There has been such an emphasis on this listserv about the mantra "z is for proportions, t is for means" that we may have gone a little overboard! The more careful mantra is probably something like "when doing inference, z is for proportions and t is for means (unless we somehow know the population sigma)". That parenthetical is important in questions about determining sample size when planning a study in which students are expected to use a given value that represents the population standard deviation. And yes, there have been both free response and multiple choice questions of that sort on the AP exam.
If a question is being posed about calculating a probability involving the sampling distribution of x-bar, it's quite likely that a value will be given for the population mean and standard deviation. In that setting (assuming we have either a Normal population distribution or a large sample size so the CLT applies), students should be using the standard Normal (z) distribution to find the desired probability. Using the t distribution might yield a similar answer, but there isn't a valid theoretical reason to do so, which would probably mean that a student wouldn't receive full credit for using t.
Hope this helps.
Daren Starnes Math Department Chair & Master Teacher The Lawrenceville School
david indelicato <email@example.com> 02/18/2011 09:20 AM Please respond to david indelicato <firstname.lastname@example.org>
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Subject [ap-stat] when sigma is known
Hey stats teachers,
I came across a multiple choice question about a sampling distribution for sample means in which a student is asked to compute an area. In the problem, it is stated that the population standard deviation is known. The question expects the student to use the z-distribution instead of the t-distribution. The solution using z (.046) and the solution using t (.050) are both given as choices. So if the student uses t, they will get the wrong answer on this multiple choice question.
I was wondering, if this situation showed up in an open response question and a student used t instead of z, would their solution be close enough to get full credit? Is it wrong to use t when we know the population standard deviation or just not as accuarate as using z? Thanks in advance for any thoughts.