8.2.1 Hypothesis testing: basic ideas 假设检验：基本思想在线视频

妈妈 我要买彩票
Mom, I want to buy a lottery

买彩票啊
Buy a lottery

二块钱一注吧
Two yuan per stake

行 就买一注玩玩吧
Okay, just go ahead

等我找二块钱给你
Wait and I will give you two yuan

你会买吗
Do you know how to buy it

要先想好7个29以内的数字
You should first have in mind seven numbers within 29

我会
I can do it

不就是想7个数字吗
Simply thinking about seven numbers

难不倒我
That could not perplex me

姐 要中个大奖
Sister. Should you win a big prize

你最想买什么
what would you want to buy mostly

中大奖
Win a big prize

小妞
Little girl

你知道中大奖的概率有多大不
do you know how high the probability of winning a big prize is

不知道啊
No, I don’t

电视里不也演有人中大奖吗
Isn’t such a scene shown on TV

等姐来算算吧
Let me work it out

保准不算不知道
I’m sure you will be surprised

一算吓你一跳
at the result

算出来了
I have worked it out

中大奖的概率
The probability of winning a big prize

大概是156万分之一
is roughly 1/1560000

156万分之一
1/1560000

这也太小了吧
That’s too small

比天上掉馅饼
Even smaller than

正好砸在自己头上的概率还小呢
the probability that a pie falls from the sky and hits one’s head exactly

这就叫小概率事件
This is the so-called small probability event

放心
Feel assured

馅饼不会正好砸到你头上的
that no pie would fall exactly onto your head

在刚才的动画中
In the cartoon just now

我们看到中彩票（大奖）
we saw winning a lottery (big prize)

是一个小概率事件
is a small probability event

那么在概率论中
In the probability theory

我们把概率很接近于0的事件
we call any event whose probability is very close to zero

称为小概率事件
a small probability event

比如说雷电伤人
For instance, thunder injury

是个小概率事件
is a small probability event

汽车的轮胎飞出
The tire flying off a car

也是小概率事件
is also a small probability event

吃饭被鱼刺卡到喉咙
Getting stuck at the throat by fishbone during meal

都是小概率事件
is a small probability event

那么一般地
Generally

我们把概率小于0.05的事件
we define any event whose probability is smaller than 0.05

就确定为小概率事件
a small probability event

而且在某些非常重要的
Moreover, in or on some highly important

实验或者场合
experiments or occasions

当事件发生会产生
if the emergence of an event will likely

非常严重后果的时候
have a very severe consequence

比如说
say

飞机的失事 沉船等等
an aircraft accident, a ship founder, etc.

