们在第六章的
was mentioned

最后的这个抽样分布当中
in the sampling distribution

跟大家提过
in Chapter VI

那么在这种情况下
In such a case

我们就使用一个卡方分布
we use a test statistic

服从卡方分布的检验统计量
obeying the χ2-distribution

来完成假设检验的整个过程
to go through the whole hypothesis testing procedure

下面我们再来看一个
Now let’s examine another

具体的例子
specific example

啤酒生产企业
A beer manufacturer

采用自动生产线罐装啤酒
adopts automatic production lines for canned beers

每瓶的装填量为640毫升
The filling volume per can is 640 mL

但由于受某些
Due to some

不可控因素的影响
uncontrollable factors

每瓶的装填量会有差异
There may be variance in the filling volume per can

此时
Now

不仅每瓶的平均装填量很重要
what matters is not only the average filling volume per can

装填量的方差同样很重要
but also the variance of filling volumes

如果方差很大
If the variance is significant

会出现装填量太多或太少的情况
the case may arise that the filling volume is either too much or too little

这样要么生产企业不划算
As a consequence, either the manufacturer feels cost-inefficient

要么消费者不满意
or consumers feel discontented

假定生产标准规定
Assume the production standard specifies

每瓶装填量的标准差
the standard deviation of filling volume per can

不应超过4毫升
should not exceed 4 mL

企业质检部门抽取了
The corporate department of quality assurance samples

10瓶啤酒进行检验
10 cans of beers to test

得到的样本标准差
finding the standard deviation of the sample

为s等于3.8毫升
s=3.8 mL

那么现在
Now

以0.05的显著性水平
at significance level of 0.05

来检验装填量的标准差
test whether the standard deviation of filling volumes

是否符合要求
meets the requirement

那么这里
Here

我们提到了
we highlight

就是按照规定
by the specification

每瓶装填量的标准差
the standard deviation of filling volume per can

不应超过4毫升
should not exceed 4 mL

现在我们来检验这个标准差
Now let’s test whether this standard deviation

是否符合要求
meets the requirement

所以事实上
In fact

我们想搜集证据予以证明的
what we hope to demonstrate by collecting evidence

就是它这个方差不符合要求
is the proposition that the variance falls short of the requirement

所以我们提出备择假设
So we propose the alternative hypothesis

就是σ平方大于4的平方
that δ2 is greater than 42

其实就是σ大于4
which is in substance δ is greater than 4

那么确定了备择假设
Now that the alternative hypothesis has been decided on

那么接下来
we can move on

我们就可以确定原假设
to decide on the null hypothesis

原假设当然就是
which naturally is

σ平方小于等于4的平方
σ2 is smaller than or equal to 42

这明显是一个
This is obviously a

右侧检验的问题
right-sided test problem

那么这里
Here

α等于005
α=0.005

自由度等于10-1=9
and degree of freedom is 10–1 = 9

那么我们大家看到
Then everyone can notice

在题目当中它的前提
the premise in the problem

就是总体服从正态（分布）
is that the population obeys the normal distribution

那么这里
Here

我们就可以构造
we can construct

我们的服从卡方分布的
the test statistic

检验统计量
obeying the χ2-distribution

这里按照我们刚才
According to

给大家的公式
the formula we have just presented to everyone

也就是n-1
namely n – 1

就是（公式如上）
it is (the formula as above)

其实这里就是3.8的平方
Actually, here it is 3.8 square

除以
divided by

大家注意
Everyone pays attention

这里除是除σ的平方
the divisor here is δsquare

而现在我们原假设是认为
While our null hypothesis suggests

σ的平方是等于4的平方的
δsquare is equal to 4 square

所以这样
Thus

我们把数据代进去
as we substitute the data

那么这个卡方检验统计量的取值
the value of the χ2-test statistic

现在算出来是8.1225
is worked out to be 8.1225

那么由于这是一个右侧检验
Since this is a right-sided test

所以它的拒绝域在右边
its rejection region is on the right side

那么这里
Here

也需要大家去查
everyone also needs to look up

卡方分布的分布表
the distribution table of χ2-distribution

来确定临界值
to determine the critical value

也就是说
In other words

当它这个右侧这块的面积
when the area of the right side

是005的时候
is 005

这个临界值是多少
what is the critical value?

那么很明显
It is very clear

我们检验统计量的取值
that the value of the test statistic

没有出现在拒绝域
is outside the rejection region

所以我们最终的结论
Hence our final conclusion

是不拒绝原假设
is not to reject the original hypothesis

那么当然这个结论就是指
Of course, this conclusion indicates

没有证据表明
there is no evidence that

装填量的标准差
the standard deviation of filling volumes

不符合要求
falls short of the requirement

那么这样我们就把一个总体比例
At this point, we have covered

和方差的假设检验讲完了
the hypothesis testing on a population proportion and variance

谢谢大家
Thank you, everyone