8.4.1 Example analysis: single population mean test 例题解析：单个总体均值检验在线视频

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

这一讲
In this lecture

我要给大家介绍
I will introduce to everyone

一个总体均值的
the issue of hypothesis testing

假设检验的问题
on a population mean

那么一个总体均值的假设检验
To begin

我们首先可以区分两大类情况
we can distinguish two major types of situations

一个是在大样本的情况下
One is the situation of a large sample

一个是在小样本的情况下
the other being the situation of a small sample

那么大样本的界定
In defining a large sample

我们一般把
we typically

样本容量大于等于30的样本
call a sample whose size is greater than or equal to 30

称其为大样本
a large sample

那么在第六章当中
In Chapter VI

我们学习了
we learned

中心极限定理
the central limit theorem

那么中心极限定理告诉我们
which tells us that

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

不论总体服从什么样的分布
whatever distribution the population obeys

只要样本容量足够大
as long as the sample size is large enough

那么样本均值
the sample mean

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

服从什么样的分布呢
What is the distribution obeyed

X_bar近似服从均值为μ
X_bar approximately obeys

也就是μ就是总体的均值
the normal distribution

方差为（公式如上）
with μ, the population mean

那么σ平方
and δ2, the variance (as in the formula above)

就是总体的方差
of the population

这样的正态分布
Such is the normal distribution

那么接下来
Next

我们在确定检验统计量的时候
we would use

就会用到
this conclusion we have learned

我们学习过的这个结论
to determine test statistics

当σ平方已知
If δ2 is known

也就是总体方差
namely in the case where the population variance

已知的情况下
is known

（公式如上）
(as in the formula above)

那么这个时候
at this moment

我们说这个随机变量
we say this random variable

它是服从标准正态分布的
obeys the standard normal distribution

那么我们就会用到
Then we would use

这样一个服从标准
such a test statistic that obeys the standard

正态分布的检验统计量
normal distribution

我们往往把这个检验统计量
We typically call this test statistic

称其为z检验统计量
z-test statistic

那么另外一种情况
In another case

是总体方差未知
where the population variance is unknown

总体方差未知
the population variance is unknown

我们和总体方差已知的情况
This is similar to

是相似的
if the population variance is known

那么X_bar在转化成
When converting X_bar into

服从标准正态分布的
the z-statistic

z统计量的时候
obeying the standard normal distribution

我们说σ不知道
we say δ is unknown

我们是在大样本的情况下
in the circumstance of large sample

可以用样本的标准差
where the standard deviation of samples

s来代替σ
s can be used to replace δ

换句话说
In other words

在大样本σ平方
in the case that δ2 of a large sample

未知的情况下
is unknown

我们的检验统计量
our z-test

z检验统计量
statistic

可以写成（公式如上）
can be written as (in the formula above)

