University-数学讨论区-Probability and Statistic I, II

~HeBe~_@

Probability and Statistic I, II

看到你们也这样开了STPM，SPM的数学讨论区

希望有任何难题可以在此帖讨论哦！


[ 本帖最后由 ~HeBe~_@ 于 27-4-2007 07:06 PM 编辑 ]

~HeBe~_@

A researcher wants to test if the proportion of the students at acertain university who own at least one credit card is lower than thereported 0.8. To do this, he tests the following hypotheses.

        
        H_o : p = 0.8  Versus  H_1 : p < 0.8


In order to use the large sample test for a population proportion, asample of size 120 students was drawn. Out of a random sample of 120students, he finds that 84 students own at least one credit card.

i) What is the approximation distribution for the sample proportion if H_o is true?

ii) The researcher decides to reject H_o if and only if the sampleproportion of the students who own at least one credit card not exceed75%. Approximately what will be the probability of type I error, theprobability of rejecting H_o when H_o is true?

iii)Suppose from the prior research, p=0.7 under H_1, what will be theapproximate probability of type II error, the probability of acceptingHo when H_1 is true associated with the decision rule?

(允许我打英文，谢谢）

[ 本帖最后由 ~HeBe~_@ 于 1-5-2007 05:28 PM 编辑 ]

dunwan2tellu

是 hypothesis testing 的 topic 吧？如果我还记得的话，他的 distribution 是 X ~ Ps(0.8,0.8(0.2)/120)

P(Type I error) = P( X =< 0.75| Ps(0.8,1/750)) = P( Z =< (0.75-0.8)/sqrt[1/750])
=P(Z =< ...)
= ...

P(Type II error) = P(X >= 0.75|Ps(0.7,0.00175)) = P(Z >= (0.75-0.7)/sqrt[0.00175])
= P(Z >= ...)
= ..

~HeBe~_@

对了，你答对了。

DADDY_MUMMY

有没有关于moment & momemnt-generating function 的资料及其应用？
还有关于sampling distribution。。。谢谢

~HeBe~_@

你参考一下，不知道是否有帮到你。

http://ocw.mit.edu/OcwWeb/Mathematics/
http://ocw.mit.edu/OcwWeb/Mathem ... tureNotes/index.htm

DADDY_MUMMY

谢谢你给的资料！这里有很多我还没看过的。。。

~HeBe~_@

不用客气。。。。

DADDY_MUMMY

今天lecture向我们讲解了关于distribution的东东，我不大懂。。。
她说T-distribution & normal distribution看 means?
(normal > means finit/infinite,T > means is unknown)
sampling distribution看 different between population & sample?
F-distribution & Chi-square distriution看 variance?
什么意思？

dunwan2tellu

原帖由 DADDY_MUMMY 于 26-9-2007 03:29 AM 发表
今天lecture向我们讲解了关于distribution的东东，我不大懂。。。
她说T-distribution & normal distribution看 means?
(normal > means finit/infinite,T > means is unknown)
sampling distribution看 dif ...

T-test(T-distribution) and Z-test(normal distribution) is a test which compare the difference between the mean of two groups of data.

F-test (F distribution) : A test to compare the variance between 2 groups of data .

Chi-Square test : A test to determine whether there is a relation between 2 variable (usually for non-parametric) . X^2 = Sum{(O-E)^2/E} <=== analog with variance

Sampling distribution : A type of distribution which drawn from some samples from a population .

yaahoo

这些分布对初学者而言是抽象了一些，主要的原因是因为大多数的课程都跳过了推导的过程，只讨论怎么用而不提为什么。当然推导的过程是相当的复杂。如果各位有兴趣懂得透彻一些，我就滥竽充数的为各位解答疑难也不妨；但是请问较Specific的问题，别说整个都不懂，因为打字是很花时间的。

基本来说，当我们知道某种‘量’的分布(distribution)时，就可以计算它出现的概率。如果出现的概率太小，我们有理由怀疑开始所假设的一些数据可能不对，这就是所谓的假设检定（hypothesis testing)。

DADDY_MUMMY所问的问题是学生的问题（student's problem)，当然该由学生的定理(Student's Theorem)来解决。

DADDY_MUMMY

Student Theorem?

yaahoo

是啊！当年Gosset用笔名Student发表了t分布的成果也就是t-test的基础。

从头说起吧.

如果X_i为Bernoulli分布,也就是只取值0和1;其中出现1的概率为p.那么在n次试验中的1的个数如果设为Y,则你可以自己想象到Y的分布会是B(n,p).

这个例子说明如果有一个随机变数(random variable)的分布已知,我们可以推出其函数的分布.

另一个简单的例子是常态分布(normal distribution), 若X_i,i=1...n为N(mu,sigma^2),则平均统计量X_bar的分布为N(mu,sigma^2/n).

所以同理我们可以推导若X_i,i=1..n为常态分布,其中变异数(Variance)未知,则统计量 T= (X_bar-mu)/(S/sqrt(n)会有t分布.

如果我们观察到的值计算出来的X_bar 与我们所猜测的mu差得比较大,则T的绝对值(absolute value)也大,那么t分布末端的概率将会很小,如果它小于预设的显著水平(significance level)则我们有理由相信该常态分布的mean不是所假设的mu.这就是统计中假设检定的基本思想.


[ 本帖最后由 yaahoo 于 30-9-2007 10:18 AM 编辑 ]

~HeBe~_@

看了后，想知道下，谁是Gosset？

DADDY_MUMMY

你所说的和dunwan2tellu说的有什么区别 ?
你说方法推导的过程复杂,有什么方法更容易let us来明白推导的方法?

yaahoo

你如果真的很有兴趣想知道，那请你找mathematical statistics or statistical inference 的书参考，恕我没有兴趣将证明打出来。另外补充一点，学生必须用点脑筋去了解自己学的科目，不是每样东西都必须讲到你完全明白，只能点到为止。

http://en.wikipedia.org/wiki/Student's_t-distribution

DADDY_MUMMY

~HeBe~_@ 又给了我一些相关网站来看，但有些我还是不大明白。
希望你们能指点，而不是像你所说的全都讲给我。。。

~HeBe~_@

其实简单来讲...

首先你要analysis那个问题是属于哪种问题,可能是population mean,可能是variance，或有可能proportion。

a)
接着，若问题是population mean, mu

你要考虑它给的variance is known or variance is unknown.

若variance is known，then use Z-test
若variance is unknown，then use t-test

b)
若问题是variance,then use Chi-square test


C)
若问题是proportion, z-test.

新力爱力新

a.        Are there differences in the employment status among the five groups of graduates?
b.        Are there differences in income among the five groups of graduates?
c.        Among Business Administration graduates, is there a difference in income between accounting and non-accounting graduates?

要怎样看题目决定用hypothesis test,ANOVA,还是chi-square test?
我很头晕..

mad1427

请问什么是anova呢。。。？