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:

    shapiro.test(precisionH4U$H4U)
    

    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?

    Thanks!

  • Henry

    Henry Correct answer

    9 years ago

    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) )
    

    qqplot

    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

    Actually evidence shows that SW test is more powerful than KS test (based on QQplot). You plot QQ, but without confidence intervals.

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Content dated before 6/26/2020 9:53 AM