tpaired1tlow.pl

:- module(tpaired1tlow, []).

:- use_module(library(http/html_write)).
:- use_module(session).
:- use_module(table).
:- use_module(r_session).
:- use_module(library(mcclass)).
:- use_module(mathml).

:- use_module(navbar).
navbar:page(tpaired1tlow, ["paired ", i(t), "-test onetailed (2)"]).

task(tratio).
task(pvalue).
task(cipaired).

:- discontiguous intermediate/2, expert/5, buggy/5, feedback/4, hint/4, r_hook/1.

% Prettier symbols for mathematical rendering
math_hook(d, overline('D')).
math_hook(s_d, subscript(s, 'D')).
math_hook(n, 'N').
math_hook(t0, overline("T0")).
math_hook(s_t0, subscript(s, "T0")).
math_hook(eot, overline("EOT")).
math_hook(s_eot, subscript(s, "EOT")).
math_hook(s2p, subscript(s, "pool")^2).
math_hook(paired(D, Mu, S_D, N), fn("paired", [D, Mu, S_D, N])).
math_hook(alpha, greek("alpha")).
math_hook(t(DF), fn(t, [DF])).

% R definitions
r_hook(t).
r_hook(d).
r_hook(mu).
r_hook(s_d).
r_hook(n).
r_hook(t0).
r_hook(s_t0).
r_hook(eot).
r_hook(s_eot).
r_hook(lo).
r_hook(incr).

r_hook('<-'/2).
r_hook(var_pool/4).
mono((var_pool)/4, [+, /, +, /]).

 
% Task description
render
--> { start(item(_T0, _S_T0, _EOT, _S_EOT, _D, _S_D, N, _Mu, _Alpha)) },      % by adding the parameter _Incr, the task description won't appear anymore
    html(
      div(class(card), div(class('card-body'),
        [ h1(class('card-title'), "Evaluation study on working memory training"),
          p(class('card-text'),
            [ "In an evaluation study, ",
              \mmlm(N = r(N)), " first-graders screened for low working memory 
              (i.e., under the 15th percentile in the Automated Working Memory Assessment)
               received 25 sessions of the computerized Cogmed working memory training.
               The primary outcome was the score on the Wide Range Achievement Test 
               (WRAT, standard mean score = 100, SD = 15) with higher scores indicating 
               higher performance. The significance level is set to ",
              \mmlm([alpha = perc(0.05), "."]), 
              " A performance increase should result in a positive ", 
              \mmlm(hyph(t, "value."))]),     
          div(class(container),
            div(class("row justify-content-md-center"),
              div(class("col-6"),
                \htmltable(
                   [ em("Table 1. "), "Observed WRAT scores at Pretest, Posttest, ",
                    "and ", \mmlm('D' = "Posttest" - "Pretest") ],
                  [ "Average", "SD" ],
                  [ "WRAT", "Pretest", "Posttest", \mmlm(d) ],
                  [ [ \mmlm([digits(1)], r(t0)),
                      \mmlm([digits(1)], r(eot)),
                      \mmlm([digits(1)], r(d1)) ],
                    [ \mmlm([digits(1)], r(s_t0)),
                      \mmlm([digits(1)], r(s_eot)),
                      \mmlm([digits(1)], r(s1_d)) ]
                  ])))),
          \download(tpaired1tlow)
        ]))).

% Question for the t-ratio
task(tratio)
--> { start(item(_T0, _S_T0, _EOT, _S_EOT, _D, _S_D, _N, Mu, _Alpha)),
      session_data(resp(tpaired, tratio, Resp), resp(tpaired, tratio, '#.##'))
    },
    html(\htmlform([ "Does the Cogmed training lead to a relevant improvement (i.e., more ",
        "than ", \mmlm([digits(1)], Mu = r(Mu)), " units) in mean WRAT ",
        "scores between Pretest and Posttest? ",
        "Please report the ", \mmlm(hyph(t, "ratio.")) ], tratio, Resp)).

