Degrees of freedom (statistics) - Quiz
  • 1. What is the number of degrees of freedom in statistics?
A) The number of values in the final calculation of a statistic that are free to vary
B) The number of variables ignored in a linear model
C) The total sample size minus the standard deviation
D) The constant value used in a hypothesis test
  • 2. How are the degrees of freedom of an estimate generally calculated?
A) Sum of all observations divided by two
B) Number of parameters minus the sample mean
C) Total sample size multiplied by the number of variables
D) Number of independent scores minus the number of parameters used as intermediate steps
  • 3. What is the degrees of freedom when estimating the variance from a random sample of N independent scores?
A) N - 1
B) N
C) N + 1
D) N - 2
  • 4. Mathematically, what does degrees of freedom represent?
A) The number of dimensions of the domain of a random vector
B) The total length of a random vector
C) The sum of squares of a sample
D) The probability of a type I error
  • 5. In which context is the term 'degrees of freedom' most often used?
A) Discrete probability trees
B) Bayesian prior estimation
C) Linear models
D) Non-linear calculus
  • 6. What does the dimension of the subspace represent in linear models?
A) The mean of the distribution
B) The sample size
C) The variance estimate
D) The number of degrees of freedom
  • 7. What actually defines degrees of freedom according to the text?
A) Hypothesis testing rules
B) Sample variance formulas
C) The underlying geometry
D) Introductory textbooks
  • 8. Who first elaborated the modern definition and usage of degrees of freedom?
A) William Sealy Gosset
B) Carl Friedrich Gauss
C) Ronald Fisher
D) Student's teacher
  • 9. In what year was Gosset's article introducing the concept published?
A) 1922
B) 1950
C) 1821
D) 1908
  • 10. Under what pen name did William Sealy Gosset publish his 1908 article?
A) Biometrika
B) Student
C) Fisher
D) Gauss
  • 11. Who popularized the term 'degrees of freedom' beginning with his 1922 work on chi squares?
A) Ronald Fisher
B) William Sealy Gosset
C) Carl Friedrich Gauss
D) Christensen
  • 12. What is the typical symbol for degrees of freedom in equations?
A) μ
B) n
C) ν (lowercase Greek letter nu)
D) σ
  • 13. What abbreviation is commonly used for degrees of freedom in text and tables?
A) d.f.
B) fre
C) dfom
D) deg
  • 14. What does modern statistical usage typically reserve the letter 'n' for?
A) Sample size
B) Degrees of freedom
C) Number of parameters
D) Variance
  • 15. How are degrees of freedom typically noted when reporting statistical test results?
A) At the very beginning of the research paper
B) Beside the test statistic as a subscript or in parentheses
C) Multiplied by the sample variance
D) As an exponent of the mean
  • 16. How many degrees of freedom does an n-dimensional random vector have when it can lie anywhere in n-dimensional space?
A) n
B) 1
C) n + 1
D) n - 1
  • 17. What constraint applies to the residual vector components in a one-sample mean decomposition?
A) They must equal the sample mean
B) They must sum up to zero
C) They must equal N minus one
D) They must be strictly positive
  • 18. How many degrees of freedom does the residual vector have in an n-dimensional sample mean decomposition?
A) 1
B) n - 2
C) n - 1
D) n
  • 19. What distribution does the residual sum of squares follow when data points are normally distributed with mean 0 and variance σ²?
A) A standard normal distribution with n degrees of freedom
B) A Student's t-distribution with n degrees of freedom
C) A scaled chi-squared distribution with n - 1 degrees of freedom
D) An F-distribution with 1 degree of freedom
  • 20. Under what condition does the one-sample t-test statistic follow a Student's t distribution with n - 1 degrees of freedom?
A) When the data is not normally distributed
B) When the variance is equal to zero
C) When the hypothesized mean is correct
D) When the sample size is less than three
  • 21. In the comparison of three means with equal sample sizes, what is the total number of degrees of freedom for the observation vector?
A) 3n - 3
B) n
C) 3n
D) n - 1
  • 22. In the three-group mean decomposition, how many degrees of freedom does the vector for the overall mean have?
A) 3n
B) 1
C) 2
D) 3
  • 23. In the three-group mean decomposition, how many degrees of freedom does the vector representing the group means minus the overall mean have?
A) n - 1
B) 1
C) 2
D) 3
  • 24. In the three-group mean decomposition, how many degrees of freedom are in the residual vector?
A) 3n
B) 2
C) n - 1
D) 3n - 3
  • 25. Who was Carl Friedrich Gauss in relation to the history of degrees of freedom?
A) A German astronomer and mathematician whose work recognized the basic concept as early as 1821
B) The creator of Student's t-distribution
C) The author of Christensen (2002)
D) An English statistician who popularized the term
  • 26. What is the residual vector's geometric role in the sample mean decomposition?
