I have a doubt as to what is the assumption for normality
in a t-test.
Two possibilities:
I think it is for the sample data I have, while a dear
friend says it relates to the population and Central Limit
Theorem. I wrote:
"I went back to the reference on the t-test normality
assumptions and the assumption refers to the individual
values, not the averages. In part it (Introduction to
Statistical Quality Control, Montgomery, 2nd Ed., pg 79,
section 3-3.2 Tests on Means of Normal Distributions,
Variance Unknown) reads:
"Suppose that x is a normal random variable with unknown
mean m and unknown variance s2. We wish to test the
hypothesis that the mean equals standard value mo. Because
the variance is unknown, we must make the additional
assumption that the random variable is normally
distributed. The normality assumption is needed to
formally develop the statistical test, but moderate
departures from normality will not seriously affect the
results."
He wrote:
"As I understand it, the assumption is that the variable
(population) distribution is normal, but not necessarily
the distribution of your sample.
If the variable (population) distribution is normal, then
one can be assured that the central limit theorem applies:
that is, the true mean of the variable (population)
distribution equals the true mean of the sampling
distribution of the mean (on which, I believe, is developed
the t-test). Of course, if the variable (population)
distribution is normal, then it is reasonable to expect
that the central tendency of any random sample will
approach the mean value of the sampling distribution of the
mean and also that any departure from normality will be
small, but could be present ( ! ! ) and most probably will
on small samples (N<30). My point the other day was, that
you should not be so concern if rigorous non-normality test
reveals your sample data as non-normal, if you have some
knowledge that your variable (population) distribution
should be normal."
Any comments?
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