这里我们要提醒大家一点
Here we shall remind everyone that

我刚才提到的
the z-test statistic

z检验统计量
I have mentioned just now

那么我们构造检验统计量
is always constructed

始终是在原假设
under the premise that

成立的这个前提下构造的
the original hypothesis holds

所以我们刚才
So we mean

说减μ的地方
subtracting μ0

事实上
in fact

我们就是减μ0
where we said subtracting μ0

因为我们的原假设
After all, our original hypothesis

就是μ等于μ0
is μ is equal to μ0

当然有些同学说
Of course, some students would say

那你可能是一个单侧检验
you may be doing a one-sided test

那么我们大家都知道
As is known to all

左侧检验
in the left-sided test

我们的原假设是
the original hypothesis is

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

右侧检验的原假设
whereas in the right-sided test the original hypothesis

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

那么我们都只需要考虑
Then the only thing we need to do is consider

μ等于μ0的情况
the case where μ is equal to μ0

那么也就是说
In other words

不论是双侧检验还是单侧检验
be it a two-sided test or one-sided test

我们在构造检验统计量的时候
when constructing a test statistic

我们都是认为
we always believe

原假设是成立的
the original hypothesis is true

也就是说
That is to say

在检验统计量当中
in test statistics

μ都是等于μ0的
μ is always equal to μ0

那么这就是
This is all about

我们在单个总体均值的假设检验
the hypothesis testing on a single population mean

在大样本的这个前提下
Under the premise of large sample

我们分两种情况考虑
we consider in two separate cases

那么出现的
Then the two emerging

这两个检验统计量
test statistics

都是z检验统计量
are both z-test statistics

接下来我们来看一个例子
Next let’s examine an example

那么这个例子
This example

是根据美国高尔夫球协会的准则
involves the codes of The United States Golf Association

只有射程和滚动距离
by which only golf whose range and rolling distance

平均正好是280码的高尔夫球
are exactly 280 yards on average

才可以在比赛中使用
can be used in the tournament

假定某公司
Suppose some company

最近开发了一种
has recently developed a

高技术生产方法
high-tech production method

用这种方法生产的
by which the golfs are produced

高尔夫球的射程和滚动距离
to have an average range and rolling distance

平均为280码
of 280 yards

现在抽取一个
Now a

有36个高尔夫球的随机样本
random sample with 36 golfs is drawn

来检验该公司的陈述是否为真
to test whether the company’s statement is true

那么数据如下表
Data are shown in the following table

显著性水平规定是005
the level of significance being 005

那么这个问题
To approach this problem

我们首先分析
what we shall analyze first

当然属于大样本的情况
is the case of a large sample

因为他抽取了一个
Since such a random sample

36个高尔夫球的
with 36 golfs

这样一个随机的样本
is drawn

那么这个是一个大样本的情况
this is the case of large sample

那么它的总体的方差
Is the population variance

知不知道呢
known

很明显
Obviously

在这个题目当中
no information about population variance

我们说它没有给出
is provided

关于总体方差的任何信息
in this problem

但是它给了我们样本的数据
but the sample data are provided

我们可以通过
We can

这个样本的数据
calculate the standard deviation, s, of the sample

去计算样本标准差s
via the data on this sample

这里提醒大家
Here everyone is reminded

我们样本标准差的计算
that the standard deviation of the sample is calculated

当然是在样本方差的基础上开方
by, of course, extracting the square root on the base of sample variance

但是样本方差
but that

我们在除的时候
we divide the sample variance

我们不是除n而是除n-1
not by n but by n-1

这个是我们无偏性的一个要求
This is a requirement for Unbiased

接下来我们就可以
Following this, we can

来完成这个假设检验的过程
complete this hypothesis testing procedure

那么根据刚才题目的陈述
According to the statement of the problem just now

我们可以明显的发现
we can clearly notice

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

因为标准要求就是280码
Given the standard requirement of 280 yards

这样的话
we shall begin

我们就首先提出
by proposing

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

原假设是μ等于280
The original hypothesis is μ is equal to 280

备择假设
whereas the alternative hypothesis

是μ不等于280
is μ is unequal to 280

那么规定的显著性水平
Then the specified level of significance

α等于0.05
α is equal to 0.05

样本容量是36
and the sample size is 36

那么我们根据之前的分析
Based on the previous analysis

在这种情形下
under such a circumstance

我们使用z检验统计量
we shall use the z-test statistic

那么z检验统计量
In the z test statistic

我们具体的做法就是（公式如上）
(as shown in the formula above)

