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
They are not oversimplifying, they are complexifying. To quote Polya
"How to solve it" : "If you need rules, use this one first: 1) Use your
own brains first".
Sigma is hardly ever known, so you must use t. Then why not simply tell
the students: "use the t table as far as it goes, (usually around
n=120), and after that, use the n=\infty line (which corresponds to the
normal distribution). Then there is no need for a rule for "when to use
z, when to use t".
Kjetil Halvorsen
>
> 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.
>
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