Which of the following statements about the chi-square distribution is…
2024
Which of the following statements about the chi-square distribution is incorrect?
Answer: C. Its variance is three times the number of degrees of freedom. — ConceptA chi-square random variable with k degrees of freedom can be represented as the sum of the squares of k independent standard normal variables. Its…
- A.
For a large number of degrees of freedom, the chi-square distribution approaches the normal distribution.
- B.
The chi-square distribution provides the reference distribution for a chi-square test statistic.
- C.
Its variance is three times the number of degrees of freedom.
- D.
Its variance is twice the number of degrees of freedom.
Show answer & explanation
Correct answer: C
Concept
A chi-square random variable with k degrees of freedom can be represented as the sum of the squares of k independent standard normal variables.
Its mean is k and its variance is 2k. It is non-negative and right-skewed for small k, while its standardized shape approaches normality as k grows.
Application
The statement assigning variance 3k conflicts with Var(X) = 2k. The statement assigning variance 2k agrees with the distribution's spread, while the remaining statements describe its large-k approximation and its use in hypothesis testing.
Contrast
Large-k normal approximation: the distribution's skewness decreases as k increases.
Test-statistic use: a chi-square statistic is compared with chi-square critical values under the null hypothesis.
Variance 3k: this coefficient does not follow from the sum-of-squares definition.
Variance 2k: variances of k independent squared standard-normal terms add to 2k.
Cross-check
Let k = 4.
Using Var(X) = 2k gives 2 × 4 = 8.
The expression 3k gives 3 × 4 = 12.
The values 8 and 12 differ, so the 3k statement is inconsistent with the variance formula.
Result
Therefore, the incorrect statement is: “Its variance is three times the number of degrees of freedom.”