8.1.1 Hypothesis testing: proposing hypotheses 假设检验：提出假设在线视频

哎呀 你看
Oops! Look

怎么还没到一天
How did the cellphone run out of power

手机就没电了
within less than a day

买的时候销售员还告诉我
When I was buying it, the salesperson told me

这款手机能待机120个小时
this cellphone could stand by for 120 hours

都是骗人的
All this is hype

是不是今天接打电话比较多
Are there many calls today

没有啊
No

哎呦
Whoops

真想投诉一下他们
I really want to complain about them

虚假广告
for their false advertising

说不定是你运气不好
Perhaps it is your bad luck to blame

正好拿到一个品质
It happened that you received an

没有达标的手机
inferior cellphone

万一因证据不足
In case of inadequate evidence

人家还要告我们诬陷啊
they would possibly accuse us of framing them up

那怎样才证据充足
So how to ensure sufficient evidence

好像质监部门要先进行抽查
It looks like the department of quality assurance will conduct a spot check first

根据抽查的结果
Not until the result of the spot check comes out

才能判断是不是虚假广告
can we tell whether it is a false advertising

爸爸妈妈
Dad and Mom

你们在讨论什么
what are you discussing

质监部门 虚假广告
The department of quality assurance, and false advertising

是说质量检测吧
Are you talking about the quality test

对对对 就是你说的
Right. It is just what you are saying

什么什么检测
What test

可以验证手机宣传里说的
It can be verified whether what was said in the advertisement in the cellphone

是不是真的
Is true or false

质量检测
The quality test

好像听说过
sounds unfamiliar to us

一定要质检部门吗
Is it sure to undergo the inspection of the department of quality assurance

大妞你会吗
Are you able to, big girl

你帮妈妈检测检测
You help Mom have a test

看看他们的宣传
Look at their advertisement

是不是真的
Is it true

妈妈
Mom

质量检测的内容
I have not learned

我还没学呢
anything about quality test

不过央视315晚会
But didn’t the March 15 evening party of CCTV

不也播了吗
play it

公布检测结果之前
Before releasing the test result

都要抽查一些产品的
they would conduct a spot check on some products

可能要买30部
Maybe you have to buy 30

或者更多手机来测哦
or more cellphones for test

你确定要吗
Are you sure you need them

啊
Ah

那还是等你学会了再说吧
Let’s talk about it until you have learned

大家好
Hello everyone

欢迎来到轻松学统计的课堂
Welcome to the Easy Learning Statistics Class

今天我们要给大家讲的是
Today I will give everyone a lecture

第八章的内容
of Chapter VIII

假设检验
Hypothesis Testing

假设检验呢
Hypothesis testing

我们说有参数检验
as we say, is classified into parametric testing

和非参数检验之分
and non-parametric testing

那么我们这章
This chapter

主要讲的就是参数检验的内容
mainly deals with parametric testing

参数检验呢
Parametric testing is a statistic procedure

是先对总体的参数
in which a certain hypothesis is proposed

提出某种假设
for the parameter(s) of population

然后利用样本信息
before the sample information is used

判断原假设
to judge whether the original hypothesis

是否成立的统计方法
is true or false

之前的第七章
In the previous Chapter VII

我们讲了区间估计的内容
we talked about interval estimation

那么大家现在可以回想一下
So now everyone can recall

区间估计的做法
the practice of interval estimation

区间估计
In interval estimation

我们往往前提
a typical premise

是对总体参数未知
is that the population parameter(s) is/are unknown to us

那么我们
So we

就利用样本的信息
use the sample information

来对总体的参数
to make a point estimation

给出一个点估计
or an interval estimation

或者是给出一个区间估计
about the population parameter(s)

