Which of the following statements about test statistics is true?

Question

BlackTom AI · Accepted Answer

Chi-square distributions are sums of squared standard normal variables, so they only take nonnegative values; similarly, F distributions are ratios of two scaled chi-square variables and also take only nonnegative values. Because neither distribution has negative support, neither can be symmetric about zero.

1) Relevant knowledge points and concepts
- Chi-square(ν) = sum of ν independent Z_i^2 where Z_i ~ N(0,1). Its support is [0, ∞). Mean = ν, variance = 2ν.  
- F(d1,d2) = ( (X1/d1) / (X2/d2) ) with X1 ~ χ^2(d1), X2 ~ χ^2(d2). Its support is [0, ∞). Mean = d2/(d2 − 2) for d2 > 2.  
- A distribution symmetric about zero must have support on both positive and negative values and, if the mean exists, mean = 0.

2) Step-by-step reasoning
- Since each Z_i^2 ≥ 0, their sum χ^2(ν) ≥ 0 always; therefore χ^2 cannot have any mass below zero and cannot be symmetric about zero.  
- The F statistic is a ratio of nonnegative quantities and so is also nonnegative; therefore it cannot be symmetric around zero.  
- Both distributions are right-skewed (long tail to the positive side), which is the opposite of symmetry about zero.

3) Calculations / intermediate formulae (illustrative)
- χ^2 mean = ν (positive), variance = 2ν. These positive means show mass is concentrated on positive side.  
- F mean = d2/(d2 − 2) (when d2 > 2), which is > 1 and positive. Positive expected values are inconsistent with symmetry about zero.

4) Why the correct answer is correct
- Symmetry about zero requires negative values; χ^2 and F have no negative values. Therefore "Neither the χ^2 distribution nor the F distribution are symmetric around zero" is true.

5) Why other typical options would be incorrect
- Statements claiming χ^2 or F are symmetric are false because of their nonnegative support.  
- Statements claiming all test statistics are nonnegative are false (e.g., z and t statistics can be negative and their distributions are symmetric about zero).  
- Statements claiming χ^2 or F can be negative or centered at zero contradict their definitions and moments.

6) Conclusion
The correct statement is C: Neither the χ^2 distribution nor the F distribution are symmetric around zero. This matches the provided answer.

Which of the following statements about test statistics is true?Single choice

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