Re: Student's t vs. z tests
I don't believe anyone has bothered to define what they mean by a z-test. There are two issues that must be dealt with: (1), What statistic is to be used and (2), what distribution is to be used to assess the size of that statistic. I contend that a z "statistic," viz., (Ybar-mu0)/(sigma/sqrt(n)), is pretty useless since it almost never is a "statistic," i.e., it involves an unknown parameter, sigma. So I spend maybe 2 minutes of class time on that "statistic" and go on to (Ybar-mu0)/(s/sqrt(n)) which I would call a t statistic. I then would say if n is large we may use the normal distribution to assess the size of this statistic (using an _extension_ of the Central Limit Effect). Finally, I get to the t distribution for assessing the size of the t statistic when dealing with smaller sample sizes (and under certain additional assumptions). Jon Cryer At 08:46 AM 4/20/01 +1000, you wrote: >There is at least two very good pedagogical reasons for teaching z >tests. Both the z and t tests are based on normality - the t test is >used only because the model standard deviation is unknown or rather, >there is no assumed value for it. Whether or not this is in practice >'always' the case is irrelevant from the point of view of understanding >what is going on. 'In an ideal world' we would do z tests! But in >practice we usually cannot assume a value for sigma, so we are forced to >use a t test. This is less powerful than the z test. So we pay a price >for the lack of knowledge - if we don't know sigma, we pay for this in >lowered power of the test. > >As a general principle, this is a fundamental aspect of statistics - I >rate it as one of the reasons why students learn statistics! Lack of >knowledge costs! > >So the two good reasons are - that the z test is the basis for the t, >and the understanding that knowledge has a very direct value. > >I hasten to add that 'knowledge' here is always understood to be >'assumed knowledge' - as it always is in statistics. > >My eight cents worth. > >Alan > > >-- >Alan McLean ([EMAIL PROTECTED]) >Department of Econometrics and Business Statistics >Monash University, Caulfield Campus, Melbourne >Tel: +61 03 9903 2102Fax: +61 03 9903 2007 > > >= >Instructions for joining and leaving this list and remarks about >the problem of INAPPROPRIATE MESSAGES are available at > http://jse.stat.ncsu.edu/ >===== > > ___ --- | \ Jon Cryer, Professor [EMAIL PROTECTED] ( ) Dept. of Statistics www.stat.uiowa.edu/~jcryer \\_University and Actuarial Science office 319-335-0819 \ * \of Iowa The University of Iowa dept. 319-335-0706 \/Hawkeyes Iowa City, IA 52242 FAX319-335-3017 |__ ) --- V "It ain't so much the things we don't know that get us into trouble. It's the things we do know that just ain't so." --Artemus Ward = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: Student's t vs. z tests
Why not introduce hypothesis testing in a binomial setting where there are no nuisance parameters and p-values, power, alpha, beta,... may be obtained easily and exactly from the Binomial distribution? Jon Cryer At 01:48 AM 4/20/01 -0400, you wrote: >At 11:47 AM 4/19/01 -0500, Christopher J. Mecklin wrote: >>As a reply to Dennis' comments: >> >>If we deleted the z-test and went right to t-test, I believe that >>students' understanding of p-value would be even worse... > > >i don't follow the logic here ... are you saying that instead of their >understanding being "bad" it will be worse? if so, not sure that this >is a decrement other than trivial > >what makes using a normal model ... and say zs of +/- 1.96 ... any "more >meaningful" to understand p values ... ? is it that they only learn ONE >critical value? and that is simpler to keep neatly arranged in their mind? > >as i see it, until we talk to students about the normal distribution ... >being some probability distribution where, you can find subpart areas at >various baseline values and out (or inbetween) ... there is nothing >inherently sensible about a normal distribution either ... and certainly i >don't see anything that makes this discussion based on a normal >distribution more inherently understandable than using a probability >distribution based on t ... you still have to look for subpart areas ... >beyond some baseline values ... or between baseline values ... > >since t distributions and unit normal distributions look very similar ... >except when df is really small (and even there, they LOOK the same it is >just that ts are somewhat wider) ... seems like whatever applies to one ... >for good or for bad ... applies about the same for the other ... > >i would be appreciative of ANY good logical argument or empirical data that >suggests that if we use unit normal distributions and z values ... z >intervals and z tests ... to INTRODUCE the notions of confidence intervals >and/or simple hypothesis testing ... that students somehow UNDERSTAND these >notions better ... > >i contend that we have no evidence of this ... it is just something that we >think ... and thus we do it that way > > > >= >Instructions for joining and leaving this list and remarks about >the problem of INAPPROPRIATE MESSAGES are available at > http://jse.stat.ncsu.edu/ >========= > > ___ --- | \ Jon Cryer, Professor [EMAIL PROTECTED] ( ) Dept. of Statistics www.stat.uiowa.edu/~jcryer \\_University and Actuarial Science office 319-335-0819 \ * \of Iowa The University of Iowa dept. 319-335-0706 \/Hawkeyes Iowa City, IA 52242 FAX319-335-3017 |__ ) --- V "It ain't so much the things we don't know that get us into trouble. It's the things we do know that just ain't so." --Artemus Ward = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
Re: convolution of binomials
Of course, if p1=p2 the answer is Binomial(n1+n2,p1). Otherwise, there is no "easy" answer (i.e., no standard distribution). Jon Cryer At 05:45 AM 2/5/01 GMT, you wrote: >Kumara Sastry wrote in message <[EMAIL PROTECTED]>... >>Suppose, X ~ Binomial(n1,p1), Y~Binomial(n2,p2) , and X and Y are >>independent. Also, Z = X+Y. Can anyone please comment on what the pdf >>of Z is? >> >>Thanks >>Kumara >> > >Pr(Z=z) = Sum from l to u of p1(r).p2(z-r) > >where p1 is the first binomial probability, and p2 is the second. >The upper & lower limits of summation, l & u, are not necessarily >0 and z but: >l = max(0, z-n2) and u = min(z, n1) > >I hope I have got that right. >I doubt if the sum simplifies much. > > >-- >Alan Miller, Retired Scientist (Statistician) >CSIRO Mathematical & Information Sciences >Alan.Miller -at- vic.cmis.csiro.au >http://www.ozemail.com.au/~milleraj >http://users.bigpond.net.au/amiller/ > > > > > >= >Instructions for joining and leaving this list and remarks about >the problem of INAPPROPRIATE MESSAGES are available at > http://jse.stat.ncsu.edu/ >===== > > ___ --- | \ Jon Cryer, Professor [EMAIL PROTECTED] ( ) Dept. of Statistics www.stat.uiowa.edu/~jcryer \\_University and Actuarial Science office 319-335-0819 \ * \of Iowa The University of Iowa dept. 319-335-0706 \/Hawkeyes Iowa City, IA 52242 FAX319-335-3017 |__ ) --- V "It ain't so much the things we don't know that get us into trouble. It's the things we do know that just ain't so." --Artemus Ward = Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =
