########################################################################
## DESCRIPTION
## A WeBWorK problem that asks students to conduct a hypothesis test 
## involving a proportion.
## WeBWorK problem written by JoAnne Taormina, <joanne(dot)taormina(at)ncc(dot)edu>
## ENDDESCRIPTION
##
## KEYWORDS('hypothesis test, 'proportion', '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;

$prop = random(.65,.80,.01);
$prop_percent = $prop*100;
$sample = random(100, 200, 10);
$decrease = random(.05,.10,.01);
$p_hat = $prop-$decrease;
$x = $p_hat*$sample;  
# round $x to nearest whole number
$x = int(1*$x+.5*($x <=> 0))/1;

#calculate a new p_hat given x and sample size
$p_hat = $x/$sample;
# round $p_hat to 4 decimal places
$p_hat_rounded = int(10000*$p_hat+.5*($p_hat <=> 0))/10000;

$alpha = 5;

@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 proportion");

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

@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 proportion");

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

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

# 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 ", "\( \mu_{\hat{p}} \)");
$radio_samp_dist_mean->extra("\( \mu \)", "\( \bar{x} \)", "\( \hat{p} \)",  "\( \mu_{\bar{x}} \)",  "\( p \)");

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

# set up for a multiple choice problem.
$radio_dist_reason = new_multiple_choice();
$radio_dist_reason->qa("Why?", "np > 5 and n(1-p) > 5");
$radio_dist_reason->extra("We are provided with \( \sigma \).", "We are not provided with \( \sigma \).", "np > 30 and n(1-p) > 30.");

@choices = ("less than or equal to 0.05", "greater than or equal to 0.05", "less than or equal to 0.0005", "greater than or equal to 0.0005","equal to 0.05", "equal to 0.0005");
@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 0.05");

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

$standard_error = sqrt($prop*(1-$prop)/$sample);
$p_value = normal_prob("-infty", $p_hat, mean=>$prop,deviation=>$standard_error );
# round $p_value to 4 decimal places
$p_value = int(10000*$p_value+.5*($p_value <=> 0))/10000;

@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 <= .05)
{
  $correct_h = "Reject Ho and accept Ha";
  $correct_i = "Yes, the vet is correct, because the proportion of people who got relief is significantly less than $prop_percent%.";
  $extra_i[0] = "No, the vet is not correct, because the proportion of people who got relief is not significantly less than $prop_percent%."; 

}
else
{
  $correct_h = "Fail to reject Ho";
  $correct_i = "No, the vet is not correct, because the proportion of people who got relief is not significantly less than $prop_percent%.";
  $extra_i[0] = "Yes, the vet is correct, because the proportion of people who got relief is significantly less than $prop_percent%."; 
}
$popup_conclusion = PopUp(["Choose:",$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 vet correct?", $correct_i);
$extra_i[1] = "Yes, the vet is correct, because the proportion of people who got relief is not significantly less than $prop_percent%.";
$extra_i[2] = "No, the vet is not correct, because the proportion of people who got relief is significantly less than $prop_percent%.";
$radio_answer->extra($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 proportion of people who get relief with the medicine $BBOLD really is $prop_percent%, but we incorrectly rejected $prop_percent%.$EBOLD");
$radio_type1_error->extra("the proportion of people who get relief with the medicine $BBOLD really is not $prop_percent&, but we incorrectly rejected $prop_percent%.$EBOLD", "the proportion of people who get relief with the medicine $BBOLD really is $prop_percent%, but we incorrectly failed to reject $prop_percent%.$EBOLD", "the proportion of people who get relief with the medicine $BBOLD really is not $prop_percent%, but we incorrectly failed to reject $prop_percent%.$EBOLD");

# 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 proportion of people who get relief with the medicine $BBOLD really is not $prop_percent%, but we incorrectly failed to reject $prop_percent%.$EBOLD");
$radio_type2_error->extra("the proportion of people who get relief with the medicine $BBOLD really is not $prop_percent%, but we incorrectly rejected $prop_percent%.$EBOLD", "the proportion of people who get relief with the medicine $BBOLD really is $prop_percent%, but we incorrectly failed to reject $prop_percent%.$EBOLD","the proportion of people who get relief with the medicine $BBOLD really is $prop_percent%, but we incorrectly rejected $prop_percent%.$EBOLD");

$alpha = Compute($alpha);
$prop = Compute($prop);
$p_hat_rounded = Compute($p_hat_rounded);
$p_value = Compute($p_value);
	   
BEGIN_TEXT
A certain medicine is said to be \($prop_percent%\) effective in giving relief to people with allergic reactions to cats and dogs.  A vet believes this claims is too high.  In a random sample of \($sample\) people with allergies, \($x\) people got relief.  Is the vet correct at \( \alpha = $alpha% \)?
$PAR
$BBOLD (a) $EBOLD State the hypotheses: $BR $BR
\(H_0:\) $SPACE The \{ $popup_H0_mean->menu() \} $SPACE of people who get relief from the medicine \{ $popup_H0_direction->menu() \} $SPACE \{ $popup_H0_value->menu() \}. $BR $BR
\(H_a:\) $SPACE The \{ $popup_Ha_mean->menu() \} $SPACE of people who get relief from the medicine \{ $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 $BITALIC rounded to 4 decimal places$EITALIC: \{ ans_rule(5) \} $BR 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( $radio_samp_dist_mean->correct_ans() ), 4 );
WEIGHTED_ANS($prop->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( $radio_samp_stat->correct_ans() ), 4 );
WEIGHTED_ANS($p_hat_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.