########################################################################
## DESCRIPTION
## A WeBWorK problem that asks questions about the correlation between two sets of 
## data values.
## WeBWorK problem written by JoAnne Taormina, <joanne(dot)taormina(at)ncc(dot)edu>
## ENDDESCRIPTION
##
## KEYWORDS('linear regression', 'correlation', 'coefficient of determination', 'correlation coefficient', 'scatter diagram')
##
## Author('JoAnne Taormina')
## Institution('Nassau Community College')
########################################################################

DOCUMENT();
loadMacros(
"PGstandard.pl",
"MathObjects.pl",
"PGgraphmacros.pl",
"weightedGrader.pl",
"PGchoicemacros.pl",
"unionTables.pl",
);

install_weighted_grader();

# make sure we're in the context we want
Context("Numeric");

# the data; x is hours watching TV, y is GPA
@datax = ();
@datay = ();

# create the first 4 data values 
for($i = 0; $i < 4; $i ++)
{
  $datax[$i] = random(2, 12, 1);
  $datay[$i] = random(2.9, 4.0, .1);

  $mockdatax[$i] = random(2, 12, 1);
  $mockdatay[$i] = random(2.9, 4.0, .1);

}
# create the second 4 data values 
for($i = 4; $i < 8; $i ++)
{
  $datax[$i] = random(10, 18, 1);
  $datay[$i] = random(2.2, 3.2, .1);

  $mockdatax[$i] = random(10, 18, 1);
  $mockdatay[$i] = random(2.2, 3.2, .1);
}
# create the third 4 data values 
for($i = 8; $i < 12; $i ++)
{
  $datax[$i] = random(16, 25, 1);
  $datay[$i] = random(1.0, 2.5, .1);
  $mockdatax[$i] = random(16, 25, 1);
  $mockdatay[$i] = random(1.0, 2.5, .1);
}

$n = 12;
$sum_x = 0;
$sum_x2 = 0;
$sum_y = 0;
$sum_y2 = 0;
$sum_xy = 0;
for ($i = 0; $i<12; $i++)
{
  $sum_x += $datax[$i];
  $sum_y += $datay[$i];
  $sum_x2 += $datax[$i]**2;
  $sum_y2 += $datay[$i]**2;
  $sum_xy += $datay[$i]*$datax[$i];
}
$r = ($n*$sum_xy - $sum_x*$sum_y)/(sqrt($n*$sum_x2-$sum_x**2)*sqrt($n*$sum_y2-$sum_y**2));

$r_squared = $r**2;

$a = ($n*$sum_xy - $sum_x*$sum_y)/($n*$sum_x2-$sum_x**2); 
$b = ($sum_y-$a*$sum_x)/$n;

$a = int(100*$a+.5*($a <=> 0))/100;

$b = int(100*$b+.5*($b <=> 0))/100;

$r = int(100*$r+.5*($r <=> 0))/100;

$r_squared = int(100*$r_squared+.5*($r_squared <=> 0))/100;

# set up for a multiple choice problem.
$radio_correlation_coefficient = new_multiple_choice();
$radio_correlation_coefficient->qa("The correct symbol for the correlation coefficient found in part (a) is ", "\( r \)");
$radio_correlation_coefficient->extra("\( \sigma \)", "\( \rho \)", "\( \bar{x} \)", "\( r^2 \)");

# set up for a multiple choice problem.
$radio_coefficient_det = new_multiple_choice();
$radio_coefficient_det->qa("The correct symbol for the coefficient of determination found in part (c) is ", "\( r^2 \)");
$radio_coefficient_det->extra("\( \sigma \)", "\( \rho \)", "\( \bar{x} \)", "\( r \)");

$random_coef_det = random(.50, .70, .01);
$random_coef_det_squared = $random_coef_det**2;
$random_coef_det_squared = int(100*$random_coef_det_squared+.5*($random_coef_det_squared <=> 0))/100;

$random_coef_det_percent = $random_coef_det*100;
$random_coef_det_squared_percent = $random_coef_det_squared*100;

$random_coef_det_percent_remaining = 100 - $random_coef_det_percent;
$random_coef_det_squared_percent_remaining = 100 - $random_coef_det_squared_percent;

# set up for a multiple choice problem.
$coef_det_interpretation = new_multiple_choice();
$coef_det_interpretation->qa("Suppose that the coefficient of determination for another sample was found to be $random_coef_det.  How should the meaning be interpreted?", "$random_coef_det_percent% of the variation in a student's GPA is influenced by the number of hours the student watches television. The remaining 
$random_coef_det_percent_remaining% is due to unexplained factors outside of the context of this problem.");
$coef_det_interpretation->extra(
"$random_coef_det_percent% of the variation in the number of hours a student watches television is influenced by the student's GPA. The remaining $random_coef_det_percent_remaining% is due to unexplained factors outside of the context of this problem.", 
"$random_coef_det_squared_percent% of the variation in a student's GPA is influenced by the number of hours the student watches television. The remaining $random_coef_det_squared_percent_remaining% is due to unexplained factors outside of the context of this problem.",
"$random_coef_det_squared_percent% of the variation in the number of hours a student watches television is influenced by the student's GPA. The remaining $random_coef_det_squared_percent_remaining% is due to unexplained factors outside of the context of this problem.");

$random_hours= random(5, 12, 1);
$calc_GPA = $a * $random_hours + $b;
# round $calc_GPA to the nearest tenth
$calc_GPA = int(10*$calc_GPA+.5*($calc_GPA <=> 0))/10;

$random_hours1= random(13, 25, 1);
$calc_GPA1 = $a * $random_hours1 + $b;

# round $calc_GPA1 to the nearest tenth
$calc_GPA1 = int(10*$calc_GPA1+.5*($calc_GPA1 <=> 0))/10;

$r = Compute($r);
$r_squared = Compute($r_squared);
$a = Compute($a);
$b = Compute($b);
$calc_GPA = Compute($calc_GPA);
$calc_GPA1 = Compute($calc_GPA1);

$aplus1 = $a + .03;
$aminus1 = $a-.03;
$bplus1=$b+1;
$bminus1=$b-1;
$f[0] = $aplus1 . "x+" . $b . " for x in <-3,35> using color:black and weight:2";
$f[1] = $aminus1 . "x+" . $b . " for x in <-3,35> using color:black and weight:2";
$f[2] = $aminus1 . "x+" . $b . " for x in <-3,35> using color:black and weight:2";
$f[3] = $aplus1 . "x+" . $b . " for x in <-3,35> using color:black and weight:2";
$f[4] = $a . "x+" . $b . " for x in <-3,35> using color:black and weight:2";
$f[5] = $a . "x+" . $b . " for x in <-3,35> using color:black and weight:2";

for ($i = 0; $i <= 5; $i++){
	$graph1[$i] = init_graph(-2,-.5,30,5,'axes'=>[0,0]);
	$graph1[$i]->lb('reset');
	$graph1[$i]->lb(new Label(28,.4,"x",'black','right','top'));
	$graph1[$i]->lb(new Label(2,4.8,"y",'black','right','top'));
	$graph1[$i]->h_ticks(0,"black",2,4,6,8,10,12,14,16,18,20,22,24,26,28);
	$graph1[$i]->v_ticks(0,"black",.5,1,1.5,2,2.5,3,3.5,4,4.5);
	$graph1[$i]->lb(new Label(-1,1,1,'black','left','middle'));
	$graph1[$i]->lb(new Label(-1,2,2,'black','left','middle'));
	$graph1[$i]->lb(new Label(-1,3,3,'black','left','middle'));
	$graph1[$i]->lb(new Label(-1,4,4,'black','left','middle'));

	$graph1[$i]->lb(new Label(4,-.5,4,'black','center','bottom'));
	$graph1[$i]->lb(new Label(8,-.5,8,'black','center','bottom'));
	$graph1[$i]->lb(new Label(12,-.5,12,'black','center','bottom'));
	$graph1[$i]->lb(new Label(16,-.5,16,'black','center','bottom'));
	$graph1[$i]->lb(new Label(20,-.5,20,'black','center','bottom'));
	$graph1[$i]->lb(new Label(24,-.5,24,'black','center','bottom'));
	$graph1[$i]->lb(new Label(28,-.5,28,'black','center','bottom'));
};

@points = ();

for($i=0; $i<$n; $i++)
{
  	$points[$i]=closed_circle($datax[$i],$datay[$i],blue);
}
$graph1[1] -> stamps(@points);
$graph1[3] -> stamps(@points);
$graph1[5] -> stamps(@points);

for($i=0; $i<$n; $i++)
{
  	$points[$i]=closed_circle($mockdatax[$i],$mockdatay[$i],blue);
}
$graph1[0] -> stamps(@points);
$graph1[2] -> stamps(@points);
$graph1[4] -> stamps(@points);

plot_functions($graph1[0],$f[0]);
plot_functions($graph1[1],$f[1]);
plot_functions($graph1[2],$f[2]);
plot_functions($graph1[3],$f[3]);
plot_functions($graph1[4],$f[4]);
plot_functions($graph1[5],$f[5]);

$fig1[0] = image(insertGraph($graph1[0]),width =>250,height =>200,tex_size =>310);

$fig1[1] = image(insertGraph($graph1[1]),width =>250,height =>200,tex_size =>310);

$fig1[2] = image(insertGraph($graph1[2]),width =>250,height =>200,tex_size =>310);

$fig1[3] = image(insertGraph($graph1[3]),width =>250,height =>200,tex_size =>310);

$fig1[4] = image(insertGraph($graph1[4]),width =>250,height =>200,tex_size =>310);

$fig1[5] = image(insertGraph($graph1[5]),width =>250,height =>200,tex_size =>310);

