A probability distribution of the sum of squares of several normally distributed variables is used to test hypotheses about categorical data and model fit. Which distribution is this?

• A. Uniform distribution
• B. Exponential distribution
• C. Chi-square distribution
• D. Normal distribution

Answer: (C)

Chi-square distribution arises when you sum the squares of independent standard normal variables. If you have k such variables Z1, Z2, ..., Zk ~ N(0, 1) and form Q = sum(Zi^2), then Q follows a chi-square distribution with k degrees of freedom. This statistic is exactly what’s used to test hypotheses about categorical data and model fit: you compare observed counts to expected counts, and the sum of squared standardized differences (under the null hypothesis) has a chi-square distribution with the appropriate degrees of freedom. That’s why chi-square is the go-to distribution for goodness-of-fit tests, tests of independence in contingency tables, and assessing how well a model fits the data. The other options don’t capture this sum-of-squares-of-normals idea: the uniform and exponential distributions describe different kinds of data, and the normal distribution is for a single variable rather than the sum of squares of several.