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Answer: As degrees of freedom increase, the chi-square approaches a lognormal distribution and the F distribution approaches a gamma distribution
**Explanation:** - **Option A**: Correct - The chi-square distribution is indeed used for hypothesis testing about population variance. - **Option B**: Incorrect - As degrees of freedom increase, the chi-square distribution approaches a normal distribution, not a lognormal distribution. The F distribution approaches a normal distribution as well, not a gamma distribution. - **Option C**: Correct - The F distribution is used to test the overall significance of a multiple regression model. - **Option D**: Correct - There is a mathematical relationship between the F statistic and R² in multiple regression analysis. Therefore, option B is the exception as it contains incorrect statements about the asymptotic behavior of these distributions.
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Each of the following is true about the chi-square and F distributions EXCEPT:
A
The chi-square distribution is used to test a hypothesis about a sample variance; i.e., given an observed sample variance, is the true population variance different than a specified value?
B
As degrees of freedom increase, the chi-square approaches a lognormal distribution and the F distribution approaches a gamma distribution
C
The F distribution is used to test the joint hypothesis that the partial slope coefficients in a multiple regression are significant; i.e., is the overall multiple regression significant?
D
Given a computed F ratio, where F ratio = (ESS/df)/(SSR/df), and sample size (n), we can compute the coefficient of determination (R^2) in a multiple regression with (k) independent variables (regressors)
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