Which of the following best describes the possible values for a chi-square statistic in a goodness of fit test?

• A. The chi-square statistic can take any value greater than or equal to $0$.
• B. The chi-square statistic can take any real value, both positive and negative.
• C. The chi-square statistic can only take values between $0$ and $1$.
• D. The chi-square statistic can only take integer values.

Answer: (A)

Verified step by step guidance

1. Recall that the chi-square statistic in a goodness of fit test is calculated using the formula: \[\chi^2 = \sum \frac{(O_i - E_i)^2}{E_i}\] where \(O_i\) are observed frequencies and \(E_i\) are expected frequencies.
2. Note that each term in the sum, \(\frac{(O_i - E_i)^2}{E_i}\), is a squared difference divided by a positive expected frequency, so it is always greater than or equal to zero.
3. Since the chi-square statistic is a sum of non-negative terms, the overall value of \(\chi^2\) cannot be negative; it must be zero or positive.
4. Understand that the chi-square statistic is not restricted to integer values; it can take any non-negative real number depending on the observed and expected frequencies.
5. Therefore, the possible values for the chi-square statistic are all real numbers greater than or equal to zero.