% Question for the p-value
task(pvalue)
--> { start(item(_T0, _S_T0, _EOT, _S_EOT, _D, _S_D, _N, Mu, _Alpha)),
      session_data(resp(tpaired, pvalue, Resp), resp(tpaired, pvalue, '.###'))
    },
    html(\htmlform([ "Does the Cogmed training lead to a relevant improvement (i.e., more ",
        "than ", \mmlm([digits(1)], Mu = r(Mu)), " units) in mean WRAT ",
        "scores between Pretest and Posttest? ",
        "Please report the ", \mmlm(hyph(p, "value")) ], pvalue, Resp)).

% Question for the confidence interval
task(cipaired)
--> { start(item(_T0, _S_T0, _EOT, _S_EOT, _D, _S_D, _N, _Mu, _Alpha)),
      session_data(resp(tpaired, cipaired, Resp), resp(tpaired, cipaired, '#.# to #.#'))
    },
    html(\htmlform([ "Determine the confidence interval for the change in ",
        "the children\u0027s WRAT scores." ], cipaired, Resp)).





%
%% Expert rules for the t-ratio task
%
% t-test for paired samples
intermediate(tratio, item).
start(item(t0, s_t0, eot, s_eot, d, s_d, n, mu, alpha)).

% First step: Extract the correct information for a paired t-test from the task
% description
intermediate(tratio, paired).
expert(tratio, stage(2), X, Y, [step(expert, paired, [])]) :-
    X = item(_, _, _, _, D, S_D, N, Mu, _Alpha),
    Y = { '<-'(t, paired(D, Mu, S_D, N)) }.

feedback(paired, [], Col, F)
 => F = [ "Correctly recognised the problem as a ",
          \mmlm(Col, hyph(t, "test")), " for paired samples." 
        ].

hint(paired, [], Col, F)
 => F = [ "This is a ", \mmlm(Col, hyph(t, "test")), " for paired ",
          "samples." 
        ].

% Second step: Apply the formula for the t-ratio. dfrac/2 is a fraction in
% "display" mode (a bit larger font than normal)
expert(tratio, stage(2), X, Y, [step(expert, tratio, [D, Mu, S_D, N ])]) :-
    X = paired(D, Mu, S_D, N),
    Y = tstat(dfrac(D - Mu, S_D / sqrt(N))).

feedback(tratio, [_D, _Mu, _S_D, _N], Col, F)
 => F = [ "Correctly identified the ", \mmlm(Col, hyph(t, "ratio")), " for ",
          "paired samples." 
        ].

hint(tratio, [D, Mu, S_D, N], Col, F)
 => F = [ "The ", \mmlm(Col, hyph(t, "ratio")), " ",
          "is ", \mmlm(Col, [dfrac(D - Mu, S_D / sqrt(N)), "."])
        ].









%
%% Buggy-Rules for the for the t-ratio task
%
% Buggy-Rule: Omit the null hypothesis Mu
% Misconception: Run the paired t-test against zero, that is, just test for a
% decrease in symptoms. This is a frequent misconception.
% The problem is known as "regression to the mean": 
% Scores at T0 tend to be systematically too low (patients present themselves
% at the hospital when they feel particularly ill). At EOT, the measurement 
% is not biased by self-selection. Therefore, we tend to see an improvement 
% even in the absence of any therapeutical effect. 
% This misconception is even built into SPSS, because the paired samples t-test
% in SPSS only allows for mu = 0.  
buggy(tratio, stage(2), X, Y, [step(buggy, mu, [Mu])]) :-
    X = paired(D, Mu, S_D, N),
    Y = tstat(dfrac(omit_right(mu, D - Mu), S_D / sqrt(N))).

feedback(mu, [Mu], Col, F)
 => F = [ "The result matches the ", \mmlm(hyph(t, "ratio,")), " when the null",
	   " hypothesis ", \mmlm(Col, color(mu, Mu)), " has been omitted.",
	   " Please do not forget ", \mmlm(Col, color(mu, Mu)), " in the ",
	   \mmlm(hyph(t, "ratio."))
        ].

hint(mu, [Mu], Col, F)
 => F = [ "Do not omit the null hypothesis ", \mmlm(Col, color(mu, Mu)),
          " in the ", \mmlm(hyph(t, "ratio.")) 
        ].