这里s是样本标准差
Where s is the standard deviation of the sample

那么这个检验统计量
Then we can substitute the data

我们把数据代进去算一下
into this test statistic and calculate

那么X_bar是275.8
X_bar is 275.8

然后减μ0 280
Then subtract μ0 280

然后除以样本标准差
and then divide the result by the standard deviation of the sample

通过计算
Through calculation

我们算出来等于12
we work it out to be 12

然后除以根号n
Next, we divide the result by the square root of n

就是根号36
namely the square root of 36

那么我们做出来的
So we figure out

这个z检验统计量的具体的取值
the exact value of the z-test statistic

是负的075
to be -0.75

那么这个时候
At this moment

大家可以对照
everyone can contrast

我们的这个图形来看一下
against this graph and take a look

在这个图形当中
In this graph

我们说因为α等于005
we say since α=0.05

所以我们的临界值
the critical value

就是正负196
is ±1.96

这个是我们可以查表确定的
which we can make sure by looking up the table

那么这里
Here

我们说我们这个
when it comes to the

负的075这个取值
value, -0.75

我们说出现在了接受域
we say it falls in the acceptance region

没有出现在拒绝域
not in the rejection region

所以我们最终的决策
So our final decision

是不拒绝H0
is not to reject H0

也就是说
In other words

没有证据表明
there is no evidence that

该公司的陈述有问题
the company’s statement is problematic

接下来我们再看一个例子
Next, let’s examine another example

这个例子
This example

是某市的一家公司
relates a company in a city

他生产一种新型的轮胎
which produces a novel tire

这种新型的轮胎的设计规格
The novel tyre is designed by the specification

是平均行驶里程
that the average mileage

至少为28000英里
shall be at least 28000 miles

他随机抽取了30只轮胎
The sampler draws 30 tyres at random

作为一个样本进行检测
as a sample to test

结果样本均值是27500英里
It turns out that the sample mean is 27500 miles

样本标准差是1000英里
and that the standard deviation of the sample is 1000 miles

那么采用005的显著性水平
At the level of significance of 0.05

检验是否有足够的证据
is there enough evidence for the test

拒绝轮胎的平均行驶里程
to reject the statement that

至少为28000英里的陈述
the average mileage of the tyres is at least 28000 miles

那么在这个问题上
On this problem

我们说提出假设
we say proposing the hypothesis

是一个难点
is a difficulty

大家通过仔细地
Everyone can read this question

阅读这个题目
carefully

我们说他想搜集证据
We say what the sampler wants to reject

予以拒绝的
by collecting evidence

是轮胎的平均行驶里程
is the statement that the average mileage of the tyres

至少为28000英里
is at least 28000 miles

所以我们的原假设
So our original hypothesis

是μ大于等于28000
is μ is greater than or equal to 28000

这是它这个题目当中
as indicated

告诉我们的
in the question

所以我之前也说
This is why I have previously told

大家先确定备择假设
everyone to decide on the alternative hypothesis first

就是说他拒绝的东西
That means what he rejects

已经非常明显了
is very obvious

那他支持的
and thus what he supports

当然就是
is definitely

平均行驶里程小于28000
the point that the average mileage is smaller than 28000

那么原假设和备择假设确定了
Now that the original hypothesis and the alternative hypothesis have been decided on

