8.3.1 Hypothesis testing: basic steps 假设检验：基本步骤在线视频

爸爸
Dad

你怎么关注起
How do you pay attention to

女生喝茶的问题了
the problem of lady tasting tea

这是一本非常经典的
This is a very classic

统计学书籍.
book of statistics

里面有很多有趣的故事
There are many interesting stories about it

想不想听听女士品茶的故事
Do you want to listen to a story about The Lady Tasting Tea

好的
Yes

那是二十世纪二十年代后期
It was in Cambridge, the United Kingdom

在英国剑桥
in the late 1920s

一个夏日的午后
After the noon of a summer day

一群大学的绅士
a group of university gentlemen

和他们的夫人们
were drinking the afternoon tea

还有来访者
with their ladies

享用着下午茶
and visitors

在品茶过程中
While tasting the tea

一位女士坚称
a lady asserted that

把奶加进茶里
the tea would taste

或把茶加进奶里
different

不同的做法会使茶的味道
if treated in different ways, such as pouring milk into the tea

品起来不同
or pouring the tea into milk

在场的一帮科学精英们
The scientific elites present

对这位女士的胡言乱语
turned up their nose at

嗤之以鼻
babble of the lady

这怎么可能呢
How can this be

他们不能想象
They could not imagine

仅仅因为加茶
the mere difference in the order

加奶的先后顺序不同
or adding tea or milk

茶就会发生不同的化学反应
would cause different chemical reactions in the tea

然而在座的一个身材矮小
However, a scrub

带着厚眼镜的先生
gentleman wearing a pair of thick glasses there

也就是著名的费希尔先生
namely the well-known figure Mr. Fisher

却不这么看
did not think so

他对这个问题很感兴趣
He was very interested in this problem

他兴奋地说到
excitedly saying

让我们来检验这个命题
“Let’s verify this proposition

并开始策划一个实验
and start to hatch an experiment”

