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?单项选择题

类似问题

A renewable energy project involves the construction of a solar power plant. The project team is using Monte Carlo simulation for risk modeling. What is the primary benefit of using Monte Carlo simulation in this context?

In which of the following situations does random allocation NOT occur?

The NSW census indicates that graduates of USyd are more likely to have a job than graduates of UNSW. This indicates:

If we were trying to select a sample that is representative of a population, under which of the following circumstances would we need to be LEAST careful about how we selected the sample from the population?

For the following 3 scatterplots, order them in terms of strength of correlation from the STONGEST to the WEAKEST:

The alternative hypothesis is

What is the probability that a randomly selected BBus student does not intend to enrol in postgraduate studies and identifies as female?[Fill in the blank]

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单