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