那么假设检验
So we say the common ground

它跟区间估计
between hypothesis testing

我们说它的共同点
and interval estimation

都是想利用样本的信息
is to infer the information about the population

来推断总体的信息
using the information about the sample

那么假设检验
The underlying idea

它的基本的思想
of hypothesis testing

就是我们对总体的参数不知道
is that the parameter(s) about the population is/are unknown to us

但是我们可以提出某种假设
but that we can propose some hypothesis

然后我们利用样本的信息
before using the information about the sample

对我们所做的假设作出判断
to make a judgment on our hypothesis

那么假设检验
So hypothesis testing

逻辑上采用了反证法
logically adopts the proof by contradiction

统计上
Statistically

依据了小概率原理
the small probability principle lays a basis

小概率原理就是指
The small probability principle refers to the viewpoint that

概率很小的事件
some event with very small probability

在一次的实验中
is almost unlikely to happen

几乎是不可能发生的
in a trial

假设检验的思想
The thought of hypothesis testing

我们在生活中经常会遇到
is often encountered in our life

比如有一个专家组
For example, a panel

要对我们学校进行教学评估
will conduct a teaching evaluation at our school

那么他来学校检查之前
Before they arrive at our school for inspection

在他的这个心里
we say they may foster a hypothesis

我们说他可能会有一个假设
in their mind

比如说
For instance

他认为江西财经大学
they believe Jiangxi University of Finance and Economics

是一个很好的学校
is a very good university

所以学风很好
where the academic atmosphere is very fine

这是他的一个开始的假设
This is a hypothesis they raise at the beginning

如果他这个假设成立
If his assumption holds

大家想一下
everyone can imagine

那么他在学校里检查
during their inspection in the school

遇到一些学风不好的情况
the academic atmosphere turns out bad for such cases

比如说经常打游戏
as playing games very often

上课看手机这种现象
and watching mobile phones in class

那么这个事件
then the probability that this event

事实上发生的概率
actually happens

应该是很小的
should be very small

但是假如
But provided that

他随意在学校
they pay arbitrary

走访了几间教室
visits to several classrooms in the school

那么就发现这个看手机
to find such phenomena as watching mobile phones

打游戏这种现象
and playing games

很频繁地发生了
happen very frequently

那么这个时候
at this moment

大家想一下
everyone can guess

他当然要怎么样
what they will surely do

推翻他开始的假设
subvert their initial hypothesis.

这实质上就是
This in essence is

假设检验的一个基本的思想
a basic thought of hypothesis testing

也就是说我们
In other words

是在原假设成立的情况下
we have defined a small probability event

先界定了小概率事件
under the condition that the original hypothesis holds

那么正常的情况下
So in normal cases

这个小概率事件
this small probability event

我们说在一次实验当中
we say, is surely unlikely to happen

当然是不可能发生的
in a single trial

但是如果发生了
But should it happen

我们就会认为这个
we would not think

事实上这个事件
this event

它就不是小概率事件
to be a small probability event

它为什么不是小概率事件呢
Why isn’t it a small probability event

就是因为原假设是有问题的
Because the original hypothesis is problematic

所以我们要推翻原假设
We shall subvert it

这就是假设检验的
This is an essential thought of

一个基本的思想
hypothesis testing

在后续的讲解当中
In the subsequent lectures

我们还会给大家
we will make

再仔细地去解读
further interpretation in more details to everyone

假设检验的一个基本的原理
about an essential principle of hypothesis testing

接下来
Next

我们对大家来讲
we address another

也是一个难点问题
difficult issue to everyone

就是关于在假设检验当中
namely the issue of how to propose the original hypothesis in a proper way

