Sampling Error Explained
A chi-square distribution is an asymmetrical family of distributions. A chi-square distribution with \(v\) degrees of freedom is the distribution of the sum of the squares of \(v\) independent standard normally distributed random variables. Intuitively, chi-square distributions take only non-negative random variables.
A chi-square distribution is used to test the variance of a population that is distributed normally.
In a summary, the following are the properties of a chi-square distribution:
An F-distribution is used to test the equality of variances of two normally distributed populations from two independent random samples.
The following are the properties of an F-distribution:
The F-distribution is the ratio of two chi-square distributions with degrees of freedom \(m\) and \(n\), respectively, where each chi-square has first been divided by its degrees of freedom, i.e.,
Where \(m\) is the numerator degrees of freedom and \(n\) is the denominator degrees of freedom.
Which of the following are most likely common characteristics of F-distribution and chi-square distribution?
- Both can take only positive value.
- Both are defined by two parameters.
- Both are negatively skewed distributions.
The correct answer is A.
Both F-distribution and chi-square distribution can only take non-negative values.
B is incorrect. A chi-square distribution is defined by one parameter (i.e., n-1 degrees of freedom), while an F-distribution is defined by parameters, i.e., degrees of freedom of the numerator (m) and degrees of freedom of the denominator (n).
C is incorrect. Both the F-distribution and the chi-square distribution are positively skewed distributions.
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