# Goodness of fit test for a normal distribution

In the example from the web site I was trying out this problem in page 107 about Goodness of fit test for a normal distribution.Question is about analysis of fat content of hamburgers.

I understand that we need to have the two intervals -$\infty$< x< 26 and $40< x<$$\infty to find the class probabilities of intervals 26< x <28 and 38< x<40. But why should the expected frequencies of the two intervals -\infty< x< 26 and 40< x<$$\infty$ be also calculated and used for chi Squared value. These two categories are not there in the given data set.

Is it necessary to break our intervals such that they start from -infinity and end from infinity as this is a normal distribution?

• Suppose there were no observations in one of the middle categories--say, in the interval $(32,34]$. That would be a glaring departure from a Normal distribution, wouldn't it? So should you or should you not include that interval in your calculation? – whuber Dec 9 '13 at 19:59
• I think that should be included and since expected frequency is less than 5, then it should be combined and create a common category (30,34] – clarkson Dec 9 '13 at 20:14
• You're right that it's because of the normal distribution being on $(-\infty,\infty)$. They're included because if the data are actually normal, those classes are possible... they just had observed frequency zero in the sample. You divide the continuous null distribution up to compute the expecteds, not the observed distribution. If you exclude those classes your test doesn't have the desired properties. Incidentally, the chi-square test has terrible power as a test of normality. If you come to edit your question, you might like to fix the typo in 'hamburger'. – Glen_b Dec 9 '13 at 22:27
• Now its "hamburger". Thanks for the explanation.So even for a Poisson distribution, say for example the variable x is the number of malfunctions per model and the given data set has values for x=0,1,2,3,4,5 when applying chi squared test since Poisson is valid for all positive values of x, we should include the category x>=6 right? – clarkson Dec 10 '13 at 8:57
• Then your lecturer was wrong. The neglected term may not make much difference in practice, but it should be there. Also, 7 cells and 6 d.f. – Nick Cox Dec 10 '13 at 18:05