那么我们这个
we should select

小概率事件的概率
the probability of the small probability event

就应该选的更小一些
smaller

比如说0.0001
say 0.0001

那么甚至更小
or even smaller

否则我们可以适当地大一些
Otherwise, we can make it slightly bigger

一般我们的小概率标准
Generally, the standard of small probability

往往取0.01
is 0.01

0.05这两个水平
or 0.05

那么也就是说
That is to say

事件发生的概率
any event whose probability

在0.01或者0.05以下的事件
is below 0.01 or 0.05

我们都称其为小概率事件
is called a small probability event

那么小概率原理
Regarding the small probability principle

我们刚刚提到
we just mentioned

这个是假设检验
this is hypothesis testing

我们说
We say

它所依据的一个基本的原理
a rationale it relies on

那么小概率原理
is the small probability principle

也被称为小概率事件的
which is also referred to as

实际不可能性原理
the practical impossibility principle

它的内容
It

指的是实际不可能事件
means a practically impossible event

在一次试验当中
is in fact unlikely to happen

实际上是不可能发生的
in a trial

我们在现实生活当中
In our real life

大家也在大量使用小概率原理
the small probability principle is vastly used

只是大家有时候
only except that everyone

可能没有意识到
may be unconscious of its use

比如说在世界上
In the world, for example

火车跟汽车相撞的事件
the event of a train colliding with a car

时有发生
happens at times

但是我们几乎没有人
But almost nobody of us

由于担心火车与汽车相撞
does not dare to ride a train or car

而不敢去乘火车 去乘汽车
for fear of any collision between the train and car

为什么呢
Why

因为大家都知道
Because it is known to all

火车与汽车
that the probability of a train

相撞的这个可能性
colliding with a car

实在是太小了
is really too small

那么大家也会认为
Then everyone would think

不会发生在自己身上
it unnecessary to be frightened about something

所以我们就不会害怕
that will never drop on himself

去坐火车 去坐汽车
and thus going straightforward to ride the train or car

小概率原理
The small probability principle

是我们在假设检验当中
is a basis on which we

决定推翻
decide whether to subvert

还是接受原假设的一个依据
or accept the original hypothesis in a hypothesis test

也是我们在长期实践当中
It is also a highly pragmatic principle

总结出来的
we have concluded

一条实用性非常强的原理
in the long-term practice

那么这就是
That is all about

我们给大家介绍的
the small probability principle

小概率原理
we introduce to everyone

两类错误与显著性水平
Two types of errors and level of significance

在假设检验当中
In hypothesis testing

我们当然希望
we definitely hope

总是能够作出正确的决策
the proper decision can always be made

但由于我们的决策
However, since our decision-making

是建立在样本信息的基础上
is based upon the sample information

而样本又是随机的
while the sample is randomly drawn

因而我们就有可能犯错误
we are prone to making errors

我们之前讲的中奖的例子
Reconsider the aforementioned example about winning a prize

比如说我
Were I the decision-maker

我就不会去买彩票
I would not buy any lottery

为什么呢
Why

因为我知道
Because I am aware

中彩票（大奖）的这个概率太小了
that the probability of winning a lottery (big prize) is too small

