Comments on: When the normal approximation for Student t isn’t good enough http://www.johndcook.com/blog/2008/11/12/normal-approximation-for-student-t-distribution-isnt-good-enough/ The blog of John D. Cook Sat, 11 Feb 2012 22:42:11 -0500 http://wordpress.org/?v=2.8.4 hourly 1 By: John http://www.johndcook.com/blog/2008/11/12/normal-approximation-for-student-t-distribution-isnt-good-enough/comment-page-1/#comment-9289 John Wed, 12 Nov 2008 22:20:26 +0000 http://www.johndcook.com/blog/?p=850#comment-9289 I agree, the one-sample z-test can be left out of a stat class. The t-test is only slightly more complicated. In fact, it may even be easier to teach the t-test first since students wouldn't be distracted by wondering how you could possibly know the variance without knowing the mean. (Assuming they're tracking well enough to be <a href="http://www.johndcook.com/blog/2008/02/27/confusion/" rel="nofollow">confused</a>.) However, I think it's worthwhile to teach the two-sample z-test. The difference of two normals is a normal; the difference between two t distributions is only <i>approximately</i> a t distribution, and there's no simple way to say what the appropriate degrees of freedom are for the approximating t distribution. So in this case, the t-test is sufficiently complicated that it's worthwhile to derive a z-test for a warm-up. I agree, the one-sample z-test can be left out of a stat class. The t-test is only slightly more complicated. In fact, it may even be easier to teach the t-test first since students wouldn’t be distracted by wondering how you could possibly know the variance without knowing the mean. (Assuming they’re tracking well enough to be confused.)

However, I think it’s worthwhile to teach the two-sample z-test. The difference of two normals is a normal; the difference between two t distributions is only approximately a t distribution, and there’s no simple way to say what the appropriate degrees of freedom are for the approximating t distribution. So in this case, the t-test is sufficiently complicated that it’s worthwhile to derive a z-test for a warm-up.

]]> By: John Venier http://www.johndcook.com/blog/2008/11/12/normal-approximation-for-student-t-distribution-isnt-good-enough/comment-page-1/#comment-9287 John Venier Wed, 12 Nov 2008 22:08:27 +0000 http://www.johndcook.com/blog/?p=850#comment-9287 I think it is also interesting / instructive to think of the slightly wider confidence intervals as the price you pay for having to estimate the variance as well as the mean, instead of just the mean. I always thought it was hokey how introductory texts ease you into things by first (or second) posing the questions in terms of estimating the population mean from a sample when the population variance is known. Of course it is done this way for pedagogical reasons, but who has ever heard of such a situation, where the variance is known exactly but the mean is unknown?[1] If I recall correctly most of my students were more confused than enlightened by this strategy. But they were taking my class because they wanted to avoid as much math as possible. I think this helps students more if they are more conversant with math. [1] I have since then thought of at least one plausible situation. Suppose you are measuring some constant quantity (say, length or mass of some object) and the measurement device is known to have an absolute error distributed normally with zero mean and some specific, known variance. Then the results you would get with repeated measurments would have an unknown mean (the quantity of interest) but be normally distributed with known variance. I think it is also interesting / instructive to think of the slightly wider confidence intervals as the price you pay for having to estimate the variance as well as the mean, instead of just the mean.

I always thought it was hokey how introductory texts ease you into things by first (or second) posing the questions in terms of estimating the population mean from a sample when the population variance is known. Of course it is done this way for pedagogical reasons, but who has ever heard of such a situation, where the variance is known exactly but the mean is unknown?[1]

If I recall correctly most of my students were more confused than enlightened by this strategy. But they were taking my class because they wanted to avoid as much math as possible. I think this helps students more if they are more conversant with math.

[1] I have since then thought of at least one plausible situation. Suppose you are measuring some constant quantity (say, length or mass of some object) and the measurement device is known to have an absolute error distributed normally with zero mean and some specific, known variance. Then the results you would get with repeated measurments would have an unknown mean (the quantity of interest) but be normally distributed with known variance.

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