7.6.2 Determination of sample size: a prelude to sampling (2) 样本容量的确定：抽样的前奏（二）在线视频

在建立总体比例的
In constructing the interval estimation

区间估计时
of population proportion

确定样本容量的原理
the principle for sample size determination

和刚才前面的均值估计时
is similar to the principle of sample size determination

样本容量的确定原理相类似
of the mean estimation in the foregoing description

只不过此处
Except that

仍然用P乘以（1-P）
we still use P times 1-P）

置换了前面的σ平方
to replace the previous σ squared

如果总体的标准差已知的话
If the standard deviation of the population is known

那么这个时候的P乘以（1-P）
at this time, the p times (1-P)

就是总体的方差 大写的(P)
is the population variance P (upper-cased)

如果是总体的成数
If the population percentage

如果是总体的比例
If the population proportion

一开始是不知道的
is unknown at first

那么同样地
similarly,

我们要用样本的方差
we use the sample variance

去估计总体的方差
to estimate the variance of the population

那符号相应地都换成了
The symbol then will be changed accordingly

小写的p乘以（1-p）
to p (lowercased) times (1-p)

所以总体比例估计的时候
So in population proportion estimation

样本容量的确定式子
the formula for sample size determination is

从构造上来讲
structurally

和均值的估计的时候是一样的
the same as for mean estimation

还是等于（公式如上）
It is still equal to (formula as above)

这个式子是一样的
The formulae are the same

只不过由于比例的特殊性
Just because of the particularity of proportion

它的σ平方
the σ squared

被p乘以（1-p）所置换了
is replaced by p times (1-p)

这个时候
At this time

你就要用样本的方差
you use the sample variance

去估计总体的方差
to estimate the population variance

同样地我们来看一个例子
Let’s have a look at an example

国家质检总局
The AQSIQ

最近三年对旅行箱包
has in recent three years conducted

每年进行了质量检测
annual quality inspection of travel bags and suitcases

2015年抽查了
In 2015, the spot inspection of

118批次产品的结果显示
118 batches of products showed that

64.4%的箱包
64.4% of the bags and suitcases

符合质量标准
conformed to the quality standard

2016年的抽查又将来临
The 2016 spot check will come again

假定研究者希望极限误差不超过8%
Assume that the investigator wants the limiting error to be no more than 8%

应抽取多少批次的
How many batches of travel bags and suitcases

旅行箱包组成样本
should be checked to form the sample

假定区间估计的时候
Suppose in interval estimation,

置信水平为95.45%
the confidence level is 95.45%

首先我们分析题目里边
Let’s first analyze the information disclosed

告诉我们的信息
in the question

有1-α是等于95.45%
1-α is equal to 5.45%

那么给定这个1-α的情况下
Then, when the 1-α is given

（公式如上）等于2
(Formula as above) is equal to 2

去年的调查
the previous year's survey

对于今年来讲
for this year

就是一个参考
is a reference

就是以往的调查显示
What the previous survey showed

所以P就相当于
is equivalent to

已经告诉我们了
telling us that P

等于64.4%
is equivalent to 64.4%

这一次抽查
In the coming sampling inspection

我希望极限误差不要超过8%
I hope the limiting error will not be more than 860

也就是E等于0.08
So E is equal to 0.08

有这些信息告诉我们
With such information known to us

必要的样本容量
the necessary sample size

很快可以计算出来
can be computed quickly

（公式如上）
(Formula as above)

