In a chi-square test, the test statistic can be written as $X=\sum \frac{{(O_i-E_i)}^2}{E_i}$. When will this chi-square statistic equal $0$?

• A. When the degrees of freedom equal $0$
• B. When $O_i=E_i$ for every category $i$
• C. When at least one category has $O_i=E_i$
• D. When the sample size is small (for example, $n<5$)

Answer: (B)

Verified step by step guidance

1. Recall the formula for the chi-square test statistic: \[X = \sum \frac{(O_i - E_i)^2}{E_i}\] where \(O_i\) is the observed frequency and \(E_i\) is the expected frequency for category \(i\).
2. Understand that the chi-square statistic measures the overall difference between observed and expected frequencies across all categories.
3. For the chi-square statistic to equal zero, each term in the summation must be zero because all terms are non-negative (squares and division by positive expected counts).
4. This means that for every category \(i\), the observed frequency must exactly equal the expected frequency: \[O_i = E_i \quad \text{for all } i.\]
5. Therefore, the chi-square statistic equals zero only when there is a perfect match between observed and expected counts in every category.