True or False: The shape of the chi-square distribution depends on the degrees of freedom.

Verified step by step guidance

1. Understand what the chi-square distribution is: it is a probability distribution that arises from the sum of the squares of independent standard normal random variables.
2. Recall that the chi-square distribution is characterized by its degrees of freedom (df), which corresponds to the number of independent standard normal variables squared and summed.
3. Recognize that the shape of the chi-square distribution changes as the degrees of freedom change; for example, with low degrees of freedom, the distribution is highly skewed, and as the degrees of freedom increase, it becomes more symmetric and approaches a normal distribution.
4. Therefore, the degrees of freedom directly influence the shape of the chi-square distribution, affecting its skewness and spread.
5. Conclude that the statement 'The shape of the chi-square distribution depends on the degrees of freedom' is True.

Key Concepts

Chi-Square Distribution

The chi-square distribution is a continuous probability distribution commonly used in hypothesis testing and confidence interval estimation for variance. It is defined as the distribution of a sum of the squares of independent standard normal random variables.

Degrees of Freedom

Degrees of freedom refer to the number of independent values or quantities that can vary in an analysis without breaking any constraints. In the chi-square distribution, degrees of freedom typically correspond to the number of independent standard normal variables squared and summed.

Effect of Degrees of Freedom on Distribution Shape

The shape of the chi-square distribution changes with degrees of freedom: for low degrees, it is highly skewed to the right, while as degrees increase, the distribution becomes more symmetric and approaches a normal distribution. Thus, the degrees of freedom directly influence the distribution's shape.