Interpretation of Shapiro-Wilk test
I'm pretty new to statistics and I need your help.
I have a small sample, as follows:
H4U 0.269 0.357 0.2 0.221 0.275 0.277 0.253 0.127 0.246
I ran the Shapiro-Wilk test using R:
and I got the following result:
W = 0.9502, p-value = 0.6921
Now, if I assume the significance level at 0.05 than the p-value is larger then alpha (0.6921 > 0.05) and I cannot reject the null hypothesis about the normal distribution, but does it allow me to say that the sample has a normal distribution?
No - you cannot say "the sample has a normal distribution" or "the sample comes from a population which has a normal distribution", but only "you cannot reject the hypothesis that the sample comes from a population which has a normal distribution".
In fact the sample does not have a normal distribution (see the qqplot below), but you would not expect it to as it is only a sample. The question as to the distribution of the underlying population remains open.
qqnorm( c(0.269, 0.357, 0.2, 0.221, 0.275, 0.277, 0.253, 0.127, 0.246) )
the qqplot looks pretty like normal i think... you can try `qqnorm(rnorm(9))` several times...
@Tomas: Perhaps better to say "the qqplot looks as if it could have come from a normal population". It might instead have come from a distribution with heavier tails.
Yes, `qqnorm(runif(9))` can produce similar result. So we cannot actually say anything...
what is the difference between "the sample has a normal distribution" and "the sample comes from a population which has a normal distribution"?
A normal distribution is a continuous distribution over all the reals. A sample (finite or even countably infinite) cannot have this kind of distribution itself, even if it is drawn from a population having this distribution.
Totally agree with the interpretation for the p value! Cannot reject != normality