% Buggy-Rule: t-test for independent samples
% Misconception: Run the t-test for independent samples despite the correlated
% measurements.
% First step of the Buggy-Rule: t-test for independent samples
% gathering the important data needed to solve the t-test for independent samples
% (both correct and incorrect solutions) from the task description
intermediate(tratio, indep).
buggy(tratio, stage(2), X, Y, [step(buggy, indep, [])]) :-
    X = item(T0, S_T0, EOT, S_EOT, D, S_D, N, Mu, _Alpha),
    Y = { '<-'(t, instead(indep, indep(T0, S_T0, N, EOT, S_EOT, N), 
            paired(D, Mu, S_D, N))) 
        }.

feedback(indep, [], Col, F)
 => F = [ "The problem was mistakenly identified as a ",
          \mmlm(Col, hyph(t, "test")), " for independent samples." 
        ].

hint(indep, [], Col, F)
 => F = [ "Do not calculate a ", \mmlm(Col, hyph(t, "test")), " for ",
          "independent samples here." 
        ].

% Second step of the Buggy-Rule: t-test for independent samples
% Determine the test statistic for the t-test for independent samples.
% The step itself is correct, although it is only needed
% if a wrong decision has been made before (bug indep).
expert(tratio, stage(2), X, Y, 
        [step(expert, tratio_indep, [T0, S_T0, N, EOT, S_EOT])]) :-
    X = indep(T0, S_T0, N, EOT, S_EOT, N),
    P = denote(s2p, var_pool(S_T0^2, N, S_EOT^2, N), "the pooled variance"),
    Y = tstat(dfrac(T0 - EOT, sqrt(P * (1/N + 1/N)))).

feedback(tratio_indep, [_T0, _S_T0, _N, _EOT, _S_EOT], Col, F)
 => F = [ "Correctly identified the ", \mmlm(Col, hyph(t, "ratio")),
          " for independent samples." 
        ].

hint(tratio_indep, [T0, S_T0, N, EOT, S_EOT], Col, F)
 => P = denote(s2p, var_pool(S_T0^2, N, S_EOT^2, N), "the pooled variance"),
    F = [ "The ", \mmlm(Col, hyph(t, "ratio")), " for independent samples ",
          "would be ", \mmlm(Col, [dfrac(T0 - EOT, sqrt(P * (1/N + 1/N))), "."])
        ].

% The following mistake cannot occur in the paired t-test, but is again only
% possible if the student has already made the wrong decision to calculate the
% t-test for independent samples. We need it anyway, to be able to diagnose the
% numeric result.

% Buggy-Rule: Forgot school math [1/N1 + 1/N2 is not 1/(N1 + N2)]
% For mysterious reasons, everyone falls into this trap at least once,
% including me and the student assistants. I have coined it "school",
% since it is an example in which the person has forgotten school math.
buggy(tratio, stage(2), X, Y, [step(buggy, school1, [N1, N2])]) :-
    dif(N1, N2),
    X = 1/N1 + 1/N2,
    Y = frac(1, color(school1, N1 + N2)).

feedback(school1, [A, B], Col, F)
 => F = [ "The result matches the expression for the ",
	   \mmlm(Col, hyph(t, "ratio"))," for independent samples with ",
	   \mmlm(Col, [frac(1, color(school, color("black", A) + color("black", B))), "."]), 
	   " Please keep in mind that ", 
	   \mmlm(Col, [color(school, color("black", frac(1, A)) + color("black", frac(1, B)))
		 =\= frac(1, color(school, color("black", A) + color("black", B))), "."])
        ].

hint(school1, [N1, N2], Col, F)
 => F = [ "Please do not forget school ",
          "math, ", \mmlm(Col, [frac(1, color(school1, N1)) +
            frac(1, color(school1, N2)) =\= frac(1, color(school1, N1+N2)), "."])
        ].