那么很明显
it is very clear

这是一个左侧检验的问题
that this a left-sided test problem

我们再来看一看已知的条件
Let’s take a further look at the known condition

我们现在已知的就是
What is known to us now is

n现在等于30
n is equal to 30 at present

是一个大样本
implying a large sample

然后X_bar知道
is also known

然后样本的标准差
and we are also informed

也告诉我们了
of the standard deviation of the samples

α等于005
α=0.05

因为是左侧检验
Since it is a left-sided test

那么我们的拒绝域
the rejection region

就在左边
is on the left side

大家可以通过查表来确定
Everyone can determine the critical value of the rejection region

拒绝域的这个临界值
by looking up the table

就是负的1.645
This value is -1.645

下面我们就来看一看
Now let’s see whether

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

有没有出现在拒绝域
falls in the rejection region

那么在这个问题上
On this problem

因为它是大样本
given the large sample

仍然是总体方差未知的情况
case remains that the population variance is unknown

我们使用z检验统计量
We use the z-test statistic

我们把数据代进去算一下
Let’s substitute the data and calculate

那么X_bar是27500减去μ
Then X_bar is 27500 minus μ

当然我们是假定μ就等于μ0了
Of course, we postulate μ=μ0

所以减去28000
hence subtracting 28000

然后除以s是1000
before dividing the result by s to get 1000

然后比上根号n
Next, we divide the result by the square root of n

就是根号30
namely the square root of 30

那么我们做出来的
So we work out

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

是负的274
to be -274

很明显 这个取值
Obviously, we say

我们说出现在了
this value appears

拒绝域里面
in the rejection region

那么这样的话
Such being the case

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

也就是说
In other words

我们不能接受
We cannot accept

该公司关于轮胎的陈述
the company’s statement about the tires

那么我们再来看一个例子
Let’s examine another example

某一小麦品种的平均产量
The average yield of some wheat variety

是5200千克
is 5200 kg

一家研究机构
A research institution

对小麦品种进行了改良
ameliorates the wheat variety

以期提高产量
in order to boost the yield

为检验改良后的新品种产量
To test whether the yield of the ameliorated new variety

是否有显著提高
has boosted significantly

随机抽取了36个地块进行试种
the sampler randomly draws 36 land blocks for trial planting

得到的样本
The sample turns out that

平均产量为5275千克
the average yield is 5275 kg

标准差为120
and that the standard deviation is 120

试检验改良后的新品种产量
Try to test whether the yield of the ameliorated new variety

是否有显著提高
has boosted significantly

α等于0.05
given α=0.05

那么在这个问题上
On this problem

我们说一个难点就是
we say a difficulty is

怎么样提出假设
how to propose the hypothesis

那么在这个问题的陈述当中
In the statement of this problem

我们会发现
we find

那么研究者
what the researcher

想收集证据
hopes to demonstrate

予以证明的东西
by collecting evidence

是改良后的新品种产量
is the statement that the yield of the ameliorated new variety

是有显著地提高
has boosted significantly

那么也就是说
In other words

它是一个右侧检验的问题
this is a right-sided test problem

那么我们首先提出
Still, we begin by proposing

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

H0就是μ小于等于5200
H0 is μ is smaller than or equal to 5200

H1就是μ大于5200
whereas H1 is μ is greater than 5200

α等于005 n等于36
α=0.05, and n=36

那么还是我们刚才所说的
This remains the case of the large sample

大样本的情形
as we have mentioned just now

那么在这个问题当中
In this problem

我们仍然把已知的信息代进去
we still substitute the known information

算一下z检验统计量的取值
to figure out the value of z-test statistic

现在算出来
Now we work it out

我们是375
to be 375

那么很明显
Evidently

这个375
the value 375

大于我们的临界值1645
is greater than the critical value 1645

这里大家可以发现
Everyone can find here

右侧检验的拒绝域
the rejection region of right-sided test

肯定在右边
must be on the right side

所以我们最终的判断
So our final decision

是拒绝原假设
is to reject the original hypothesis

也就是说
In other words

改良后的新品种产量
the yield of the ameliorated new variety

有显著的提高
has boosted significantly

那么再接下来
Following this

我们要讨论小样本的情况
we shall discuss the circumstance of small sample

那么小样本的情况
Regarding the case of small sample

我们这个地方
we actually

我们的前提条件
have two

事实上有两个
preconditions

一个是总体服从正态（分布）
One is that the population obeys the normal distribution

并且是小样本
the other being that the sample size is small

再接下来
Further

我们就区分两种具体情况
we have two specific cases to distinguish:

一个是总体的方差已知
One is that the population variance is known

一个是总体的方差未知
the other being that the population variance is unknown

那么在总体方差已知的情况下
In the case that the population variance is known

大家想一想
everyone just think over

我们在第六章
In Chapter VI

也给过大家一个结论
we have provided a conclusion that

如果总体服从正态（分布）
if the population obeys the normal distribution

那么X_bar
then X_bar

它是一定服从正态（分布）的
must obey the normal distribution

那么它服从均值为μ
one where

方差为（公式如上）
the mean is μ

这样的正态分布
and the variance is (as in the formula above)

所以在σ平方已知的情况下
Given that δ2 is known

那么我们仍然构造的是
what we still construct

一个z检验统计量
is a z-test statistic

也就是说
In other words

我们用（公式如上）
we use (the formula as above)

那么这个检验统计量
Thus the test statistic

它是服从标准正态分布的
obeys the standard normal distribution

跟我们之前探讨的
Much similar to

大样本的情况
the case of large sample

是非常相似的
we have explored previously

那么在这个问题上
on this problem

我们说最为特殊的情况
we say the most special case

就是在σ平方未知的情况下
is what test statistic we should use

我们应该使用什么检验统计量
if δ2 is unknown

在这里大家要注意
Here everyone shall pay attention

事实上
in fact

我们是包含了三个条件
we involve three conditions

这三个条件
none of which

是缺一不可的
is dispensable

也就是说首先
In other words, in the first place

我们总体要服从正态（分布）
the population shall obey the normal distribution

接下来
Next

我们面对的是一个小样本
what we face is a small sample

并且我们总体的方差未知
and the population variance is unknown

在这种情况下
Under such a circumstance

我们用（公式如上）
we use (the formula as above)

