########################################################################
## DESCRIPTION
## A WeBWorK problem that asks students to conduct a hypothesis test with an 
## unknown population standard deviation and a raw data set.
## WeBWorK problem written by JoAnne Taormina, <joanne(dot)taormina(at)ncc(dot)edu>
## ENDDESCRIPTION
##
## KEYWORDS('hypothesis test, 'sample standard deviation', 'p-value')
##
## Author('JoAnne Taormina')
## Institution('Nassau Community College')
########################################################################

DOCUMENT();        # This should be the first executable line in the problem.

loadMacros(
"PG.pl",
"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGstatisticsmacros.pl",
"PGnumericalmacros.pl",
"weightedGrader.pl",
"PGstandard.pl",
"MathObjects.pl",
"parserPopUp.pl",
);
install_weighted_grader();

TEXT(beginproblem());
$showPartialCorrectAnswers = 1;

$mu = random(8290,8490,10);
$sample = 15;

# get the data values
@data=();
@deviation_from_mean=();
@deviation_from_mean_squared=();

$sum= 0;
for($i = 0; $i < $sample; $i++)
{
  $data[$i] = random(-100,190,1)+$mu;
  $sum += $data[$i];
}
$x_bar = $sum/$sample;

for($i=0; $i<$sample; $i++){
 $deviation_from_mean[$i] = $data[$i] - $x_bar;
 $deviation_from_mean_squared[$i] = $deviation_from_mean[$i]**2; 
 $sum_deviation_from_mean_squared += $deviation_from_mean_squared[$i];
}
$sample_variance = $sum_deviation_from_mean_squared/($sample-1);
$s = sqrt($sample_variance);

$alpha = 1;

# round $s to 4 decimal places
$s_rounded = int(10000*$s+.5*($s <=> 0))/10000;

# round $x_bar to 2 decimal places
$x_bar_rounded = int(100*$x_bar+.5*($x_bar <=> 0))/100;

@choices = ("population mean", "sample mean", "population proportion", "sample proportion");
@choices = @choices[shuffle(4)];
$popup_H0_mean = PopUp(["Choose:",$choices[0], $choices[1], $choices[2], $choices[3]], "population mean");

@choices = ("is", "is greater than", "is less than");
@choices = @choices[shuffle(3)];
$popup_H0_direction = PopUp(["Choose:",$choices[0], $choices[1], $choices[2]],"is");

@choices = ($s_rounded, $sample, $mu, $x_bar_rounded,"1%");
@choices = @choices[shuffle(5)];
$popup_H0_value = PopUp(["Choose:",$choices[0], $choices[1], $choices[2], $choices[3], $choices[4]],$mu);

@choices = ("population mean", "sample mean", "population proportion", "sample proportion");
@choices = @choices[shuffle(4)];
$popup_Ha_mean = PopUp(["Choose:",$choices[0], $choices[1], $choices[2], $choices[3]], "population mean");

@choices = ("is", "is greater than", "is less than");
@choices = @choices[shuffle(3)];
$popup_Ha_direction = PopUp(["Choose:",$choices[0], $choices[1], $choices[2]],"is greater than");

@choices = ($s_rounded, $sample, $mu, $x_bar_rounded,"1%");
@choices = @choices[shuffle(5)];
$popup_Ha_value = PopUp(["Choose:",$choices[0], $choices[1], $choices[2], $choices[3], $choices[4]],$mu);

# set up for a multiple choice problem.
$radio_samp_dist_mean = new_multiple_choice();
$radio_samp_dist_mean->qa("The correct symbol for mean of the sampling distribution is ", "\( \mu_{\bar{x}} \)");
$radio_samp_dist_mean->extra("\( \mu \)", "\( \bar{x} \)", "\( \hat{p} \)",  "\( \mu_{\hat{p}} \)",  "\( p \)");

# set up for a multiple choice problem.
$radio_dist = new_multiple_choice();
$radio_dist->qa("Choose the correct distribution", "T-Distribution");
$radio_dist->extra("Normal Distribution");

# set up for a multiple choice problem.
$radio_dist_reason = new_multiple_choice();
$radio_dist_reason->qa("Why?", "We are not provided with \( \sigma \).");
$radio_dist_reason->extra("We are provided with \( \sigma \).", "We are provided with \( \bar{x} \).", "We are not provided with \( \bar{x} \).");

