Which of the following is an appropriate description of the $\chi -\mathrm{square}$ distribution?

• A. It is a continuous distribution that is always bell-shaped regardless of the degrees of freedom.
• B. It is a discrete distribution used for modeling the number of successes in a fixed number of trials.
• C. It is a symmetric distribution defined for all real numbers.
• D. It is a continuous probability distribution that is skewed to the right and defined only for non-negative values.

Answer: (D)

Verified step by step guidance

1. Step 1: Understand the nature of the chi-square distribution. It is a continuous probability distribution, meaning it can take on any value within a range, not just specific discrete values.
2. Step 2: Recognize the shape of the chi-square distribution. Unlike the normal distribution, it is not symmetric or bell-shaped. Instead, it is skewed to the right, especially for lower degrees of freedom.
3. Step 3: Note the domain of the chi-square distribution. It is defined only for non-negative values (i.e., values greater than or equal to zero) because it is based on the sum of squared standard normal variables.
4. Step 4: Differentiate the chi-square distribution from other distributions mentioned: it is not discrete (so it cannot model counts like the binomial distribution), and it is not symmetric or defined over all real numbers (unlike the normal distribution).
5. Step 5: Conclude that the correct description of the chi-square distribution is that it is a continuous probability distribution, skewed to the right, and defined only for non-negative values.