我认为
I don’t think

这么小概率的事件
such a small probability event

不会发生在我的身上
would happen to me

但是有些同学说
But some students may ask

你为什么不买呢
why not buy one

你如果万一中了奖呢
and what if you should win the prize

那么也就是说
That is to say

其实假设检验
in hypothesis testing

我们是有可能犯错误的
we are prone to making errors

就是我们认为
As we believe

在原假设成立的条件下
under the premise that the original hypothesis holds

小概率事件
a small probability event

在一次实验当中
would never happen

是不会发生的
in a trial

如果发生了
If it should happen

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

但实际上
But in fact

在原假设成立的这个前提下
under the premise that the original hypothesis holds

这个小概率事件
such a small probability event

也可能会发生
is also likely to happen

如果按照我们刚才所讲的
According to the small probability principle

小概率原理
we have just said

那么我们就拒绝原假设
we should reject the original hypothesis

但事实上
But in fact

原假设是正确的
the original hypothesis is correct

所以这样的话
Such being the case

我们就会犯错误
we would likely make errors

那么在我们假设检验当中
In hypothesis testing

大家都知道
as is known to all

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

是不能同时成立的
must not hold simultaneously

那么我们最终的决策的结果
So our final decision

要么就是拒绝原假设
is either to reject the original hypothesis

要么就不拒绝原假设
or not to

我们在决策的时候
Upon making a decision

总是希望
we always hope

当原假设正确的时候
we are not rejecting it

我们没有拒绝它
when the original hypothesis is correct

当原假设不正确的时候
or we can reject it

我们能够拒绝它
when the original hypothesis is incorrect

但我刚才给大家
But from the example

介绍的这个例子
I have introduced to everyone

大家会发现
you may find

实际上
in fact

我们在假设检验的过程当中
we are prone to making errors

是有可能犯错误的
in the process of hypothesis testing

就是我们认为
Namely, we believe

很小概率的事件
a very small probability event

在一次实验当中不会发生
would never happen in a trial

但事实上
when in fact

它也有微弱的可能会发生
it has a slight possibility to happen

如果我们严格按照
If we conduct the hypothesis test

小概率原理
in strict accordance to

去做这个假设检验
the small probability principle

那么我们就可能会犯错误
we will likely make errors

那么我们犯的错误
Then the errors we make

我们总结为两类错误
are summarized as two types

那么第一类错误呢
Type I errors

就是当原假设是正确的时候
stem from our rejection of the original hypothesis

我们拒绝了它
when it is correct

那么第一类错误的概率
The probability of Type I errors

我们常常记为α
is usually denoted as α

也称为显著性水平
also called the level of significance

那么第二类错误
Type II errors

其实就是我们所常说的纳伪错误
are actually what we usually call the false-accepting errors

就是当原假设为错误的时候
That is, we choose not to reject it

我们却没有拒绝原假设
when the original hypothesis is false

那么第二类错误的概率
The probability of Type II errors

我们记为β
is denoted as β

下面我们给大家的这个表
Below the table for everyone

就展示了
presents

我们第一类错误和第二类错误
Type I errors versus Type II errors

它发生的这个情形
Everyone can notice

那么在这张表当中
the circumstances in which it happens

大家可以看到
in this table

当H0为真的时候
When H0 is true

我们拒绝H0
we make Type I errors

就是犯了第一类错误
also called the true-rejecting errors

也叫弃真错误
If we reject H0

犯第一类错误的概率记为α
The probability of making Type I errors is denoted as α

当H0为假的时候
When H0 is false

我们如果没有拒绝
if we choose not to reject

这样一个假的原假设的话
such a false original hypothesis

那我们就是犯了第二类错误
we make Type II errors

那么犯第二类错误的概率
The probability of making Type II errors

记为β
is denoted as β

第二类错误也称为取伪错误
Type II errors are also called false-accepting errors

那么在样本容量
Under the premise

一定的这个前提下
that the sample size is constant

我们犯这两类错误的概率
the probabilities that we make both types of errors

是互为消长的
are mutually trading off

一般来说
Generally

α和β这个关系
the relation between α and β

就像翘翘板
is just like a rocker

如果α小呢
When α is small

β就大
β is great

α大呢
and

β就小
vice versa

那么在样本容量一定的情况下
Given that the sample size is constant

我们不能同时减少这两类错误
we cannot reduce both types of errors simultaneously

那么唯一的办法
The only way

就是扩大样本容量
is to enlarge the sample size

下面我给大家举一个例子
Below I will give everyone an example

来说明两类错误
to explain the two types of errors

首先我们的原假设
To begin, our original hypothesis

是一个真心爱你的男生
is a boy who loves you sincerely

备择假设
while the alternative hypothesis

