Question

Choose the correct statements about chi-square probability density function.
LIST:
A. They start at 0.
B. They are Symmetrical.
C. They have a mean equal to their degree of freedom.
D. They have variance equal to degree of freedom.
E. They have variance equal to 2 $\times$ degree of freedom.

• A. A, C and E only
• B. A, B and D only
• C. A, B and C only
• D. A, C and B only

Hint:

Chi-square distribution: Mean = df, Variance = 2 $\times$ df, starts at 0, positively skewed.

Answer 1

The Correct Option is A

Step 1: Chi-Square Distribution - General Properties.
The chi-square distribution is fundamental to hypothesis testing, particularly for independence and goodness-of-fit tests. Its domain is restricted to non-negative values.

Step 2: Key Characteristics.

• The chi-square distribution has a lower bound of 0, excluding negative values.
• It is positively skewed, not symmetrical, although it approximates a normal distribution as degrees of freedom increase.
• The mean of a chi-square distribution is equivalent to its degrees of freedom ($df$).
• The variance of a chi-square distribution is $2 \times df$.

Step 3: Statement Evaluation.

• (A) Correct: The chi-square distribution begins at 0.
• (B) Incorrect: It is skewed, not symmetrical.
• (C) Correct: The mean equals the degrees of freedom.
• (D) Incorrect: The variance is not equal to the degrees of freedom.
• (E) Correct: The variance is equal to $2 \times df$.

Step 4: Final Determination.
Consequently, statements A, C, and E are accurate.