那么这个统计量
So we say the statistic

我们说它不再服从
no longer obeys

标准正态分布
the standard normal distribution

也就是说在大样本的情况下
Put it another way, in the case of the large sample

我们可以用样本标准差
we can substitute the standard deviation of the sample

去代掉总体标准差
for the standard deviation of the population

但是在小样本的情况下
while in the case of small sample

你做了这种代换之后
once you substitute this way

它这个新的统计量
the new statistic

它的分布不再服从
no longer obeys

标准正态分布
the standard normal distribution

而是服从自由度为n-1的t分布
but the t-distribution with degree of freedom of n–1

也就是说
In other words

在这种情况下
under such a circumstance

我们要使用一个
we shall use a

t检验统计量
t-test statistic

来进行检验
to conduct the test

下面我们也给大家
Below we take a few examples

举一些例子
for everyone

某工厂生产的铁丝
The tensile resistance of the iron wires produced by some factory

抗拉力服从正态分布
obeys the normal distribution

且已知其平均抗拉力
Known are their average tensile resistance

为570公斤
of 570 kg

标准差为8公斤
and standard deviation of 8 kg

由于更换原材料
The raw materials are changed

虽然标准差不会有变化
Despite no variation in the standard deviation

但不知其抗拉力
it is unknown whether their tensile resistance

是否与原来一样
is the same as before

现从生产的铁丝中
Now 10 samples are drawn

抽取10个样品
from the produced iron wires

求得其平均抗拉力
whose average tensile resistance is found

为575公斤
to be 575 kg

试问能否认为平均抗拉力
Can it be considered that the average tensile resistance

无显著的变化
has no significant variation

下面我们来分析一下这道题
Now let’s analyze this problem

那么从这个题目当中
In the problem

我们可以看出
we can see

这明显是一个
it is obviously a

双侧检验的问题
two-sided test problem

因为它在题目当中说
Since it is told in the problem that

原来是570
the original average tensile resistance is 570

问现在有没有显著的变化
and the question asks whether there is significant variation

所以我们这道题目的原假设
the original hypothesis of this problem

就是μ等于570
is μ=570

备择假设
whereas the alternative hypothesis

是μ不等于570
is μ=570

α等于005 n等于10
Given α=0.05 and n=10

这明显是一个小样本的问题
this obviously is a small sample problem

那么在这个题目当中
So in this problem

我们说有一个非常重要的条件
we say a very important condition

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

并且σ已知
Given δ is known

那么在这种情况下
in such a circumstance

根据我们刚才的分析
according to our analysis just now

我们仍然是采用
we still adopt

z检验统计量
the z-test statistic

那么也就是说
In other words

（公式如上）
(the formula is shown as above)