% Buggy-Rule: Forgot school math (Same for N1 = N2)
buggy(tratio, stage(2), X, Y, [step(buggy, school2, [N])]) :-
    X = 1/N + 1/N,
    Y = frac(1, color(school2, 2*N)).

feedback(school2, [N], Col, F)
 => F = [ "The result matches the expression for the ", 
	   \mmlm(Col, hyph(t, "ratio")), " for independent samples with ",
	   \mmlm(Col, [frac(1, color(school2, 2*N)), "."]), " Please keep in mind that ",
	   \mmlm(Col, [frac(1, color(school2, N)) + frac(1, color(school2, N)) =\= frac(1, color(school2, 2*N)), "."])
        ].

hint(school2, [N], Col, F)
 => F = [ "Please do not forget school math, ",
          \mmlm(Col, [frac(1, color(school2, N)) + frac(1, color(school2, N)) =\= frac(1, color(school2, 2*N)), "."])
        ].

% Buggy-Rule: Forgot parentheses
% Forget parentheses in numerator and denominator of X / Y, with X = A - B and
% Y = C / D. That is, calculate A - (B / C) / D instead of (A - B) / (C / D).
% 
% This is the first buggy rule that ever came to my attention, therefore the
% name, bug1.
buggy(tratio, stage(2), X, Y, [step(buggy, bug1, [D, Mu, S, SQRT_N])]) :-
    X = dfrac(D - Mu, S / SQRT_N),
    M0 = drop_left(bug1, D - Mu),
    S0 = drop_right(bug1, S / SQRT_N),
    Y = add_left(bug1, 
        D - add_right(bug1, dfrac(M0, S0) / SQRT_N)).

feedback(bug1, [D, Mu, S, SQRT_N], Col, F)
 => F = [ "The result matches the fraction without parentheses around the ", 
	   "numerator and the denominator, ", \mmlm([error(correct) | Col], 
	       [dfrac(color(bug1, paren(color("#000000", D - Mu))), 
	           color(bug1, paren(color("#000000", S / SQRT_N)))), "."]),
	   " Please do not forget the parentheses around the numerator and the ",
	   "denominator of a fraction."
        ].

hint(bug1, [D, Mu, S, SQRT_N], Col, F)
 => F = [ "Do not forget the parentheses around the numerator and ",
          "the denominator of a fraction, ",
          \mmlm([error(correct) | Col],
            [dfrac(color(bug1, paren(color("#000000", D - Mu))), 
              color(bug1, paren(color("#000000", S / SQRT_N)))), "."])
        ].

% One challenging aspect of word problems ("Textaufgaben") is that students
% have trouble to extract the correct information from the task description.
 
% Buggy-Rule: Use the mean of TO instead of the mean of D
% The depends means: This bug is limited to the paired t-test and co-occurs
% with s_t0.
buggy(tratio, stage(1), X, Y, 
        [step(buggy, t0, [d, t0]), depends(s_t0), depends(paired)]) :-
    X = d,
    Y = instead(t0, t0, d).

feedback(t0, [D, T0], Col, F)
 => F = [ "The result matches the ", \mmlm(Col, hyph(t, "ratio")), " with the", 
	   " T0 average ", \mmlm(Col, color(t0, T0)), " instead of the average", 
	   " change score ", \mmlm(Col, [color(t0, D), "."]), " Please insert the average",
	   " change score ", \mmlm(Col, color(t0, D)), " into the ",
	   \mmlm(Col, hyph(t, "ratio."))
	].

hint(t0, [_D, T0], Col, F)
 => F = [ "Do not insert the T0 average ", \mmlm(Col, color(t0, T0)), " ",
          "into the ", \mmlm(Col, hyph(t, "ratio.")), " Use the change ",
          "scores instead." 
	].

