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If your conclusion differs whether you use t or z, your decision is "at the edge". The total uncertainty (T) in a decision has two parts, sampling error (S) , and everything else (N). We can get a rough handle on the sampling error which the t or z help put in perspective. The "nonsampling" uncertainty has to be taken into account subjectively. However, think of both as "nonnegative". So the sampling uncertainty (S) is the bare minimum of total uncertainty (T). If you always use t, your confidence intervals will be a little wider. You will make slightly more conservative decisions. The question you have to ask yourself is whether with 30 cases it really makes a difference if , for example, your margin of error is +/- $1000 or is +/- $1042. If you are doing "back of the envelope" calculations, you can (a) look up a t, (b) use 1.96 or (c) commit heresy for both camps and use 2 which easier to multiply by. If you are writing a program as an exercise use t. Run the following SPSS syntax to get a handle on how far off you might be by using t instead of z. *your margin of error (one side of confidence interval) * by using t instead of z *will be a small percentage wider. see the variable mult. new file. input program. loop df = 30 to 200. compute t=idf.t(.975,df). compute z=idf.normal(.975,0,1). end case. end loop. end file. end input program. compute mult = (100*t)/z. formats t z (f6.2) mult (pct8.2). list. execute. cache. Ronny Richardson wrote: > A few weeks ago, I posted a message about when to use t and when to use z. > In reviewing the responses, it seems to me that I did a poor job of > explaining my question/concern so I am going to try again. > > I have included a few references this time since one responder doubted the > items to which I was referring. The specific references are listed at the > end of this message. > > Bluman has a figure (2, page 333) that is suppose to show the student "When > to Use the z or t Distribution." I have seen a similar figure in several > different textbooks. The figure is a logic diagram and the first question > is "Is sigma known?" If the answer is yes, the diagram says to use z. I do > not question this; however, I doubt that sigma is ever known in a business > situation and I only have experience with business statistics books. > > If the answer is no, the next question is "Is n>=30?" If the answer is yes, > the diagram says to use z and estimate sigma with s. This is the option I > question and I will return to it briefly. > > In the diagram, if the answer is no to the question about n>=30, you are to > use t. I do not question this either. > > Now, regarding using z when n>=30. If we always use z when n>=30, then you > would never need a t table with greater than 28 degrees of freedom. (n<=29 > would always yield df<=28.) Bluman cuts his off at 28 except for the > infinity row so he is consistent. (The infinity row shows that t becomes z > at infinity.) > > However, other authors go well beyond 30. Aczel (3, inside cover) has > values for 29, 30, 40, 60, and 120, in addition to infinity. Levine (4, > pages E7-E8) has values for 29-100 and then 110 and 112, along with > infinity. I could go on, but you get the point. If you always switch to z > at 30, then why have t tables that go above 28? Again, the infinity entry I > understand, just not the others. > > Berenson states (1, page 373), "However, the t distribution has more area > in the tails and less in the center than down the normal distribution. This > is because sigma is unknown and we are using s to estimate it. Because we > are uncertain of the value of sigma, the values of t that we observe will > be more variable than for Z." So, Berenson seems to me to be saying that > you always use t when you must estimate sigma using s. > > Levine (4, page 424) says roughly the same thing, "However, the t > distribution has more area in the tails and less in the center than does > the normal distribution. This is because sigma is unknown and we are using > s to estimate it. Because we are uncertain of the value sigma, the values > of t that we observe will be more variable than for Z." > > So, I conclude 1) we use z when we know the sigma and either the data is > normally distributed or the sample size is greater than 30 so we can use > the central limit theorem. > > 2) When n<30 and the data is normally distributed, we use t. > > 3) When n is greater than 30 and we do not know sigma, we must estimate > sigma using s so we really should be using t rather than z. > > Now, every single business statistics book I have examined, including the > four referenced below, use z values when performing hypothesis testing or > computing confidence intervals when n>30. > > Are they > > 1. Wrong > 2. Just oversimplifying it without telling the reader > > or am I overlooking something? > > Ronny Richardson > > References > ---------- > (1) Basic Business Statistics, Seventh Edition, Berenson and Levine. > > (2) Elementary Statistics: A Step by Step Approach, Third Edition, Bluman. > > (3) Complete Business Statistics, Fourth Edition, Aczel. > > (4) Statistics for Managers Using Microsoft Excel, Second Edition, Levine, > Berenson, Stephan. > > ================================================================= > Instructions for joining and leaving this list and remarks about > the problem of INAPPROPRIATE MESSAGES are available at > http://jse.stat.ncsu.edu/ > ================================================================= --------------F89CEF3F1CDF5660163AA634 Content-Type: text/x-vcard; charset=us-ascii; name="Arthur.Kendall.vcf" Content-Transfer-Encoding: 7bit Content-Description: Card for Art Kendall Content-Disposition: attachment; filename="Arthur.Kendall.vcf" begin:vcard n:Kendall;Art tel;work:301-864-5570 x-mozilla-html:FALSE adr:;;;;;; version:2.1 email;internet:[EMAIL PROTECTED] fn:Art Kendall end:vcard --------------F89CEF3F1CDF5660163AA634-- ================================================================= Instructions for joining and leaving this list and remarks about the problem of INAPPROPRIATE MESSAGES are available at http://jse.stat.ncsu.edu/ =================================================================