在实验中
In the experiment

坚持茶有不同味道的那位女士
the lady who had insisted the tea would taste different

被奉上一连串的
was offered a series of

已经调制好的茶
well mixed teas

其中有的是先加奶
some of which were made

后加茶制成的
in the tea-after-milk order

有的则是先加茶
whereas some were made

后加奶制成的
in the milk-after-tea order

费希尔先生调配出了
Mr. Fisher mixed

八杯其他条件一模一样
eight cups of tea with the same conditions

而仅仅是倒茶
only except that the orders of pouring tea

倒奶顺序相反的茶
and milk were opposite

其中两类各有四杯
There were four cups in each of the two types

然后他让女士品尝之后
Then he let the lady have a taste before

告诉他哪四杯
telling him which four cups

是先加奶后加茶制成的
of tea were made in the tea-after-milk order

当然
Of course

剩下的就是
the rest of them

先加茶后加奶制成的了
were made in the milk-after-tea order

在分析实验结果的时候
While analyzing the experiment result

他运用了这样的逻辑
he used such a logic

他首先假设
He began by hypothesizing

女士没有这个能力区分
the lady could not make a distinction

这个假设被称为原假设
This hypothesis is called the original hypothesis

然后如果女士很好地
If the lady identified

鉴别了这八杯茶
those eight cups of tea inerrantly

那就说明
it shows that

在原假设成立的情况下
something eccentric happened

发生了非常反常的现象
in the case that the original hypothesis holds

以至于说明
so much so that

原假设是令人怀疑的
the original hypothesis is doubtful

从统计上来说
Statistically

如果在原假设成立的前提下
if a very small probability event happens

发生了非常小概率的事件
under the premise that the original hypothesis holds

那我们就有理由怀疑
we have the reason to doubt

原假设的真实性
the veracity of the original hypothesis

费希尔先生的原假设是
The original hypothesis made by Mr. Fisher is that

女士没有这样的能力
the lady had no such ability

但是实际这位女士猜对了
While the lady guessed

所有的八杯茶
all the eight cups of tea right

如果在原假设成立的前提下
If the lady was guessing blindly without any ground

女士是毫无根据的瞎猜
under the premise that the original hypothesis holds

她猜对的概率是1/70
and the probability she guessed right was 1/70

这个概率很小
a very small probability

只能说明
it only shows

女士是具有这个能力的
the lady had such an ability

如果女士没有这个能力
If the lady had no such ability

但是她侥幸靠瞎猜
but she Passed the test

通过了测试
by guessing blindly

使我们错误地认为
so that we misbelieve

她有这个能力
she had such ability

这个概率只有1/70
the probability is only 1/70

这种错误
and such errors

被称为第一类错误
are called Type I errors

一般来说
Generally

不希望这个错误
it is hoped that the probability

发生的概率超过5%
such errors occur does not exceed 5%

刚才的故事告诉我们
The story just now tells us

假设检验
hypothesis testing

是先对总体的参数
is a statistical procedure that starts with

提出某种假设
the proposal of some hypothesis about the parameter(s) of the population

然后我们利用样本的信息
before we utilize the information on a sample

来判断假设是否成立的统计方法
to judge whether the hypothesis is true

它在逻辑上
Logically

运用的是反证法
it employs the proof by contradiction

那么统计上
Statistically

依据的是小概率原理
it relies on the small probability principle

那么总结刚才故事当中
To summarize the basic steps

体现出来的假设检验的
of hypothesis testing manifested

基本步骤
in the story just now

那么我们可以
we can

把这个假设检验的步骤
generalize these steps of hypothesis testing

概括为以下的四点
as the following four points

第一点
First

大家首先要建立
everyone shall set up

正确的原假设和备择假设
the knowledge point on

这个知识点
correct original hypothesis and alternative hypothesis

我们在开始的时候
At the very beginning

给大家详细的讲解过
we interpreted in details to everyone

大家注意关键点
The key points to note

就是我们先确定备择假设
are that we shall decide on the alternative hypothesis

再确定原假设
before the original hypothesis

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

那么备择假设
So the alternative hypothesis

是我们想收集证据
is the hypothesis we want to corroborate

予以支持的假设
by collecting evidence

这些关键点
Hope everyone pays attention to

希望大家重视起来
these key points

第二步
At the second step

我们要确定
we shall determine

我们假设检验的检验统计量
the test statistics in our hypothesis test

那么依据检验统计量的取值
According to the value of the test statistics

我们最终对原假设
we make the final judgment on the original hypothesis

和备择假设作出判断
and the alternative hypothesis

那么如何确定
So how to determine

检验统计量的内容呢
the content of test statistics

我们在后续的学习当中
In the subsequent learning

会给大家介绍
we will make an introduction to everyone

第三步
At the third step

就是根据事先确定的显著性水平
we determine the rejection region

确定拒绝域
according to the predetermined level of significance

第四步 就是根据我们
At the fourth step, we compare

第二步确定的
the value of the test statistics determined in the second step

检验统计量的这个取值

那么跟我们的
with

拒绝域的临界值做比较
the critical value of the rejection region

最终作出判断
thereby making the final judgment

下面我们给大家介绍一下
Below we introduce to everyone

检验统计量
something about test statistics

那么检验统计量
Test statistics

就是在假设检验当中
are the sample statistics for making a judgment on the original hypothesis

我们根据样本数据计算得到的
we work out according to the sample data

对原假设进行判断的样本统计量
during the hypothesis test

也就是说
In other words

我们怎么认为小概率事件
whether we believe a small probability event

有没有发生
has happened

事实上就是依据
relies in fact on

检验统计量的取值来确定的
the value of test statistics

那么在假设检验当中
During the hypothesis test

我们需要根据所检验的问题
We need to

样本容量以及总体的分布
determine the test statistics in a scientific way

和总体的分布
based on the problem under test

是否已知等各种条件
whether the sample size

要科学地确定检验统计量
and the distribution of population is known, and other conditions

那么在后续的学习当中
In the subsequent learning

我们会根据具体的情况
we will introduce the corresponding test statistics

给大家介绍相应的检验统计量
to everyone, contingent on the specific situation

下面我们看几张图
Next, let’s look at several graphs

那么第一张图
The first graph

是我们双侧检验的
is a schematic diagram of hypothesis test

假设检验的一个示意图
of our two-sided test

那么在这个图形当中
In this graph

大家看
everyone notices

因为是双侧检验
since it is a two-sided test

我们说不等于
we say “unequal to”