A) A vector pointing randomly in n-dimensional space
B) Least-squares projection onto the (n - 1)-dimensional orthogonal complement
C) Oblique projection onto the subspace spanned by the vector of 1's
D) A multiple of the vector of 1's
  • 27. In the bivariate regression example, what variables are considered fixed?
A) a
B) ei
C) xi
D) Yi
  • 28. In multiple regression, how many parameters are estimated including the intercept?
A) p
B) n
C) p - 1
D) n - p
  • 29. What does $b1$ represent in the multiple regression equation?
A) The intercept
B) The error term
C) The dependent variable
D) The first slope
  • 30. What are people usually interested in instead of the component vectors themselves in statistical testing problems?
A) Standard deviations
B) Squared lengths
C) Sample sizes
D) Mean values
  • 31. What is the residual or error sum-of-squares denoted as in the text?
A) SSE
B) SST
C) SSM
D) ANOVA
  • 32. In which type of setting do sums-of-squares no longer have scaled chi-squared distributions?
A) One-way Analysis of Variance
B) Simple linear regression
C) Balanced ANOVA models
D) Unbalanced split-plot designs
  • 33. Do numbers representing fractional degrees of freedom have a genuine degrees-of-freedom interpretation?
A) Only in one-way ANOVA
B) Only under the null hypothesis
C) Yes
D) No
  • 34. Where does the degrees of freedom arise from in the numerator of the normal variable example statistic?
A) The total sum-of-squares
B) The population variance
C) The sample mean
D) The residual sum-of-squares
  • 35. What type of regression methods include ridge regression and smoothing splines?
A) Ordinary least squares
B) Multivariate analysis
C) Non-linear classification
D) Non-standard regression methods
  • 36. How can the fitted values of non-standard regression procedures be expressed?
A) Jy
B) By
C) Xy
D) Hy
  • 37. What is one possible definition for regression effective degrees of freedom mentioned in the text?
A) tr(I - H)
B) tr(H'Σ)
C) tr(H)
D) tr(2I - H)
  • 38. Which of the following is also a valid expression for regression effective degrees of freedom?
A) tr(Σ - H)
B) tr(H'H)
C) tr(I - H')
D) tr(HH'H)
  • 39. What formula involving the hat matrix is listed as a regression effective degrees of freedom definition alongside tr(H'H)?
A) tr(2I – H H')
B) tr(H + HH')
C) tr(H – 2H'H)
D) tr(2H – H H')
  • 40. Which approximation is named in the text for regression effective degrees of freedom?
A) Bonferroni approximation
B) Satterthwaite approximation
C) Gaussian approximation
D) Chi-squared approximation
  • 41. In the case of linear regression, what is the hat matrix H equal to?
A) X(X 'X)−1X '
B) X(X 'X)X '
C) X'(X 'X)−1X
D) (X 'X)−1X 'X
  • 42. The trace of the hat matrix, tr(H), is equal to the sum of which elements?
A) x_ij
B) h_ij
C) h_ii
D) r_i
  • 43. What simple smoothing matrix example is used in the text to help conceptualize effective degrees of freedom?
A) A chi-squared matrix
B) A linear smoother
C) A polynomial filter
D) A Gaussian blur
  • 44. What is used to mitigate data noise when considering smoothing matrices?
A) A standard linear regression hat matrix
B) A simple smoothing matrix like a Gaussian blur
C) An observation covariance matrix
D) The Satterthwaite approximation
  • 45. If the goal is to estimate error variance, how is redf defined?
A) tr(2H - HH')
B) tr((I − H)'(I − H))
C) tr(H'H)
D) tr(I - 2H)
  • 46. In general, what serves as the numerator in the error variance formulas?
A) The trace of the hat matrix
B) The error variance itself
C) The objective function being minimized
D) The number of observations n
  • 47. What matrix does the symbol Σ represent in the context of the hat matrix?
A) A quadratic form matrix
B) An observation covariance matrix
C) A smoothing blur matrix
D) An identity matrix
  • 48. What does the error variance σ2 scale in the context of parameters?
A) The unknown parameters' a posteriori standard deviation
B) The degrees of freedom in structural equation models
C) The known input variables
D) The number of primary sampling units
  • 49. In a one-factor confirmatory factor analysis with 4 items, how many knowns are there?
A) 6
B) 8
C) 4
D) 10
  • 50. In a one-factor confirmatory factor analysis with 4 items, how many unknowns (4 factor loadings and 4 error variances) are there?
A) 10
B) 8
C) 6
D) 4
  • 51. In organizational sciences, what proportion of papers published in top journals report degrees of freedom inconsistent with the models described?
A) Nearly all
B) Zero
C) Nearly half
D) About ten percent
  • 52. How many stratum means are estimated and subtracted in the complex survey sampling design variance formula?
A) One stratum mean
B) n_h stratum means
C) L stratum means
D) m_hi stratum means
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