% Buggy-Rule: Use SD of T0 instead of SD of D
buggy(tratio, stage(1), X, Y, Flags) :-
    Flags = [step(buggy, s_t0, [s_d, s_t0]), depends(paired)],
    X = s_d,
    Y = instead(s_t0, s_t0, s_d).

feedback(s_t0, [S, S_T0], Col, F)
 => F = [ "The result matches the ", \mmlm(Col, hyph(t, "ratio")), 
	   " with the standard deviation for T0 ", \mmlm(Col, color(s_t0, S_T0)),
	   " instead of the standard deviation of the change score ", 
	   \mmlm(Col, [color(s_t0, S), "."]), " Please insert the standard deviation of the", 
	   " change score ", \mmlm(Col, color(s_t0, S)), " into the ", 
	   \mmlm(Col, hyph(t, "ratio."))
	].

hint(s_t0, [_S, S_T0], Col, F)
 => F = [ "Do not insert the standard deviation for ",
          "T0 ", \mmlm(Col, color(s_t0, S_T0)), " into ",
          "the ", \mmlm(Col, hyph(t, "ratio.")), " Use the change scores ",
          "instead." 
	].

% Buggy-Rule: Use mean EOT instead of mean D
buggy(tratio, stage(1), X, Y, [step(buggy, eot, [d, eot]), 
        depends(s_eot), depends(paired)]) :-
    X = d,
    Y = instead(eot, eot, d).

feedback(eot, [D, EOT], Col, F)
 => F = [ "The result matches the ", \mmlm(Col, hyph(t, "ratio")), " with the",
           " EOT average ", \mmlm(Col, color(eot, EOT)), " instead of the ", 
           "average change score ", \mmlm(Col, [color(eot, D), "."]), " Please insert",
           " the average change score ",\mmlm(Col, color(eot, D)), " into the ",
	   \mmlm(Col, hyph(t, "ratio."))
	].

hint(eot, [_D, EOT], Col, F)
 => F = [ "Do not insert the EOT average ", \mmlm(Col, color(eot, EOT)), " ",
          "into the ", \mmlm(Col, hyph(t, "ratio.")), " Use the change ",
          "scores instead." 
	].

% Buggy-Rule: Use SD of EOT instead of SD of D
buggy(tratio, stage(1), X, Y, Flags) :-
    Flags = [step(buggy, s_eot, [s_d, s_eot]), depends(paired)],
    X = s_d,
    Y = instead(s_eot, s_eot, s_d).

feedback(s_eot, [S, S_EOT], Col, F)
 => F = [ "The result matches the ", \mmlm(Col, hyph(t, "ratio")), " with the",
	   " standard deviation for EOT ", \mmlm(Col, color(s_eot, S_EOT)), 
	   " instead of the standard deviation of the change score ", 
	   \mmlm(Col, [color(s_eot, S), "."]), " Please insert the standard deviation of the",
	   " change score ", \mmlm(Col, color(s_eot, S)), " into the ", 
	   \mmlm(Col, hyph(t, "ratio."))
	].

hint(s_eot, [_S, S_EOT], Col, F)
 => F = [ "Do not insert the standard deviation for ",
          "EOT ", \mmlm(Col, color(s_eot, S_EOT)), " into ",
          "the ", \mmlm(Col, hyph(t, "ratio.")), " Use the change scores ",
          "instead." 
	].

% Buggy-Rule: Use of n instead of sqrt(n)
buggy(tratio, stage(2), X, Y, [step(buggy, sqrt1, [n])]) :-
    X = sqrt(n),
    Y = omit_right(sqrt1, n^(1/2)).

feedback(sqrt1, [N], Col, F)
 => F = [ "The result matches the ", \mmlm(Col, hyph(t, "ratio")), " without", 
	  " square root around ", \mmlm(Col, [color(sqrt1, N), "."]), " Please do not",
	  " forget the square root around ", \mmlm(Col, [color(sqrt1, N), "."])
        ].

hint(sqrt1, [N], Col, F)
 => F = [ "Do not forget the square root around ",
          \mmlm(Col, [color(sqrt1, N), "."])
        ].

% Buggy-Rule: Use of N instead of sqrt(N)
buggy(tratio, stage(2), X, Y, [step(buggy, sqrt2, [N])]) :-
    X = sqrt(N),
    dif(N, n),
    Y = omit_right(sqrt2, N^(1/2)).

feedback(sqrt2, [N], Col, F)
 => F = [ "The result matches the ", \mmlm(Col, hyph(t, "ratio")), " without", 
	   " square root around ", \mmlm(Col, [color(sqrt2, N), "."]), " Please do not",
	   " forget the square root around ", \mmlm(Col, [color(sqrt2, N), "."])
        ].

hint(sqrt2, [N], Col, F)
 => F = [ "Do not forget the square root around ",
          \mmlm(Col, [color(sqrt2, N), "."])
        ].