是一个不是真心爱你的男生
is a boy who does not love you sincerely

那么如果H0实际上成立
If H0 holds in fact

但是你
when you

却凭你自己的经验
reject H0

拒绝了H0
by dint of your personal experience

那也就是说
in other words

你拒绝了一个
you reject a

你认为不爱你
boy whom you do not believe to love you

而实际上真心爱你的男生
when in fact he loves you sincerely

那么你就犯了第一类错误
then you make Type I errors

如果H0实际上不成立
If H0 does not hold in fact

而你接受了H0
while you accept it

那这个意思就是
which means

你接受了一个
you accept a boy

你感觉爱你
whom you feel love you

而实际上并不爱你的男生
when in fact he does not love you

那么你就犯了第二类错误
then you make Type II errors

那么如果想同时减小
To diminish

犯第一类错误
the probabilities of making Type I errors

和第二类错误的概率
and Type II errors

那么你就只能
the only way for you is

增加恋爱的次数N
to increase the times of your love affairs, N

其实就是我们所说的样本容量
actually the so-called sample size

比如一个经历过
For instance, a girl who has experienced

100次恋爱的女生
100 love affairs

她101次恋爱
would make Type I errors

犯第一类错误
and Type II errors

和第二类错误的概率
at much smaller probabilities

就会小很多了
in her 101st time of love

这个就是我们的两类错误
This is all about the two types of errors

我们应该如何去控制两类错误呢
How should we control the two types of errors

一般来讲
Generally

对于一个给定的样本
for a given sample

如果犯第一类错误的代价
if the cost of making Type I errors

比犯第二类错误的代价
is relatively higher than

相对较高的话
that of making Type II errors

那我们就把犯
then we shall make

第一类错误的概率
the probability of making Type I errors

定得低一些
slightly lower

反之
On the contrary

如果犯第一类错误的代价
if the cost of making Type I errors

比犯第二类错误的代价
is relatively lower than

相对较低
that of making Type II errors

那么我们就把
then we shall make

第一类错误的概率
the probability of making Type I errors

定得高一些
slightly higher

也就是说
In other words

发生哪一类错误的后果更严重
We shall first control

我们就首先要控制
the probability of whichever type of errors

哪一类错误发生的概率
whose occurrence will bring about the more serious consequence

但由于犯第一类错误
However, since the probability

这个概率
of making Type I errors

是我们研究者可以控制的
is controllable by researcher

所以在假设检验当中
in hypothesis testing

我们往往先控制第一类错误
we usually control the probability

它的发生的概率
Type I errors occur

下面这个地方
Next

我们也给大家举一个例子
we shall still give everyone an example

假设我们有一个药物的生产商
Suppose we have a pharmaceutical manufacturer

它生产一种感冒药
that produces a cold cure

那么这个药物
The cold cure

我们说研制出来
shall undergo a large quantity of experiments

当然要进行大量的实验
after being developed

来证明这个药物是有效的
so as to testify its efficacy

然后投入生产
before being put into production

那么他在做这个
When conducting the hypothesis test

假设检验的时候
the manufacturer naturally

他的原假设
proposes the original hypothesis

就是药物无效
that the medicine is inefficacious

备择假设是药物有效
the alternative hypothesis being that the medicine is efficacious

那么他就开始
Now the manufacturer starts

他假设检验的过程了
the hypothesis testing procedure

那么这里
Here

我们说
we say

第一类错误是弃真错误
Type I errors are true-rejecting errors

那么弃真错误的意思
meaning

就是原假设是真的
that we reject the original hypothesis

我们却拒绝了它
when it is true

也就是说药物是无效的
In other words, we treat the medicine to be efficacious

但是我们当成了有效
before putting it into production

然后投入生产
when the medicine is inefficacious

那么第二类错误是什么呢
So what are Type II errors

第二类错误
Type II errors

是这个药物有效
are that this medicine is efficacious

但是我们当成无效
but we don’t think so

那就不生产了
then no longer produce

那么这个时候
At this moment

大家思考一下
everyone has a think

在这个情形下
which types of errors

犯哪类错误的代价更大呢
are made at a higher cost under such a circumstance

从这个厂商的角度考虑
Considering the manufacturer’s angle

当然我们说这个问题
we say of course there is no standard answer

没有一个标准的答案
to this question

大家可能有自己不同的看法
Everyone may have different opinions

但是一般来说
But generally speaking

在这个问题上面
on this question

犯第一类错误的代价
the cost of making Type I errors

可能会更大
may be higher

因为第一类错误
because Type I errors

是说这个药物无效
suggest you treat this medicine to be efficacious

但你当成有效
and put it into mass production and market

去大量生产 投入市场了
when it is inefficacious

但事实上
Since in fact

它是没有效果的
it is inefficacious

那当然也不会有效益了
definitely no benefit will come out