@perm = shuffle(6);
@inv = invert(@perm);
@fig1 = @fig1[@perm];
@eqn = ("\( \ y = 2 f(x) \)", "\( \ y = 0.5 f(x) \)");
@letter = ("A", "B", "C", "D", "E","F");
$correct = $letter[$inv[5]];

TEXT(beginproblem());
Context()->texStrings;
BEGIN_TEXT

A sample of 12 students were asked how many hours they each watched television per week, x, and their GPA, y.  
$BR $BR
$BCENTER
\{ begintable(2) \}
\{ row( "\(x : \)", "\($datax[0] \)", "\($datax[1] \)", "\($datax[2] \)", "\($datax[3] \)", "\($datax[4] \)", "\($datax[5] \)", "\($datax[6] \)", "\($datax[7] \)", "\($datax[8] \)", "\($datax[9] \)", "\($datax[10] \)", "\($datax[11] \)" ) \}
\{ row( "\(y : \)", "\($datay[0] \)", "\($datay[1] \)", "\($datay[2] \)", "\($datay[3] \)", "\($datay[4] \)", "\($datay[5] \)", "\($datay[6] \)", "\($datay[7] \)", "\($datay[8] \)", "\($datay[9] \)", "\($datay[10] \)", "\($datay[11] \)" ) \}
\{ endtable() \}
$ECENTER
$BR $BR
(a) Find the correlation coefficient for the sample data. Round to the nearest hundredth (2 decimal places).
$BR $BR $BCENTER
\{ ans_rule(5) \} 
$BR $BR $ECENTER

(b)
\{ $radio_correlation_coefficient->print_q() \}
\{ $radio_correlation_coefficient->print_a() \}
$BR$BR

(c) Find the coefficient of determination for the sample data. Round to the nearest hundredth (2 decimal places).
$BR $BR $BCENTER
\{ ans_rule(5) \} 
$BR $BR $ECENTER

(d)
\{ $radio_coefficient_det->print_q() \}
\{ $radio_coefficient_det->print_a() \}
$BR$BR

(e)
\{ $coef_det_interpretation->print_q() \}
\{ $coef_det_interpretation->print_a() \}
$BR$BR

(f) Find the regression equation for the sample data in the form \(y' = ax + b \). Round $BITALIC a $EITALIC and $BITALIC b $EITALIC to the nearest hundredth (2 decimal places).
$BR $BR $BCENTER
y'  = \{ ans_rule(5)\} x  + \{ ans_rule(5) \}  
$BR $BR $ECENTER

(g) Graph the scatter plot and the regression equation on your calcluator.  Which of the graphs A-F matches your graph?
\{ pop_up_list(["?","A","B","C","D","E","F"]) \}
$BR $BCENTER (Click on a graph to enlarge it) $ECENTER
$PAR
\{
BeginTable().
  AlignedRow([$fig1[0],$fig1[1],$fig1[2]]).
  TableSpace(5,0).
  AlignedRow(["A","B","C"]).
  TableSpace(25,6).
  AlignedRow([$fig1[3],$fig1[4],$fig1[5]]).
  TableSpace(5,0).
  AlignedRow(["D","E","F"]).
EndTable().
$BR.$BCENTER."(Click on a graph to enlarge it)".$ECENTER
\}

$BR
$BR

(h) Suppose that a student watches television $random_hours hours per week.  Use the regression equation found in part (f) to predict the student's GPA. Round GPA to the nearest tenth (1 decimal place).
$BR $BR $BCENTER
\{ ans_rule(5)\}  
$BR $BR $ECENTER

(i) Suppose that a student watches television $random_hours1 hours per week.  Use the regression equation found in part (f) to predict the student's GPA. Round GPA to the nearest tenth (1 decimal place).
$BR $BR $BCENTER
\{ ans_rule(5)\}  
$BR $BR $ECENTER

END_TEXT

Context("Numeric");

WEIGHTED_ANS( $r->cmp(tolType=>'absolute',tolerance=>0), 8 );
WEIGHTED_ANS( radio_cmp( $radio_correlation_coefficient->correct_ans() ), 4 );
WEIGHTED_ANS( $r_squared->cmp(tolType=>'absolute',tolerance=>0), 8 );
WEIGHTED_ANS( radio_cmp( $radio_coefficient_det->correct_ans() ), 4 );
WEIGHTED_ANS( radio_cmp( $coef_det_interpretation->correct_ans() ), 15 );
WEIGHTED_ANS( $a->cmp(tolType=>'absolute',tolerance=>0), 8 );
WEIGHTED_ANS( $b->cmp(tolType=>'absolute',tolerance=>0), 8 );
Context()->normalStrings;

WEIGHTED_ANS(str_cmp( $correct ),15);

Context("Numeric");
WEIGHTED_ANS( $calc_GPA->cmp(tolType=>'absolute',tolerance=>.11), 15 );
WEIGHTED_ANS( $calc_GPA1->cmp(tolType=>'absolute',tolerance=>.11), 15 );

ENDDOCUMENT();