%
% Expert-Rules for the p-value task
%
% Gathering the important data needed to solve the t-test for paired samples
% (both correct and incorrect solutions) from the task description.
intermediate(pvalue, item).

% First step: Extract the correct information for a paired t-test from the task
% description
intermediate(pvalue, paired).
intermediate(pvalue, onetailed).
expert(pvalue, stage(2), X, Y, [step(expert, paired, [])]) :-
    X = item(_, _, _, _, D, S_D, N, Mu, _Alpha),
    Y = { '<-'(t, paired(D, Mu, S_D, N)) ;
          '<-'(p, onetailed(t, N-1))
        }.



% feedback(paired, [], Col, F)
%  => F = [ "Correctly recognised the problem as ",
%           "a ", \mmlm(Col, hyph(t, "test")), " for paired samples."
%         ].

% hint(paired, [], Col, F)
%  => F = [ "This is a ", \mmlm(Col, hyph(t, "test")), " for paired ",
%           "samples."
%         ].

% Second step: Apply the formula for the t-ratio. dfrac/2 is a fraction in
% "display" mode (a bit larger font than normal)

intermediate(pvalue, tratio).
expert(pvalue, stage(2), X, Y, [step(expert, tratio, [D, Mu, S_D, N])]) :-
    X = paired(D, Mu, S_D, N),
    Y = dfrac(D - Mu, S_D / sqrt(N)).





% feedback(tratio, [_D, _Mu, _S_D, _N], Col, F)
%  => F = [ "Correctly identified the ", \mmlm(Col, hyph(t, "ratio")), " for ",
%           "paired samples."
%         ].

% hint(tratio, [D, Mu, S_D, N], Col, F)
%  => F = [ "The ", \mmlm(Col, hyph(t, "ratio")), " ",
%           "is ", \mmlm(Col, [dfrac(D - Mu, S_D / sqrt(N)), "."])
%         ].

% Third step: Determine the one-tailed p-value
expert(pvalue, stage(2), X, Y, [step(expert, pvalue, [])]) :-
    X = onetailed(T, DF),
    Y = pval(pt(T, DF, false)).

feedback(pvalue, [], Col, F)
 => F = [ "Correctly determined the one-tailed ", \mmlm(Col, hyph(p, "value.")) ].

hint(pvalue, [], Col, F)
 => F = [ "The one-tailed ", \mmlm(Col, hyph(p, "value")), " must be determined." ].


%
%% Buggy-Rules for the p-value task
%

% Buggy-Rule: report the left-tail instead of the right-tail. 
buggy(pvalue, stage(2), X, Y, [step(buggy, wrongtail, [DF])]) :-
     X = onetailed(T, DF),
     Y = pval(pt(T, DF, true)).

feedback(wrongtail, [DF], Col, F)
 => F = [ "The result matches the left-sided ", \mmlm(Col, hyph(p, "value.")), 
          " Please make sure to use the area of the ", \mmlm(Col, hyph(t(DF), "distribution")),
          " on the right side of the calculated ", \mmlm(Col, hyph(t, "value."))
        ].

hint(wrongtail, [DF], Col, F)
 => F = [ "Use the upper tail of the ", \mmlm(Col, hyph(t(DF), "distribution."))].


% Buggy-Rule: used the wrong t-value and/or degrees of freedoms
% ToDo: the matching values should include all the values from 0 to 1 except the ones for 
% the correct t-value and degrees of freedom (both left- and right-sided). So this should be a catch-all condition.
buggy(pvalue, stage(2), X, Y, [step(buggy, wrong, [T, DF])]) :-
     X = onetailed(T, DF),
     Y = pval(pt(T, DF, true)).

feedback(wrong, [T, DF], Col, F)
 => F = [ "The result is not the ", \mmlm(Col, hyph(p, "value")), 
	        " associated with the calculated ", \mmlm(Col, hyph(t, "value")), \mmlm(T = r(T)), 
          " and degrees of freedom ", \mmlm([DF = r(DF), "."]), 
          " Please make sure to look at the correct column and row on the table of the ", 
          \mmlm(Col, hyph(t, "distribution."))].

hint(wrong, [_T, _DF], Col, F)
 => F = [ "Look at the correct column and row on the table of the ", 
          \mmlm(Col, hyph(t, "distribution."))].