它是服从标准正态分布的
It obeys the standard normal distribution

那么我们把实际的样本数据
We substitute the actual sample data

代进去看一看
and see

这个z检验统计量
what is the value

它的取值是多少
of the z-test statistic

那么我们把数据代进去之后
After substituting the data

我们计算出来的
we figure out

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

是1976
to be 1976

α等于005
Given α=0.05

我们双侧检验
the critical value on both sides

那么两侧的临界值
of our two-sided test

是正负196
is ±196

所以我们可以发现
So we can find

我们检验统计量的这个值
the value of the test statistic

我们说出现在了拒绝域
appears in the rejection region

所以我们最终的这个决策
So our final decision

是拒绝原假设
is to reject the original hypothesis

也就是说
In other words

有显著的证据表明
there is significant evidence that

更换原材料后
after changing the raw materials

产品的抗拉力有显著的变化
the tensile resistance of the products has changed significantly

那么这个就是
That is all about

我们刚才的这道题目
the problem we examined just now

接下来我们再看一道题
Next, let’s examine another example

一种汽车配件的平均长度
The average length of an automobile accessory

要求为12厘米
is required to be 12 cm

高于或低于该标准
Lengths above or below this standard

均被认为是不合格的
are all deemed unqualified

汽车生产企业
While purchasing accessories

在购进配件的时候
automobile manufacturers

通常是经过招标
usually go through an invitation for bids

然后对中标的配件提供商
before testing the samples provided

提供的样品进行检验
by accessory suppliers winning the bids

以决定是否购进
to decide whether or not to purchase

现对一个配件提供商
Now 10 samples provided by

提供的10个样本
an accessory supplier

进行了检验
are tested

假定该供货商
Assume the length of the accessories

生产的配件程度
produced by this supplier

服从正态分布
obeys the normal distribution

在005的显著性水平下
At the level of significance of 0.05

检验该供货商提供的配件
test whether the accessories provided by this supplier

是否符合要求
meet the requirement

那么这个题目
We call this problem

我们说明显是一个小样本的问题
a small sample problem

那么并且
Moreover

我们在这个题目当中也发现
we also find in this problem

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

并且总体的标准差是不知道的
Plus, the standard deviation of the population is unknown

我们只知道样本的标准差
and what is only known to us is the standard deviation of the sample

另外这是明显的一个
Furthermore, this is obviously a

双侧检验的问题
two-sided test problem

那么我们的原假设
So our original hypothesis

是μ等于12
is μ=12

备择假设
whereas the alternative hypothesis

是μ不等于12
is μ≠12

那么这个时候
At this moment

我们所使用的检验统计量
the test statistic we use is

就是一个t检验统计量
a t-test statistic

因为在正态分布的前提下
Given the premise of normal distribution

那么总体的方差未知
where the population variance is unknown

并且是小样本
and the premise of small sample

这种情况
then (the formula is shown as above)

我们刚刚分析过
as we have just analyzed

那么（公式如上）
such cases

它是服从自由度
It obeys the t-distribution

为n-1的t分布的
with the degree of freedom of n–1

所以这里
So here

我们先把检验统计量的取值求出来
we first find the value of the test statistic

那么我们把数据代进去计算
By substituting the data, we figure out

那么这个检验统计量的取值
the value of this test statistic

是负的07035
to be -0.7035

接下来对于双侧检验
Next, regarding the two-sided test

我们需要查表
we need to look up the table

查t分布的表
specifically the table of t-distribution

那么这里
Here

我们服从
it obeys

自由度为n-1的t分布
the t-distribution with the degree of freedom of n–1

所以这里
So here

它的自由度就是10-1=9
its degree of freedom is 10–1 = 9

然后我们说两侧的
Next, we say the respective

拒绝域的面积各为0025
area of rejection region on either side is 0.025

然后我们把相应的临界值查出来
Then we look up the corresponding critical value

那么这个临界值
which

就是正负2.62
is ±2.62

那么我们t检验统计量的取值
The value of the t-test statistic

刚才是负的0.7035
was -0.7035 just now

那么当然
Of course

我们这个取值
this value

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

所以我们最终的决策是
so our final decision is

不拒绝原假设
not to reject the original hypothesis

也就是说
In other words

没有证据表明
there is no evidence that

该供货商提供的零件
the accessories provided by this supplier

不符合要求
fall short of the requirement

下面我们总结一下
Below let’s sum up

关于一个
the hypothesis testing

总体均值的假设检验
on a population mean

我们首先
First off, we

要区分大样本和小样本
shall make a distinction between the large sample and small sample

在大样本的情况下
In the case of large sample

无论σ是否知道
whether δ is known

我们使用的都是
what we use all the same is

z检验统计量
the z-test statistic

那么在小样本的情况下
While in the case of small sample

σ已知
δ is known

当然还有一个前提
Of course another premise is

总体服从正态（分布）
the population obeys the normal distribution

那么我们仍然使用的是
We still use

z检验统计量
the z-test statistic

比较特殊的一个情况
A relatively special case

就是我刚才给大家介绍的
is, as I have introduced to everyone just now

三个条件缺一不可
none of the three conditions is dispensable:

总体服从正态（分布）
Population obeys the normal distribution

σ未知
δ is unknown

而且是小样本
and the sample size is small

在这种情况下
In such cases

我们要使用t检验统计量
we shall use the t-test statistic

那么t检验统计量的形式
So the t-test statistic has the form

就是（公式如上）
as (shown in the above formula)

它是服从自由度为
It obeys the t-distribution

n-1的t分布的
with the degree of freedom of n–1

所以大家要特别注意
So everyone shall pay special attention

这一个特殊的情况
this is a special case

好 我们这一讲就讲到这里
Well, so much for our lecture

谢谢大家
Thank you, everyone