@choices = ("less than or equal to 0.01", "greater than or equal to 0.01", "less than or equal to 0.0001", "greater than or equal to 0.0001","equal to 0.01", "equal to 0.0001");
@perm = shuffle(6);
@choices = @choices[@perm];
$popup_p_value = PopUp(["Choose:",$choices[0], $choices[1], $choices[2], $choices[3], $choices[4], $choices[5]],"less than or equal to 0.01");

# set up for a multiple choice problem.
$radio_samp_stat = new_multiple_choice();
$radio_samp_stat->qa("Symbol of the sample statistic:", "\( \bar{x} \)");
$radio_samp_stat->extra("\( \mu \)", "\( \hat{p} \)",  "\( \mu_{\hat{p}} \)",  "\( p \)", "\( \mu_{\bar{x}} \)");

$standard_error = $s/(sqrt($sample));
$p_value=tprob($sample-1, ($x_bar-$mu)/$standard_error  );

#$p_value = normal_prob($x_bar, "infty", mean=>$mu, deviation=>$standard_error  );
# round $p_value to 4 decimal places
$p_value = int(10000*$p_value+.5*($p_value <=> 0))/10000;

$alpha = Compute(1);
$mu = Compute($mu);
$x_bar_rounded = Compute($x_bar_rounded);
$p_value = Compute($p_value);

@choices = ("Reject Ho and accept Ha", "Reject Ha and accept Ho", "Fail to reject Ho", "Fail to reject Ha");
@choices = @choices[shuffle(4)];
if($p_value <= .01)
{
  $correct_h = "Reject Ho and accept Ha";
  $correct_i = "Yes, because the mean cost of raising a child from birth to age 2 is significantly more than $mu dollars.";
  $extra_i[0] = "No, because the mean cost of raising a child from birth to age 2 is not significantly more than $mu dollars.";

}
else
{
  $correct_h = "Fail to reject Ho";
  $correct_i = "No, because the mean cost of raising a child from birth to age 2 is not significantly more than $mu dollars.";
  $extra_i[0] = "Yes, because the mean cost of raising a child from birth to age 2 is significantly more than $mu dollars."; 
}
$popup_conclusion = PopUp(["Choose:",$choices[0], $choices[1], $choices[2], $choices[3]],$correct_h);

# set up for a multiple choice problem.
$radio_answer = new_multiple_choice();
$radio_answer->qa("Is the mean cost of raising a child greater than $mu dollars?", $correct_i);
$extra_i[1] = "Yes, because the mean cost of raising a child from birth to age 2 is not significantly more than $mu dollars.";
$extra_i[2] = "No, because the mean cost of raising a child from birth to age 2 is significantly more than $mu dollars.";
$radio_answer->extra($extra_i[0], $extra_i[1],  $extra_i[2]);

# set up for a multiple choice problem.
$radio_type1_error = new_multiple_choice();
$radio_type1_error->qa("If a type I error were made in this test, it would mean", "the mean really is $mu dollars, but we incorrectly rejected $mu dollars.");
$radio_type1_error->extra("the mean really is not $mu dollars, but we incorrectly rejected $mu dollars.", "the mean really is $mu dollars, but we incorrectly failed to reject $mu dollars.", "the mean really is not $mu dollars, but we incorrectly failed to reject $mu dollars.");

# set up for a multiple choice problem.
$radio_type2_error = new_multiple_choice();
$radio_type2_error->qa("If a type 2 error were made in this test, it would mean", "the mean really is not $mu dollars, but we incorrectly failed to reject $mu dollars.");
$radio_type2_error->extra("the mean really is not $mu dollars, but we incorrectly rejected $mu dollars.", "the mean really is $mu dollars, but we incorrectly failed to reject $mu dollars.","the mean really is $mu dollars, but we incorrectly rejected $mu dollars.");
   
BEGIN_TEXT
The U.S. Department of Agriculture reports that the mean cost of raising a child from birth to age 2 in in a rural area is \( $mu\) dollars.  You believe this value is is too low, so you sample \($sample \) rural families and find the following costs (in dollars) of raising a child from birth to age 2.
$BR $BR
\[ $data[0], \ $data[1], \ $data[2], \ $data[3], \ $data[4], \ $data[5], \ $data[6], \ $data[7], \ $data[8], \ $data[9], \ $data[10], \ $data[11], \ $data[12], \ $data[13], \ $data[14] \] 
$BR
Conduct a hypothesis test based on this sample data, with \( \alpha = 1%\).  Answer the question, is the mean cost of raising a child greater than \($mu\) dollars?
$PAR
$BBOLD (a) $EBOLD State the hypotheses: $BR $BR
\(H_0:\) $SPACE The \{ $popup_H0_mean->menu() \} $SPACE cost in dollars of raising a child from birth to age 2 \{ $popup_H0_direction->menu() \} $SPACE \{ $popup_H0_value->menu() \}. $BR $BR
\(H_a:\) $SPACE The \{ $popup_Ha_mean->menu() \} $SPACE cost in dollars of raising a child from birth to age 2 \{ $popup_Ha_direction->menu() \} $SPACE \{ $popup_Ha_value->menu() \}. $BR  $BR

