##DESCRIPTION
##KEYWORDS('statistics','Inference about a population')
##
## CMMK tagged this problem

## DBchapter('Inference About a Population')
## DBsection()
## Date('07/12/2005')
## Author('Cristina Murray-Krezan')
## Institution('UVa')
## TitleText1('Statistics for Management and Economics')
## EditionText1('6')
## AuthorText1('Keller, Warrack')
## Section1()
## Problem1()


##ENDDESCRIPTION

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

loadMacros(
"PG.pl",
"PGbasicmacros.pl",
"PGchoicemacros.pl",
"PGanswermacros.pl",
"PGgraphmacros.pl",
"PGnumericalmacros.pl",
"PGstatisticsmacros.pl",
"extraAnswerEvaluators.pl"
);

TEXT(beginproblem());
$showPartialCorrectAnswers = 0;
install_problem_grader(~~&std_problem_grader);

$pval = 0;
while ($pval < 0.005) {

$mu1 = random(63,68,1);
$mu2 = random(73,78,1);

$p1[1] = $mu1 + random(-5,5,1);
$p1[2] = $mu1 + random(-5,5,1);
$p1[3] = $mu1 + random(-5,5,1);
$p1[4] = $mu1 + random(-5,5,1);
$p1[5] = $mu1 + random(-5,5,1);
$p1[6] = $mu1 + random(-5,5,1);
$p1[7] = $mu1 + random(-5,5,1);
$n1 = 7;

$p2[1] = $mu2 + random(-5,5,1);
$p2[2] = $mu2 + random(-5,5,1);
$p2[3] = $mu2 + random(-5,5,1);
$p2[4] = $mu2 + random(-5,5,1);
$p2[5] = $mu2 + random(-5,5,1);
$p2[6] = $mu2 + random(-5,5,1);
$p2[7] = $mu2 + random(-5,5,1);
$p2[8] = $mu2 + random(-5,5,1);
$n2 = 8;

$xbar1 = ($p1[1]+$p1[2]+$p1[3]+$p1[4]+$p1[5]+$p1[6]+$p1[7])/$n1;
$xbar2 = ($p2[1]+$p2[2]+$p2[3]+$p2[4]+$p2[5]+$p2[6]+$p2[7]+$p2[8])/$n2;
$s1 = sqrt( (($p1[1]-$xbar1)**2 + ($p1[2]-$xbar1)**2 + ($p1[3]-$xbar1)**2 + 
        ($p1[4]-$xbar1)**2 + ($p1[5]-$xbar1)**2 + ($p1[6]-$xbar1)**2 + 
        ($p1[7]-$xbar1)**2)/($n1-1));
$s2 = sqrt( (($p2[1]-$xbar2)**2 + ($p2[2]-$xbar2)**2 + ($p2[3]-$xbar2)**2 + 
        ($p2[4]-$xbar2)**2 + ($p2[5]-$xbar2)**2 + ($p2[6]-$xbar2)**2 + 
        ($p2[7]-$xbar2)**2 + ($p2[8]-$xbar2)**2)/($n2-1));

$df = $n1 + $n2 - 2;
$sp2 = (($n1-1)*$s1**2 + ($n2-1)*$s2**2)/$df;

$alpha = random(0.02, 0.08, 0.005);

$tstat = ($xbar1 - $xbar2)/sqrt($sp2*(1/$n1 + 1/$n2));
$tcrit = tdistr($df,$alpha);
$pval = tprob($df,-$tstat);
}

$mc = new_multiple_choice();

@ans = ("Reject \(H_0\).", "Do Not Reject \(H_0\).",
        "Reject \(H_1\).", "Do Not Reject \(H_1\).");

if ($pval < $alpha) {$tag = 0;} else {$tag = 1;}

$mc -> qa('D. Your decision for the hypothesis test:', $ans[$tag]);

$mc -> extra($ans[1-$tag],$ans[2],$ans[3]);


BEGIN_TEXT
$PAR
Random samples of resting heart rates are taken from two groups.
Population 1 exercises regularly, and Population 2 does not.  The data 
from these two samples is given below:
$PAR 

Population 1: $p1[1], $p1[2], $p1[3], $p1[4], $p1[5], $p1[6], $p1[7] 

$PAR

Population 2: $p2[1], $p2[2], $p2[3], $p2[4], $p2[5], $p2[6], $p2[7], $p2[8]

$PAR

Is there evidence, at an \(\alpha = $alpha\) level of significance,
to conclude that there those who exercise regularly have lower resting heart 
rates?  (Assume that the population variances are equal.)  Carry out an appropriate
hypothesis test, filling in the information requested.
$PAR

A. The value of the standardized test statistic: \{ ans_rule(25) \}
$PAR
$BBOLD Note:$EBOLD For the next part, your answer should use interval notation.  An
answer of the form \((-\infty, a)\) is expressed (-infty, a), an answer of the 
form \((b, \infty)\) is expressed (b, infty), and an answer of the 
form \((-\infty, a) \cup (b, \infty)\) is expressed (-infty, a)U(b, infty).
$PAR
B. The rejection region for the standardized test statistic: \{ ans_rule(25) \} $PAR 
$PAR
C. The p-value is \{ ans_rule(25) \} $PAR

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

END_TEXT

ANS(num_cmp($tstat));
ANS(interval_cmp("(-infty,-$tcrit)", sloppy=>'yes'));
ANS(num_cmp($pval,tol=>0.005));
ANS(radio_cmp($mc->correct_ans));

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