Explanation:
This question asks which statement about chi-square and F distributions is NOT true.
Analysis of each option:
Option A: ✓ TRUE - The chi-square distribution is indeed used to test hypotheses about population variance. Given a sample variance, we can test whether the true population variance differs from a specified value using the chi-square test.
Option B: ✗ FALSE - This is the correct answer. As degrees of freedom increase:
Option C: ✓ TRUE - The F distribution is used in multiple regression analysis to test the joint significance of all regression coefficients (except the intercept). This tests whether the overall regression model is significant.
Option D: ✓ TRUE - There is indeed a mathematical relationship between the F-statistic and R² in multiple regression. The formula is: F = [(R²/k) / ((1-R²)/(n-k-1))], where k is the number of independent variables and n is the sample size.
Key Points:
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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)