%
%% Expert Rules for the confidence interval task
%
% t-test and confidence intervall for paired samples 
intermediate(cipaired, item).

% First step: Extract the correct information for a paired t-test and 
% the associated confidence interval from the task description
intermediate(cipaired, paired).
expert(cipaired, stage(2), X, Y, [step(expert, paired, [])]) :-
    X = item(_, _, _, _, D, S_D, N, Mu, Alpha),
    Y = paired(D, Mu, S_D, N, Alpha).

feedback(paired, [], Col, F)
 => F = [ "Correctly recognised the problem as ",
          "a ", \mmlm(Col, hyph(t, "test")), " for paired samples and that ",
          "a confidence interval for the mean value has to be calculated." 
        ].

hint(paired, [], Col, H)
 => H = [ "This is a ", \mmlm(Col, hyph(t, "test")), " for paired ",
          "samples. Calculate the confidence interval for the mean difference." 
        ].

% Second step: Apply the formula for the confidence interval for a mean value.
intermediate(cipaired, quant).
expert(cipaired, stage(2), X, Y, [step(expert, ci_lower, [D, S_D, N, Alpha])]) :-
    X = paired(D, Mu, S_D, N, Alpha),
    Y = hdrs(neginf(D - dot(quant(D, Mu, S_D, N, Alpha), S_D / sqrt(N)))).

feedback(ci_lower, [_D, _S_D, _N, _Alpha], Col, F)					
 => F = [ "Correctly identified the formula for the lower bound of ",
           "the confidence interval for a mean value in a ",
            \mmlm(Col, hyph(t, "test."))
        ].

hint(ci_lower, [D, S_D, N, Alpha], _Col, H)
 => H = [ "The formula to calculate the lower  bound of the ",
          "confidence interval is ",
	  \mmlm([(D - (qt(1 - Alpha, N-1) * S_D / sqrt(N))), "."])
        ].

% Third step: Choose the correct quantile of the t-distribution
expert(cipaired, stage(2), X, Y, [step(expert, tquant, [N, Alpha])]) :-
    X = quant(_D, _Mu, _S_D, N, Alpha),
    Y = qt(1 - Alpha, N-1).

feedback(tquant, [_N, Alpha], Col, F)
 => F = [ "Correctly used the ", \mmlm(Col, hyph(1 - Alpha, "quantile")),
          "of the ", \mmlm(Col, hyph(t, "distribution."))
        ].

hint(tquant, [_N, Alpha], Col, H)
 => H = [ "Make sure to use the ", \mmlm(Col, hyph(1 - Alpha, "quantile")),
          "of the ", \mmlm(Col, hyph(t, "distribution."))
        ].


%
%% Buggy-Rules for the confidence interval task
%
% Buggy-Rule: Use t-statistic instead of t-quantile
buggy(cipaired, stage(2), X, Y, [step(buggy, tstat, [D, S_D, N, Mu, Alpha])]) :-
    X = quant(D, Mu, S_D, N, Alpha),
    P = denote(t, dfrac(D - Mu, S_D / sqrt(N)), ["the observed", space, t, "-statistic."]),
    Y = P. 

feedback(tstat, [_D, _S_D, _N, _Mu, _Alpha], Col, F)
 => F = [ "The result matches the confidence interval based on the observed ",
          \mmlm(Col, hyph(t, "statistic.")), " Please use the quantile ",
           "of the ", \mmlm(Col, hyph(t, "distribution")), " instead."
        ].

hint(tstat, [_D, _S_D, _N, _Mu, _Alpha], Col, H)
 => H = [ "Do not insert the observed ", \mmlm(Col, hyph(t, "statistic ")),
          "into the formula for the confidence interval. Use the quantile of ", 
	  "the ", \mmlm(Col, hyph(t, "distribution")), " instead."
        ].