如何正确提出原假设的问题
during hypothesis testing

在假设检验当中
In hypothesis testing

我们说有两个假设
we say there are two hypotheses

一个是原假设
One is the original hypothesis

一个是备择假设
and the other is an alternative hypothesis

那么原假设
The original hypothesis

我们又叫它零假设
is also referred to as null hypothesis

通常是我们研究者
which is usually the hypothesis

想收集证据
we, as researchers, hope to oppose

予以反对的假设
by collecting evidence

用H0来表示
It is denoted as H0

那么它表达的含义
typically expressing the connotation

往往就是总体的参数没有变化
that there is no variation in the parameter(s) of the population

或者变量之间没有关系
or that there is no relation between the variables

那么原假设
So the original hypothesis

在假设检验当中
is always true at the very beginning

最初都是假定成立的
of hypothesis testing

然后我们再根据样本的数据
Next, we decide whether there is enough evidence

确定我们是否有足够的证据
for us to reject the original hypothesis

来拒绝原假设
based on the data about the sample

那么原假设的形式
So

大家可以看一下
everyone can have a look at

原假设的形式
the form of original hypothesis

总是比如说H0 μ
Cases are always that H0, μ

等于某一个数值
is equal to a certain value

或者H0 μ
or that H0, μ

大于等于某一个数值
is greater than or equal to a certain value

或者H0 μ
or that H0, μ

小于等于某一个数值
is smaller than or equal to a value

总是这样的形式
Such is the regular form

那什么是备择假设呢
Then what is alternative hypothesis

备择假设又叫研究假设
Alternative hypothesis is also called the research hypothesis

那么它是研究者想收集证据
It is the hypothesis the researcher hopes

予以支持的假设
to corroborate by collecting evidence

我们把它用H1或者Ha来表示
We denote it as H1 or Hα

那么根据刚才
According to

我们对原假设的这个讲解
our interpretation about original hypothesis just now

那么备择假设呢
alternative hypothesis

事实上它跟原假设正好是相反的
is just contrary to the original hypothesis

是完全相反的一个对立的事件
and is an opposite event

那么它所表达的含义
It expresses the connotation

就是总体的参数发生了变化
that there is some variation in the parameter(s) of the population

或者变量之间有某种关系
or that there exists a certain relation between the variables

那么备择假设
So alternative hypothesis

通常都是来表达
usually expresses

研究者倾向于支持的看法
the opinion the researcher tends to support

也就是说我们要想办法收集证据
In other words, we shall manage to collect evidence

拒绝原假设
to reject the original hypothesis

以支持备择假设
in order to corroborate the alternative hypothesis

那么备择假设
The form

它的形式
of alternative hypothesis

跟刚才我们所讲的
is corresponding to

原假设的形式
the form of original hypothesis

是对应的
as we mentioned just now

那么它往往具有
It typically has

这样一下的几个形式
such forms as below

那么就是比如说
For instance

H1是μ不等于某一个数值
H1 is that μ is not equal to a certain value

或者μ大于某一个数值
or that μ is greater than a certain value

或者μ小于某一个数值
or that μ is smaller than a certain value

原假设和备择假设
The original hypothesis and the alternative hypothesis

构成了完备事件组
compose a complete event group

它们是完全对立的事件
They are totally opposite events

那么在我们假设检验当中
In hypothesis testing

你要么拒绝原假设
you must either reject the original hypothesis

要么不拒绝原假设
or accept it

那么必须
That is to say

我们作出一种选择
we must make a definite choice

假设检验
Hypothesis test

有双侧检验与单侧检验之分
falls into two-sided test and one-sided test

那么怎么样去判断
So how to judge

一个假设检验
whether a hypothesis test

它是双侧检验还是单侧检验呢
is a two-sided test or a one-sided test

我们主要来看
It mainly depends on

备择假设的符号
the sign of the alternative hypothesis

如果备择假设
If the alternative hypothesis

含有符号不等号
contains an inequality sign

那么这个假设检验
then this hypothesis test

就是一个双侧检验
is a two-sided test

或者双尾检验
or two-tailed test

如果备择假设
If the alternative hypothesis

具有特定的方向性
has particular directivity

比如说含有符号
for example, if the hypothesis test contains

大于号或者小于号的假设检验
the greater-than or smaller-than sign

我们就称为
we call it

单侧检验或者是单尾检验
a one-sided test or one-tailed test