把数字代入进去
Substitute the numbers and we get

运算结果是143.29
the result of 143.29

同样地我们要将这个结果
And again, we're going to take this result

取下一个整数
to enter the next integer

也就是144
That is 144

也就是我们的必要的样本容量
which is our necessary sample size

如果你抽144批次的产品来调查
If you take 144 batches of products in the survey

根据我们前面的信息
according to our previous information,

它计算的总体比例的极限误差
the calculated limiting error of population proportion

应该不会超过8%
should be no more than 8%

同样地还是那句话
Once again

如果条件允许的情况下
if conditions permit,

你还可以把这个样本容量再扩大
you can also expand this sample size even further

比如150批次
150 batches

比如160批次
or 160 batches

这都是可行的
Are both OK

我们现在所计算的
The necessary sample size

这个叫必要的样本容量是指
we are computing refers to

最少的样本容量了
the minimum sample size

同样地关于比例的估计
Similarly, about proportion estimation

我们做几点说明
we need to note something

由于总体比例P
As the population proportion P

在大多数的情况下
is also unknown

也是未知的
in most cases,

同样地有下面的一些方法
there are also the following methods

可以获得P的值
to enable us to get the value of P

第一
The first method is

使用有同样或者类似单元的
to use the previous sample proportion

以前样本的样本比例
with the same or similar units

这个是最常用的一种方法
This is a mostly used method

刚才例子里边就是用这个方法
It was this method we used in the foregoing example

第二
The second method is

抽取一个预备样本
to draw a preparatory sample

进行试验性研究
for experimental study

用试验性样本的比例
Use the proportion of this experimental sample

作为P的估计值
as the estimate of P

也就是在小范围里边
That is to get a sample

先抽一个
in a small scope

第三 运用对P值的判断
The third method is to use a judged

或者是最好的猜测
or best guessed value of P

假如你对这个事物
If you sufficiently know

了解得足够多
about the thing

那么你可以根据
you may, based on

你对它的了解
your understanding

来进行一个推测
conduct an inference

它可能是多少
about its possible number

比如对于我来讲
For example, for me

你如果问四六级的通过率
if you ask the passing rate of CET-4 and CET-6

那根据我的了解
according to my understanding

第一次参加四六级考试的学生
for students taking CET-4 and CET-6 for the first time

他的通过率
the passing rate is

可能大概会在75%左右
probably around 75%

那这个就是根据我的了解
That’s the number got

所得到的数字
according to my understanding

那别人的了解
But other’s understanding

他可能不一样
may be different

或者你对这个事物
Or you are fully unknown

完全不了解
about this thing

那就没办法推测了
Then, there's no way to speculate

第四个
The fourth method

如果上面的方法都不适用
if none of the above works

我们还有最后一根稻草
we have one last straw

就是P=0.5
That is p=0.5

大家可以想一想
You may think

为什么P=0.5
why P=0.5

可以来帮我们
can help us

推算样本容量呢
to figure out the sample size

回忆一下比例的方差有什么特殊的地方
Remember what's special about

有什么特殊的地方
the variance of proportion

对
Right

比例的方差是等于P乘以（1-P）
The variance of proportion is equal to P times 1- P

那P乘以（1-P）
Then P times 1- P

它是不是一个二次函数
Is it a quadratic function

是一个开口向下的二次函数
a quadratic function with a downward opening

那这个二次函数
Then, does this quadratic function

它是不是有一个最大值
have a maximum value

在什么时候可以实现
that can be realized at a certain time

是不是恰好在p=0.5的时候
Is it when p=0.5

我们的方差p乘以（1-p）
our variance p times 1-p

就可以取到最大值为1/4
can get a maximum value of 1/4

如果方差取到了最大值
If the variance gets the maximum value

意味着（公式如上）
this mean (formula as above)

这个式子里边
in this formula

方差的部分
the variance part

就已经实现了最大化
is realized maximization

其他的任意的P
Other sample sizes computed from

所计算出来的容量
any P

都会比这个式子
will be smaller than

所计算出来的容量
that computed from

要来得更小一些
this formula

也就是说
That is to say

如果我们采用P=0.5
if we adopt P=0.5

它其实是一个最保守的估计
it's actually the most conservative estimate

它可以适用任何其他的情况
It can be applied to any other situation

另外
Besides

可能还有同学会提出一个疑问
you may have a question

什么疑问呢
What question

就是 老师
That is, teacher

前面并没有说它就是大样本
it was not said to be a large sample

为什么你就使用了（公式如上）呢
why did you use the formula (as above)

前面不是讲过
It was said that

只有大样本的情形下
only in the case of a large sample

才可以用（公式如上）吗
could this formula (as above) be used

小样本的情形不是要用（公式如上）
in the case of a small sample, the formula (as above)