% Buggy-Rule: Use z-quantile instead of t-quantile. 
% This rule may be dropped because we might not be able to distinguish the results.
buggy(cipaired, stage(2), X, Y, [step(buggy, qnorm, [N, Alpha])]) :-
    X = quant(_D, _Mu, _S_D, N, Alpha),
    Y = instead(qt, qnorm(1 - Alpha) , qt(1 - Alpha, N - 1)).

feedback(qnorm, [N, Alpha], Col, F)
 => F = [ "The result matches the confidence interval based on the standard ",
          "normal distribution. ",
          "Please insert the quantile of the ", \mmlm(Col, hyph(t, "distribution")),
          \mmlm(Col, color(qnorm, qt(1 - Alpha, N - 1))), " into ",
          "the formula for the confidence interval."
        ].

hint(qnorm, [_N, _Alpha], Col, H)
 => H = [ "Do not insert the quantile of the ", \mmlm(Col, hyph(z, "distribution ")), 
          "into the formula for the confidence interval. Use the quantile of the ", 			
	  \mmlm(Col, hyph(t, "distribution")), "instead."
        ].

% Buggy-Rule: Calculating the confidence intervall with SPSS
% and forgetting to add Mu to the results of the bounds in the end.
buggy(cipaired, stage(2), X, Y, [step(buggy, spss, [Mu]), excludes(qnorm), excludes(tstat), excludes(sqrt1)]) :-
    X = paired(D, Mu, S_D, N, Alpha),
    Y = hdrs(pm(add_right(spss, D - Mu), dot(quant(D, Mu, S_D, N, Alpha), S_D / sqrt(N)))).

feedback(spss, [Mu], Col, F)
 => F = [ "The result matches the upper and lower bound calculated by SPSS. ",
	  "Please do not forget to add ", \mmlm(Col, Mu), " to the upper and lower ",
	  "bound of the confidence interval, if you calculate it with SPSS."
        ].

hint(spss, [Mu], Col, H)
 => H = [ "If you calculate the confindence intervall with SPSS, keep in mind",
	  " that SPSS subtracts ", \mmlm(Col, Mu), " from the two bounds of",
          " the CI (which must be undone)."
        ].

% Buggy-Rule: Use of N instead of sqrt(N)				
buggy(cipaired, stage(2), X, Y, [step(buggy, sqrt1, [N])]) :-			
    X = dot(quant(D, Mu, S_D, N, Alpha), S_D / sqrt(N)),
    Y = dot(quant(D, Mu, S_D, N, Alpha), S_D / omit_right(sqrt1, (2*N)^(1/2))).

feedback(sqrt1, [N], Col, F)
 => F = [ "The result matches the confidence interval without square root around ", 
          \mmlm(Col, [color(sqrt1, N), "."]), " Please do not forget the square root",
          " around ", \mmlm(Col, [color(sqrt1, N), "."])
        ].

hint(sqrt1, [N], Col, F)
 => F = [ "Do not forget the square root around ",
          \mmlm(Col, [color(sqrt1, N), "."])
        ].

% Buggy-Rule: Use of N instead of sqrt(N) in the t-ratio
buggy(cipaired, stage(2), X, Y, [step(buggy, sqrt2, [N])]) :-
    X = dfrac(D - Mu, S_D / sqrt(N)),
    Y = dfrac(D - Mu, S_D / omit_right(sqrt2, (3*N)^(1/2))).

feedback(sqrt2, [N], Col, FB)
 => FB = [ "The result matches the confidence interval based on the observed ",
	   \mmlm(Col, hyph(t, "statistic")), " without square root around ",
	   \mmlm(Col, [color(sqrt2, N), "."]), " Please do not forget the square root around ",
           \mmlm(Col, [color(sqrt2, N), "."])
         ].

hint(sqrt2, [N], Col, FB)
 => FB = [ "Do not forget the square root around ",
           \mmlm(Col, [color(sqrt2, N), "."])
         ].