A statistician is conducting a chi-square goodness-of-fit test and is limited by the cost per individual to conduct the study. The statistician selects a sample size of 39, which is the smallest sample possible that will meet the condition for large expected counts. Which of the following could not be the null hypothesis for the study?

In a chi-square goodness-of-fit test, the null hypothesis states that there is no significant difference between the observed and expected frequencies in different categories. To meet the condition for large expected counts, it is essential that the expected frequency in each category is at least 5.

If the null hypothesis posits that all categories have equal expected frequencies, then the expected frequency in each category would be the total number of observations divided by the number of categories, which in this case would be 39 individuals divided by the number of categories. If the number of categories is too large, it could result in expected counts below 5, which violates the assumption of large expected counts.

For example, if there are 10 categories, each category would be expected to have 3.9 individuals on average, which is not sufficient for a chi-square goodness-of-fit test. Therefore, the null hypothesis should not assert that all categories have equal expected frequencies to ensure that the condition for large expected counts is met.

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