那么通常
Generally

我们备择假设的方向为小于号的
a hypothesis test is called left-sided test

称为左侧检验
if the direction of the alternative hypothesis is a smaller-than sign

备择假设的方向为大于号的
or right-sided test

称为右侧检验
if the direction of the alternative hypothesis is a greater-than sign

接下来这张表
The following table

就给我们总结了
makes a summary of

刚才单侧检验和双侧检验的内容
the aforementioned one-sided test versus two-sided test

那么大家可以看
Everyone can look at

这个原假设和备择假设
the concrete forms

在双侧检验和单侧检验下
of the original hypothesis and the alternative hypothesis

它的具体的形式
under two-sided test and one-sided test

同学们在学习的过程当中
One puzzle students have encountered during the study

有一个难点问题就是
is that

不知道如何在假设检验当中
they don’t know how to propose a hypothesis properly

正确地提出假设
in hypothesis testing

那么下面
In the following part

我们会给大家
I will make an elaborate explanation

具体讲解一下这个知识点
on this point to everyone

那么正如我们刚才所讲
Just as we said

原假设和备择假设
the original hypothesis and the alternative hypothesis

它一起构成了一个完备事件组
compose a complete event group

而且相互对立
and they are antithetical to each other

那么在一项假设检验当中
It follows that in a hypothesis test

原假设和备择假设
either the original hypothesis or the alternative hypothesis

必然有一个成立
must hold

而且只有一个成立
and not both

那么我们在确定
So, when deciding on

原假设和备择假设的时候
the original hypothesis and the alternative hypothesis

建议大家先确定备择假设
everyone is advised to decide on the latter

再确定原假设
before the former

为什么这样呢
Why

因为备择假设
Because the alternative hypothesis

我们刚才说了
as I said just now

是研究者想收集证据
is the hypothesis the researcher hopes to corroborate

予以支持的假设
by collecting evidence

所以你要把你
You need to make clear

想支持什么东西搞清楚
what you hope to support

然后把它放在备择假设上面
before attaching it to the alternative hypothesis

那么当然它的对立事件
Naturally, its opposite event

就是原假设了
is the original hypothesis

另外大家还需要注意的一点就是
Besides, everyone needs to pay attention to another point that

等号总是放在原假设上
the equal sign is always attached to the original hypothesis

那么刚才
Just now

我们给大家看了
We demonstrated to everyone

单侧检验和双侧检验
one-sided test and the two-sided test

就是关于总体的均值
Regarding the mean of population

我们提出的一系列的假设
we propose a series of hypotheses

给大家总结了
We have summarized to everyone

什么情况下双侧检验
in what situation a hypothesis test is two-sided test

什么情况下左侧检验
left-sided test

什么情况下是右侧检验
or right-sided test

那么不知道大家发现了没有
I don’t know if you notice that

我们的等号总是（在）原假设上
the equal sign is always attached to the original hypothesis

就是原假设
That is, the original hypothesis

总是μ等于多少
is always about μ is equal to what

或者μ大于等于多少
or μ is greater than or equal to what

或者μ小于等于多少
or μ is smaller than or equal to what

这个是我们一个通常的做法
This is our common practice

主要也是为了方便
This is mainly to facilitate the progress of

我们假设检验
our hypothesis testing

它整体的
as a holistic

这样一个分析过程的进行
analytic process

那么接下来
Next

我们通过几个例子
we take several examples

给大家练习一下
for everyone to practice

如何正确地提出
how to propose

原假设和备择假设
an original hypothesis and an alternative hypothesis properly

首先大家看这样一个例子
Below is the first example for everyone to examine

一种零件的生产标准
The production standard of a component

是直径应为10厘米
is that the diameter should be 10 cm

为了对生产过程进行控制
In order to control the production process

质量监测人员
a quality monitoring personnel

定期对一台加工机床检查
conducts regular inspection on a processing machine

确定这台机床生产的零件
and decides whether the components produced by this machine

是否符合标准要求
meet the standard requirement

如果零件的平均直径
In either case the mean diameter of the components

大于或者小于10厘米
is greater or smaller than 10 cm

则表明生产过程不正常
the production process is considered abnormal

必须进行调整
and must be regulated

试陈述用来检验生产过程
