Fix p value in paired T-test

[ylansegal]

Thank you for making this gem available!

I believe I've found (and fixed) a bug. In the Student's T-test (Paired T-Test) wiki page, we see the following example:

left_group
# => [0.12819915256260872, 0.24345459073897613, 0.27517650565714014, 0.8522185144081152, 0.05471111219486524]
right_group
# => [0.3272414061985621, 0.2989306116723194, 0.642664937717922, 0.9476073892620895, 0.7050008194345182]
# ...
StatisticalTest::TTest.paired_test(alpha = 0.01, :two_tail, left_group, right_group)
# => {:t_score=>-2.520215812331144, :probability=>0.032670873044446366, :p_value=>1.9346582539111072, :alpha=>0.01, :null=>true, :alternative=>false, :confidence_level=>0.99}

Notice the p-value returned is greater greater than 1.

If we do the same calculation in R:

left_group <- c(0.12819915256260872, 0.24345459073897613, 0.27517650565714014, 0.8522185144081152, 0.05471111219486524)
right_group <- c(0.3272414061985621, 0.2989306116723194, 0.642664937717922, 0.9476073892620895, 0.7050008194345182)

t.test(left_group, right_group, alternative = 'two.sided', paired = TRUE, conf.level = 0.99)

#   Paired t-test

# data:  left_group and right_group
# t = -2.5202, df = 4, p-value = 0.06534
# alternative hypothesis: true mean difference is not equal to 0
# 99 percent confidence interval:
# -0.7732524  0.2261783
# sample estimates:
# mean difference
#     -0.2735371

The p-value is calculated as 0.06534.

I believe the root cause is the same as what was fixed for .perform in #24. In this PR I use that same fix for .paired_test and the specs corroborate the numbers in the R calculations.

Assuming this is accepted, the wiki should also be changed:

# => [0.12819915256260872, 0.24345459073897613, 0.27517650565714014, 0.8522185144081152, 0.05471111219486524]
right_group
# => [0.3272414061985621, 0.2989306116723194, 0.642664937717922, 0.9476073892620895, 0.7050008194345182]


StatisticalTest::TTest.paired_test(alpha = 0.01, :two_tail, left_group, right_group)
# =>
# {:t_score=>-2.520215812331144,
#  :probability=>0.03267087304444638,
#  :p_value=>0.06534174608889276,
#  :alpha=>0.01,
#  :null=>true,
#  :alternative=>false,
#  :confidence_level=>0.99}

Thank you!

[estebanz01]

Thank you for you contribution! I'll take a look.

[estebanz01]

Can you please rebase this PR with latest master (just dependabot changes) and post a full output of bundle exec rake ? I'm not sure why the Github workflows are not running for this.

[ylansegal]

Can you please rebase this PR with latest master (just dependabot changes) and post a full output of bundle exec rake ? I'm not sure why the Github workflows are not running for this.

@estebanz01 Of course! I rebased, and ran the command:

$ bundle exec rake
/Users/ylansegal/.asdf/installs/ruby/3.2.1/bin/ruby -I/Users/ylansegal/.asdf/installs/ruby/3.2.1/lib/ruby/gems/3.2.0/gems/rspec-core-3.12.1/lib:/Users/ylansegal/.asdf/installs/ruby/3.2.1/lib/ruby/gems/3.2.0/gems/rspec-support-3.12.0/lib /Users/ylansegal/.asdf/installs/ruby/3.2.1/lib/ruby/gems/3.2.0/gems/rspec-core-3.12.1/exe/rspec --pattern spec/\*\*\{,/\*/\*\*\}/\*_spec.rb
....................................................................................................................................This is an alpha version code. The generated sample is similar to an uniform distribution
.This is an alpha version code. The generated sample is similar to an uniform distribution
*..............................The composite simpson's rule needs even intervals!
......................................{:d_max=>(3/25), :d_critical=>0.18152976342488275, :total_samples=>550, :alpha=>0.05, :null=>true, :alternative=>false, :confidence_level=>0.95}
...................

Pending: (Failures listed here are expected and do not affect your suite's status)

  1) Statistics::Distribution::TStudent#random does not generate a random sample that follows an uniform distribution
     # random sample follows an uniform and t-student distribution
     Failure/Error: expect(test[:null]).to be false

       expected false
            got true
     # ./spec/statistics/distribution/t_student_spec.rb:83:in `block (3 levels) in <top (required)>'

Finished in 9.87 seconds (files took 0.07437 seconds to load)
221 examples, 0 failures, 1 pending

Thank you,

[estebanz01]

Thank you very much for your efforts! I'm going to release a new version later today with this fix.