有大于和小于两种情况
includes both greater-than and smaller-than cases

所以在双侧检验当中
So during the two-sided test

大家看
everyone notices

我们两侧的
the shaded areas

这个阴影部分的面积
on both sides

各为α/2
are α/2 each

加起来就是α
summing up to α

也就是我们所说的显著性水平
the so-called level of significance

而显著性水平
While the level of significance

就是我们人为规定的
is the artificially specified

小概率事件的这个概率
probability of a small probability event

也就是说检验统计量的取值
In other words, once the value of test statistics

一旦落在了
falls

这两个阴影部分的
inside either

这个区域的话
of the shaded areas

那么就意味着
it means that

小概率事件发生了
the small probability even happens

那么根据假设检验的反证法
Based on the proof by contradiction for hypothesis testing

和小概率原理的这样一个思想
and on the thought of small probability principle

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

我们认为小概率事件
we do not believe the small probability event

是不会发生的
would happen

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

那么我们就有理由
we have the reason

拒绝原假设了
to reject the original hypothesis

那么这个就是双侧检验的
This is such a rejection region

这样一个拒绝域
of two-sided test

就是在两侧的这个阴影部分
It is the shaded area on either side

就是我们的这个拒绝域
It is the rejection region

接下来的这个图
The next graph

它是一个左侧检验的拒绝域
is the rejection region of left-sided test

大家要注意一点
Everyone shall pay attention that

我们左侧检验的拒绝域在左边
the rejection region of left-sided test is on the left

因为左侧检验的
since the original hypothesis

它的原假设
of left-sided test

是μ大于等于μ0
is μ is greater than or equal to μ0

而备择假设
while the alternative hypothesis

是μ小于μ0
is that μ is smaller than μ0

那么我们在原假设
In the case that

成立的情况下
the original hypothesis holds

不可能发生的这个事件
we say the event unlikely to happen

我们说肯定是在小于这一侧
must fall on the smaller-than side

所以我们假设检验的
This is why the rejection region

这个拒绝域在左侧
of the hypothesis test is on the left

就是在左侧检验的时候
That is, the rejection region of the hypothesis is on the left

我们假设检验的拒绝域在左边
during the left-sided test

那么大家看
Everyone notices

左边这块阴影部分的面积
the shaded area on the left

就是α
is α

这就是我们刚刚说的
This is the level of significance

这个显著性水平
we have just mentioned

就是在这一侧
On this side

我们认为
we consider

如果检验统计量的值
if the value of a test statistic

取到了这个区间
falls within this interval

那么就意味着
it means

小概率事件发生了
the small probability event happens

那么我们同样
We have the reason to reject the original hypothesis

有理由拒绝原假设
as well

同样的道理
By the same token

我们可以看右侧检验
we can examine the right-sided test

在右侧检验当中
In the right-sided test

我们可以看到
we can notice that

右侧检验的拒绝域
its rejection region

就在右边
is on the right side

因为右侧检验的原假设
This is because the original hypothesis of the right-sided test

是μ小于等于μ0
is μ is smaller than or equal to μ0

当然它不可能发生的
which is definitely impossible

这个事件的方向
We say the direction of the event

我们说应该是在
should be

μ大于μ0的这个方向
the one in which μ is greater than μ0

这样的话
This way

我们同样
we determine

同样我们在右边
such a shaded area

确定了这样一个阴影部分
on the right as well

那么它这个阴影部分的面积就是α
The shaded area is α

就是我们人为规定的
the artificially specified

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

也就是我们的显著性水平
also the level of significance

根据刚才的三个图形
According to the three graphs just now

我们就可以确定
we can determine

我们统计量决策的这个规则
the rule for statistic decision-making

当然在给出
Of course

显著性水平α的这个前提下
Given the level of significance, α

那么我们要找出临界值
we shall find the critical value

也就是说
namely

阴影部分的那个界限
the limit to the shaded area

那么可以根据
We can

你是双侧检验或者单侧检验
determine the corresponding critical value

去确定相应的临界值
depending on whether it is a two-sided test or one-sided test

接下来
Next

我们就根据
we compare

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

跟我们确定的临界值进行比较
with the critical value we have determined

在双侧检验的情况下
In the case of two-sided test

如果我们统计量的取值
if the absolute value of the statistic

它的绝对值大于临界值
is greater than the critical value

因为双侧检验
namely

我们既要考虑左侧
the value of the statistic

又要考虑右侧
falls in the rejection region

那么这样的话
then

如果这个检验统计量的
we shall reject the original hypothesis

取值的绝对值大于临界值了
since in two-sided test

那么也就是说
we should

这个统计量的值
consider both the left side

我们说它取在了拒绝域里
and

那么我们要拒绝原假设
the right side

在左侧检验的时候
During the left-sided test

我们的拒绝域在左边
the rejection region is on the left

所以只要统计量的取值
so as long as the value of the statistic

小于我们的临界值
is smaller than the critical value

那么我们就要拒绝原假设
we shall reject the original hypothesis

那么当然
Of course

在右侧检验的时候
during the right-side test

如果我们统计量的这个取值
if the value of the statistic

大于这个临界值的话
is greater than the critical value

那么同样
we shall also reject the original hypothesis

我们也要拒绝原假设
the same way

因为这意味着
After all, this means

小概率事件发生了
given the small probability event has happened

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

那么这一讲
So much for

我们就讲到这里
this lecture

谢谢大家
Thank you everyone