Try to state an original hypothesis and an alternative hypothesis

是否正常的原假设和备择假设
for testing whether the production process is normal

在这个例子当中
In this example

我们来想一下
Let’s have a think

那么我们研究者
Our researcher

在这个例题当中
in this scenario

就是这个质量监测人员
is the quality monitoring personnel

这个质量监测人员
who

他想收集证据证实的
hopes to collect evidence to testify

其实就是这个零件的尺寸
the dimension of the components

不符合他的要求
What he hopes to verify

这是他想收集证据
is that the dimension falls short of

予以证实的东西
his requirement

所以这样的话
Such being the case

它的备择假设
its alternative hypothesis

就是μ不等于10厘米
is μ is unequal to 10 cm

那么当我们确定了
After we have decided on

这个备择假设之后
this alternative hypothesis

那么我们接下来
we can

就完全可以确定原假设了
completely decide on the original hypothesis in the following step

那么因为备择假设
Since the alternative hypothesis

是μ不等于10厘米
is μ is unequal to 10 cm

所以原假设
the original hypothesis

当然是μ等于10厘米
is definitely μ is equal to 10 cm

所以这个
So this

是一个典型的双侧检验的例子
is a typical example of the two-sided test

再接下来
Further

我们再看一个例子
Let’s examine another example

学校想购入十万只灯泡
A school wants to purchase one hundred thousand bulbs

合同规定其使用寿命
whose useful life, as specified in the contract

平均不能低于1000小时
cannot be less than 1000 hours on average

已知灯泡使用寿命
It is known that the useful life of the bulbs

服从正态分布
obeys the normal distribution

那么它的σ
and that its standard deviation δ

就是标准差为200
is 200

现从中随机抽取100只
Now 100 bulbs are sampled at random from them

测得样本均值为960小时
The sample mean is measured to be 960 hours

可否认为
Can it be considered

这批灯泡的平均使用寿命
that the mean useful life of this batch of bulbs

低于1000小时
is less than 1000 hours

那么在这个例题当中
In this example question

那么大家也来想一想
everyone just has a think

这个时候 研究者
whether the researcher, at this moment

想收集证据
hopes to testify

予以证实的
that the mean useful life of the bulbs

是灯泡的平均使用寿命
is less than 1000 hours

低于1000小时
or that it is more than 1000 hours

还是高于1000小时呢
by collecting evidence

因为合同规定
Since it has be stipulated in the contract

它的使用寿命
that its useful life

平均不能低于1000小时
cannot be less than 1000 hours on average

那么你来做这个假设检验
when you perform this hypothesis test

那么研究者
the researcher

它是想收集证据
hopes to collect evidence

来证实这个灯泡的平均寿命
to testify that the mean lifetime of these bulbs

是低于1000小时的
is less than 1000 hours

那么在这样一个分析的前提下
Under this premise of analysis

我们就可以确定备择假设
we can decide on the alternative hypothesis

备择假设就是μ小于1000
that μ is less than 1000 hours

那么当然
Of course

我们确定备择假设之后
we can decide on the original hypothesis

就可以确定原假设了
after the alternative hypothesis

那么原假设
So the original hypothesis

就是μ大于等于1000小时
is μ is greater than or equal to 1000 hours

那么大家也可以发现
Everyone can find

等号总在原假设上
that the equal sign is always attached to the original hypothesis

这个是我们通常的做法
This is our common practice

再接下来
Let’s move on

我们再看一个例子
and examine another example

一家研究机构估计
A research institution estimates

某城市中
in a city

家庭拥有汽车的比例
the proportion of households owning a car

超过30%
exceeds 30%

为验证这一估计是否正确
In order to verify whether this estimation is correct

该研究机构随机抽取了
this research institution draws

一个样本进行检验
a sample at random to test

是陈述用于检验的
Try to state an original hypothesis and an alternative hypothesis

原假设与备择假设
for the test

在这个问题当中
In this problem

大家注意 立场很重要
everyone should notice standpoint matters significantly

那么我们在这个例子当中说
We say, in this example

这个研究机构
that the research institution

他估计出来
estimates

家庭拥有汽车的比例
the proportion of households owning a car

超过了30%
exceeds 30%

是这个研究机构
Because the research institution

想搜集证据
hopes to collect evidence

支持他的这个论断
to corroborate this argument

所以他进行了这个假设检验