$BBOLD (b) $EBOLD 
\{ $radio_samp_dist_mean->print_q() \}
\{ $radio_samp_dist_mean->print_a() \} $BR $BR

$BBOLD (c) $EBOLD The value for the mean of the sampling distribution is: \{ ans_rule(5) \}. $BR $BR

$BBOLD (d) $EBOLD 
\{ $radio_dist->print_q() \}
\{ $radio_dist->print_a() \} $BR 

\{ $radio_dist_reason->print_q() \}
\{ $radio_dist_reason->print_a() \} $BR $BR

$BBOLD (e) $EBOLD  State the decision rule: $BR $BR
Reject \(H_0\) at \( \alpha \)= \{ ans_rule(5) \}% if the p-value of the sample statistic is \{ $popup_p_value->menu() \}. $BR $BR

$BBOLD (f) Experiment $EBOLD $BR
\{ $radio_samp_stat->print_q() \}
\{ $radio_samp_stat->print_a() \} $BR 
Value of the sample statistic $BITALIC rounded to 2 decimal places$EITALIC: \{ ans_rule(7) \} $SPACE
P-value of the sample statistic $BITALIC rounded to 4 decimal places$EITALIC:  \{ ans_rule(5) \} $BR $BR

$BBOLD (h) Conclusion $EBOLD $BR
\{ $popup_conclusion->menu() \} at \( \alpha \)= \{ ans_rule(5) \}% $BR $BR

$BBOLD (i) Answer the question $EBOLD $BR
\{ $radio_answer->print_q() \}
\{ $radio_answer->print_a() \} $BR $BR

$BBOLD (j) Type I Error $EBOLD $BR
\{ $radio_type1_error->print_q() \}
\{ $radio_type1_error->print_a() \} $BR $BR

$BBOLD (k) Type 2 Error $EBOLD $BR
\{ $radio_type2_error->print_q() \}
\{ $radio_type2_error->print_a() \} $BR $BR

END_TEXT
Context("Numeric");
WEIGHTED_ANS( $popup_H0_mean->cmp(),2 );
WEIGHTED_ANS( $popup_H0_direction->cmp(),2 );
WEIGHTED_ANS( $popup_H0_value->cmp(),2 );
WEIGHTED_ANS( $popup_Ha_mean->cmp(),2 );
WEIGHTED_ANS( $popup_Ha_direction->cmp(),2 );
WEIGHTED_ANS( $popup_Ha_value->cmp(),2 );
WEIGHTED_ANS( radio_cmp( $radio_samp_dist_mean->correct_ans() ), 4 );
WEIGHTED_ANS($mu->cmp(tolType=>'absolute',tolerance=>0), 4);
WEIGHTED_ANS( radio_cmp( $radio_dist->correct_ans() ), 4 );
WEIGHTED_ANS( radio_cmp( $radio_dist_reason->correct_ans() ), 4 );
WEIGHTED_ANS($alpha->cmp(tolType=>'absolute',tolerance=>0), 2);
WEIGHTED_ANS( $popup_p_value->cmp(),4 );
WEIGHTED_ANS( radio_cmp( $radio_samp_stat->correct_ans() ), 4 );
WEIGHTED_ANS($x_bar_rounded->cmp(tolType=>'absolute',tolerance=>0), 8);
WEIGHTED_ANS($p_value->cmp(tolType=>'absolute',tolerance=>0), 10);
WEIGHTED_ANS( $popup_conclusion->cmp(),10 );
WEIGHTED_ANS($alpha->cmp(tolType=>'absolute',tolerance=>0), 2);
WEIGHTED_ANS( radio_cmp( $radio_answer->correct_ans() ), 10 );
WEIGHTED_ANS( radio_cmp( $radio_type1_error->correct_ans() ), 10 );
WEIGHTED_ANS( radio_cmp( $radio_type2_error->correct_ans() ), 10 );

ENDDOCUMENT();       # This should be the last executable line in the problem.