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More generally, where there are K categories, suppose that the null hypothesis specifies pt, p2, . . . , Pk for the probabilities that an observation falls into the categories. We assume that these possibilities are mutually exclusive and collectively exhaustiveâ€”that is, each sample observation must belong to one of the categories and cannot belong to more than one. In that case, the hypothesized probabilities must sum to 1, that is

Then, if there are n sample observations, the expected numbers in each category, under the null hypothesis, will be

This is shown in Table 11.4.

We have a null hypothesis about the population that specifies the probabilities that a sample observation will fall into each possible category. The sample observations are to be used to check this hypothesis. If the numbers of sample values observed in each category are very close to those expected if the null hypothesis were true, this fact would lend support to that hypothesis. We might, in such circumstances,

TABLE 11.4 Observed and expected numbers for n observations and K categories

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