it conducts this hypothesis test

那么在这种情况下
Such being the case

我们说研究者想支持的结论
we say the conclusion the researcher hopes to uphold

就是家庭拥有汽车的比例
is that the proportion of households owning a car

是超过30%的
is over 30%

所以我们先确定备择假设
So we first decide on the alternative hypothesis

备择假设
which

就是P大于30%
is P is greater than 30%

P就表示这个拥有汽车的比例
where P signifies the proportion of owning cars

那么当然我们的原假设
Of course, our original hypothesis

也就随之确定了
is decided thereupon

那么原假设
The original hypothesis

就是P小于等于30（%）
is P is smaller than or equal to 30%

那么仍然可以发现
It can still be found

等号总在原假设上
that the equal sign is always attached to the original hypothesis

下面大家来做一个练习
Now everyone does a practice

某化肥厂
A chemical fertilizer plant

将生产的尿素装袋
bags the urea it has produced

每袋规定重量为50公斤
Specified weight per bag is 50 kg

为保证装袋质量
In order to assure the quality of bagging

质量检验人员
the quality inspector

需要在包装机工作时
needs to sample several bags of products at random

随机抽取若干袋产品
and measure their weight

测量其重量
while the packing machine is working

以判定包装机工作是否正常
so as to determine whether the packing machine operates as normal

在这个问题当中
In this problem

研究者想搜集证据
the viewpoint the researcher hopes

予以证实的这个观点
to testify

我们说非常的明确
by collecting evidence

就是要搜集证据
is that the packing machine is judged

判定这个包装机工作怎么样
not to operate as normal

不正常
which we say is very explicit

不然他也不会来做
Otherwise he would not do

这样一个检验
such a test

那么所以
It follows that

在这个问题当中
in this problem

备择假设就是
the alternative hypothesis is

μ不等于50公斤
μ is unequal to 50 kg

那么原假设
whereas the original hypothesis

就是μ等于50公斤
is μ is equal to 50 kg

因为它的标准
Since the standard

是规定的每袋的重量为50公斤
weight per bag is 50 kg

所以大于 小于
neither greater-than nor less-than

都是不符合要求的
meets the requirement

所以这个问题
So this problem

显然是一个双侧检验的问题
is one of the two-sided tests

再接下来
Further

我们再看一道练习题
let’s examine another exercise problem

某化工企业
A chemical engineering enterprise

为了提高产品的合格率
needs to purchase a batch of

需要采购一批
high-purity raw materials W

纯度较高的W原料
in order to improve the qualified ratio of products

在已经接到的报价中
Among the quoted prices received

A厂商价格合理
Manufacturer A has stated a reasonable price

并宣称产品
and claims that

平均纯度不低于98%
the mean purity of its products is no less than 98%

符合该企业
meeting the requirement of this enterprise

对于原材料纯度的要求
for the purity of raw materials

为此该企业准备进行一次检测
This enterprise is prepared to conduct a test

核实A厂商报价中
to verify whether the value of the technical indicator

W原料的平均纯度不低于98%的
that the mean purity of raw materials W is no less than 98%

技术指标数值是否属实
is true in Manufacturer A's price

然后决定是否签订采购合同
before deciding whether to sign the purchase contract

那么这里
Here

从他这个问题的表述当中
from the statement of this problem

大家可以发现
everyone can notice

我们做这个检验
in doing this test

我们是站在企业的这个角度上
we stand at the angle of the enterprise

那么我们想搜集证据
So what we hope to testify

证明的东西
by collecting evidence

就是这个A厂商
is that Manufacturer A's

他的论断不成立
argument does not stand

所以我们在这个问题当中
Hence, in this problem

备择假设就是μ小于98%
the alternative hypothesis is μ is less than 98%

那么当然原假设
Of course, then, the original hypothesis

就是μ大于等于98%
is μ is greater than or equal to 98%

等号总在原假设上
The equal sign is always attached to the original hypothesis

然后我们这个检验
Next, shall we call this test

我们说它是一个左侧检验
a left-sided test

还是右侧检验呢
or a right-sided test

按照我们刚才给大家介绍的知识
According to the knowledge we have just now introduced to everyone

那么它的备择假设的符号
the sign in its alternative hypothesis

是一个小于号
is a smaller-than sign

所以它是一个左侧检验
so it is a left-sided test

好 这一讲的内容
Well, so much

我们就讲到这里
for this lecture

谢谢大家
Thank you, everyone