9.5.4  Testing a distribution with the χ² distribution: chisquaret

The chisquaret command will use the χ² test to compare sample data to a specified distribution.

• chisquaret takes one mandatory argument and an unspecified number of optional arguments:
  • L, a list of sample data.
  • Optionally, distr. This can be one of
    • The name of a distribution (see Section 9.3.14 for a list of distributions and their parameters).
    • Another list of sample data.
    By default this will be the uniform distribution.
  • params, the parameters of the distribution distr or the keyword classes and optionally c_min and c_dim, the minimum size and default size of a statistics class (by default, class_min and class_size, which themselves default to 0 and 1; see Section 3.5.8).
• chisquaret(L ⟨,distr⟩) returns the result of the χ² test between the sample data and the named distribution or the two sample data.

Examples.

• Input:
  chisquaret([57,54])
  Output:

  Guessing data is the list of number of elements in each class,

  adequation to uniform distribution

  Sample adequation to a finite discrete probability distribution

  Chi2 test result 0.0810810810811,

  reject adequation if superior to chisquare_icdf(1,0.95)=3.84145882069 or chisquare_icdf(1,1-alpha) if alpha!=5%

  0.0810810810811

• Input:
  chisquaret([1,1,1,1,1,0,0,1,0,1,1],[.4,.6])
  Output:

  Sample adequation to a finite discrete probability distribution

  Chi2 test result 0.742424242424,

  reject adequation if superior to chisquare_icdf(1,0.95)=3.84145882069

  or chisquare_icdf(1,1-alpha) if alpha!=5%

  0.742424242424

• Input:
  chisquaret(ranv(1000,binomial,10,.5),binomial)
  Output:

  Binomial: estimating n and p from data 10 0.5055

  Sample adequation to binomial(10,0.5055,.), Chi2 test result 7.77825189838,

  reject adequation if superior to chisquare_icdf(7,0.95)=14.0671404493

  or chisquare_icdf(7,1-alpha) if alpha!=5%

  7.77825189838

• Input:
  chisquaret(ranv(1000,binomial,10,.5),binomial,11,.5)
  Output:

  Sample adequation to binomial(11,0.5,.), Chi2 test result 125.617374161,

  reject adequation if superior to chisquare_icdf(10,0.95)=18.3070380533

  or chisquare_icdf(10,1-alpha) if alpha!=5%

  125.617374161

• As an example using class_min and class_size:
  Input:

  L:= ranv(1000,normald,0,.2)

  chisquaret(L,normald,classes,-2,.25)

  or (setting class_min to −2 and class_size to −0.25 in the graphical configuration):
  chisquaret(L,normald,classes)
  Output:

  Normal density,

  estimating mean and stddev from data -0.00345919752912 0.201708100832

  Sample adequation to normald_cdf(-0.00345919752912,0.201708100832,.),

  Chi2 test result 2.11405080381,

  reject adequation if superior to chisquare_icdf(4,0.95)=9.48772903678

  or chisquare_icdf(4,1-alpha) if alpha!=5%

  2.11405080381

  In this last case, you are given the value of d² of the statistic D² = ∑_j=1^k (n_j − e_j)/e_j, where k is the number of sample classes for classes(L,-2,0.25) (or classes(L)), n_j is the size of the jth class, and e_j = n p_j where n is the size of L and p_j is the probability of the jth class interval assuming a normal distribution with the mean and population standard deviation of L.