所以这个是
So this is a

弃真错误的这样一个
case-in-point of true-rejecting errors

这个是弃真错误的代价更大
whose cost is higher

这样的一个例子
If I change another precondition

那如果我换一个前提条件
in this example

再来举这个例子
Let’s take the following example

比如说 在非典的时期
During the SARS period

我们说形势非常的严峻
the situation was terribly harsh

那么同样
Similarly

我们说一个药物的生产商
a pharmaceutical manufacturer

研制出了
developed

治疗非典的这个药物
a medicine to cure SARS

那么它也要去大量地实验
It also had to conduct a large quantity of experiments

去做这个检验
as well as the test

那么原假设
Given that the original hypothesis

仍然是药物无效
remains that the medicine is inefficacious

备择假设是药物有效
while the alternative hypothesis is that the medicine is efficacious

那么两类错误
the two types of errors

跟刚才也是一样的
remain the same as in the previous scenario

第一类错误
Type I errors

是药物无效
are that you treat the medicine to be efficacious

你却当成了有效
when it is inefficacious

第二类错误
whereas type II errors

是药物有效
are that you treat the medicine to be inefficacious

你当成了无效
when it is efficacious

那么我想
I think

我们说世界上
perhaps nothing else

可能没有任何的东西
in the world

比生命更加重要
outweighs one’s life

所以在那个非典的
So over that highly severe

非常严峻的时期
period of SARS

我想你犯
I suppose

第二类错误的代价会更大
you would have to pay a higher cost for making Type II errors

也就是说这个药物本来有效
which means you treat the medicine to be inefficacious

你却当成了无效
and decide not to put it into production

然后不投入生产
when it is inherently efficacious

所以也没有办法
So there is no way

挽救很多人的生命了
to save many lives

所以关于两类错误的控制
So everyone must value

大家一定要重视
the control of both types of errors

就是自己要来判断
Specifically, make a personal judgment

犯哪类错误的代价更大
which type of errors would be made at a higher cost

那么哪类错误的代价大
We shall make the probability of whichever type of errors

我们就把哪类错误的这个概率
made at higher cost

人为控制得小一点
A bit smaller

那么我们刚才也已经说了
As just said

我们研究者能够控制的
what we, as researchers, are capable to control

主要是第一类错误的概率
is mainly the probability of Type I errors

也就是α的大小
namely the value of α

那么α定得小一点
The smaller α is fixed

β就大一点
the greater β is

α大一点
and

β就小一点
vice versa

这就是关于两类错误
This is all about the problem

如何控制的问题
of how to control the two types of errors

接下来
Next

我们来说一说
Let’s have a discussion on

显著性水平α
the level of significance α

那么显著性水平
The level of significance

它就是事先确定的
is a predetermined

我们用于拒绝原假设H0时
necessary evidence

所必须的证据
for rejecting the original hypothesis H0

那么刚才我们也提到
Just now we also mentioned

α事实上就是犯
α is in fact the maximum probability

第一类错误的最大概率
of making Type I errors

那么我们说
So we say

在原假设为真的时候
when the original hypothesis is true

我们拒绝原假设的概率
the probability that we reject it

就是α
is α

那么α呢
The value of α

我们的取值往往是
typically ranges among

0.01 0.05 0.10
0.01, 0.05, and 0.10

那么α是由研究者事先确定的
α is predetermined by the researcher

也就是说
In other words

我们在做假设检验的时候
before conducting a hypothesis test

我们要事先规定
we shall specify

我们小概率事件的概率
the probability of a small probability event

也就是你认为多大的概率
namely how high a probability you would consider

称其为小概率
as a small probability

那么在原假设成立的情况下
In the case that the original hypothesis holds

我们规定了
we just care

小概率事件的概率之后
exactly whether

我们就来关心
such a small probability event

到底这个小概率事件
has happened

有没有发生了
after we specify the probability of it

按照反证法的思想
In the light of the thought of proof by contradiction

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

小概率事件
a small probability event

按理是不会发生的
is unlikely to happen by principle

但是如果小概率事件发生了
But should a small probability event happen

我们就认为很反常了
it would be taken as an anomaly

因为在原假设成立的情况下
Since in the case that the original hypothesis holds

小概率事件
a small probability event

按理是不会发生的
is unlikely to happen by principle

它如果发生了
if it should happen

那么我们就有理由拒绝原假设
then we have the reason to reject the original hypothesis

那么显著性水平
The level of significance

就给了我们这个小概率事件的
provides a specification of the probability

概率的这样一个规定
of small probability events

它就是我们事先规定的
which is the level of the small probability

这个小概率的水平
we have prespecified