做临界值的吗
should be used to find the critical values

对的
Right

这个问题也是非常好的一个问题
That's a very good question

那我们知道
We know

如果我们要用（公式如上）
if we want to use the formula as above

一定要先已知n
the n must be known first

才能够查它对应的自由度
in order to find its corresponding degree of freedom

所对应的临界值
its corresponding critical value

所以我们现在是推算样本容量
Now, we are computing the sample size

小n一定是未知的
and the lowercased n is sure to be unknown

那么t的值
then, the value of t

是一定没有办法来查找的
is surely unable to be found

这是第一个原因
That's the first reason

第二个原因就是
The second reason is that

在我们实际开始抽样之前
before we actually start sampling

我们就期待它是一个
we expect it to be

大样本的调查
a large sample survey

因为通过前面的学习
Because through the previous study

我们也知道
we also know that

大样本它是能够得到一个
A large sample can get a

更准确的结论的
more accurate conclusion

这就是我们前面学过的
This is the consistency evaluation criterion

估计量评价标准里边的
we learned earlier

一致性的评价标准
in the estimator evaluation criteria

因为样本容量越大
Because the larger the sample size

它的标准差是越来越小的
the smaller the standard deviation

所以基于这两个原因
So for those two reasons

我们在推算样本容量的时候
when computing the sample size

直接使用（公式如上）就可以了
we can directly use the formula as above

那接下来给大家一个课堂练习
So I'm going to give you an exercise in class

来把刚才的这些知识回顾一下
to review the knowledge we've just learned

当然最后如果你已经
Of course, if finally

成功的闯关过来的话
you've made it through

我想你都已经知道了
I think you have known

如何来计算样本容量
how to compute the sample size

这就告诉我们
That tells us

如果题目里边给定的信息
if the given information in the question

既有均值估计
include some conditions

也有比例估计时的
in both mean estimation

一些条件的时候
and proportion estimation

那么我们是分两条路
we have two paths

分别地把均值估计时的
to compute the sample size

样本容量计算出来
in mean estimation

以及比例估计时的样本（容量）计算出来
and the sample size in proportion estimation

然后在两者里边选更大的一个
Then, we choose the larger one of them

作为我们的必要的样本容量
as our necessary sample size

那么两个条件
Then, both conditions

它就都能够同时满足了
can be satisfied at the same time

所以下一次再看到这样的情形
So when we see such a situation next time

我们就不要觉得无从下手了
do not think there is no way to start

分两条路单独进行
Just to take both paths respectively

来进行样本容量的计算
to compute the sample size

就可以了
OK

如果是题目里边告诉了你
If the question gives you

三个不同的比例
three different proportions and

然后要你计算样本容量的话
ask you to compute the sample size

我想大家也应该会判断
I think you can judge

我应该怎么样来算
how to do

原则上来讲
In principle

给了你三个不同的比例
when it gives three different proportions

那就把它们分别代入到我们的
substitute them respectively into the

比例估计式的式子里面
formula of proportion estimation

去计算就可以了
and do the calculation

不过根据咱们前面的介绍
However, according to our previous introduction

比例越接近0.5的时候
the closer the proportion gets to 0.5

它对应的方差就会越大
the bigger the corresponding variance

那么方差越大
The larger the variance

自然它对应的样本容量
the bigger the corresponding sample size

也就会越大
will be

所以如果是同时给了你
So, if three different proportions are given for you

三个不同的比例要你来计算
to compute

必要的样本容量的时候
the necessary sample size

你可以先预判一下
you may judge first

哪一个离0.5更近
which is closer to 0.5 and

直接选它代入进去运算
then directly substitute into the formula

准没错的
It won’t be wrong

这就是样本容量的确定方法
That‘s is the method of sample size determination

那通过这个学习
Through the study

我想大家也应该明白了
I think you should have understood

为什么要把样本容量的确定
why we put the introduction of the sample size determination

放在最后一个问题来介绍
at last

原因是
The reason is that

在我们计算样本容量的时候
when computing the sample size

我们是使用了
we use

极限误差的计算公式
the formula for calculating the limiting error

所以我们是采用了倒叙的方法
So we're using a flashback method

来安排了整个章节的内容
to arrange the contents of the whole chapter

到这里为止
By now,

这一章的内容 参数估计
the content of this chapter, parameter estimation

这一章的内容
the total content of this chapter

就全部介绍给大家了
has been introduced to you

我们稍微总结一下
Let’s make a little summary

在这一章里边
In this chapter,

我们首先学习了
we first learned

参数估计的一般问题
the general issues of parameter estimation

包括点估计
including point estimate

包括估计量评价的优良标准
and excellence criteria for estimator evaluation

第二 我们介绍了
Second, we introduced

区间估计的基本原理
the basic principles of interval estimation

在这个基础之上
On this basis

我们分别介绍了
we introduced

总体均值 总体比例
population mean, population proportion

以及总体方差的区间估计的方法
and the interval estimation method of the population variance

最后我们介绍了
Finally, we introduced

样本容量的确定方法
the determination method of sample size

这就是我们这一章的所有的内容
That's all we have in this chapter

谢谢大家
Thank you