View of /trunk/NationalProblemLibrary/UVA-Stat/setStat212-Homework11/stat212-HW11-10.pg

    1 ##DESCRIPTION
    2 ##KEYWORDS('statistics','Inference about a population')
    3 ##
    4 ## CMMK tagged this problem
    5 
    6 ## DBchapter('Inference About a Population')
    7 ## DBsection()
    8 ## Date('07/12/2005')
    9 ## Author('Cristina Murray-Krezan')
   10 ## Institution('UVa')
   11 ## TitleText1('Statistics for Management and Economics')
   12 ## EditionText1('6')
   13 ## AuthorText1('Keller, Warrack')
   14 ## Section1()
   15 ## Problem1()
   16 
   17 
   18 ##ENDDESCRIPTION
   19 
   20 DOCUMENT();        # This should be the first executable line in the problem.
   21 
   22 loadMacros(
   23 "PG.pl",
   24 "PGbasicmacros.pl",
   25 "PGchoicemacros.pl",
   26 "PGanswermacros.pl",
   27 "PGgraphmacros.pl",
   28 "PGnumericalmacros.pl",
   29 "PGstatisticsmacros.pl",
   30 "extraAnswerEvaluators.pl"
   31 );
   32 
   33 TEXT(beginproblem());
   34 $showPartialCorrectAnswers = 0;
   35 install_problem_grader(~~&std_problem_grader);
   36 
   37 $pval = 0;
   38 while ($pval < 0.005) {
   39 
   40 $mu1 = random(63,68,1);
   41 $mu2 = random(73,78,1);
   42 
   43 $p1[1] = $mu1 + random(-5,5,1);
   44 $p1[2] = $mu1 + random(-5,5,1);
   45 $p1[3] = $mu1 + random(-5,5,1);
   46 $p1[4] = $mu1 + random(-5,5,1);
   47 $p1[5] = $mu1 + random(-5,5,1);
   48 $p1[6] = $mu1 + random(-5,5,1);
   49 $p1[7] = $mu1 + random(-5,5,1);
   50 $n1 = 7;
   51 
   52 $p2[1] = $mu2 + random(-5,5,1);
   53 $p2[2] = $mu2 + random(-5,5,1);
   54 $p2[3] = $mu2 + random(-5,5,1);
   55 $p2[4] = $mu2 + random(-5,5,1);
   56 $p2[5] = $mu2 + random(-5,5,1);
   57 $p2[6] = $mu2 + random(-5,5,1);
   58 $p2[7] = $mu2 + random(-5,5,1);
   59 $p2[8] = $mu2 + random(-5,5,1);
   60 $n2 = 8;
   61 
   62 $xbar1 = ($p1[1]+$p1[2]+$p1[3]+$p1[4]+$p1[5]+$p1[6]+$p1[7])/$n1;
   63 $xbar2 = ($p2[1]+$p2[2]+$p2[3]+$p2[4]+$p2[5]+$p2[6]+$p2[7]+$p2[8])/$n2;
   64 $s1 = sqrt( (($p1[1]-$xbar1)**2 + ($p1[2]-$xbar1)**2 + ($p1[3]-$xbar1)**2 +
   65         ($p1[4]-$xbar1)**2 + ($p1[5]-$xbar1)**2 + ($p1[6]-$xbar1)**2 +
   66         ($p1[7]-$xbar1)**2)/($n1-1));
   67 $s2 = sqrt( (($p2[1]-$xbar2)**2 + ($p2[2]-$xbar2)**2 + ($p2[3]-$xbar2)**2 +
   68         ($p2[4]-$xbar2)**2 + ($p2[5]-$xbar2)**2 + ($p2[6]-$xbar2)**2 +
   69         ($p2[7]-$xbar2)**2 + ($p2[8]-$xbar2)**2)/($n2-1));
   70 
   71 $df = $n1 + $n2 - 2;
   72 $sp2 = (($n1-1)*$s1**2 + ($n2-1)*$s2**2)/$df;
   73 
   74 $alpha = random(0.02, 0.08, 0.005);
   75 
   76 $tstat = ($xbar1 - $xbar2)/sqrt($sp2*(1/$n1 + 1/$n2));
   77 $tcrit = tdistr($df,$alpha);
   78 $pval = tprob($df,-$tstat);
   79 }
   80 
   81 $mc = new_multiple_choice();
   82 
   83 @ans = ("Reject \(H_0\).", "Do Not Reject \(H_0\).",
   84         "Reject \(H_1\).", "Do Not Reject \(H_1\).");
   85 
   86 if ($pval < $alpha) {$tag = 0;} else {$tag = 1;}
   87 
   88 $mc -> qa('D. Your decision for the hypothesis test:', $ans[$tag]);
   89 
   90 $mc -> extra($ans[1-$tag],$ans[2],$ans[3]);
   91 
   92 
   93 BEGIN_TEXT
   94 $PAR
   95 Random samples of resting heart rates are taken from two groups.
   96 Population 1 exercises regularly, and Population 2 does not.  The data
   97 from these two samples is given below:
   98 $PAR
   99 
  100 Population 1: $p1[1], $p1[2], $p1[3], $p1[4], $p1[5], $p1[6], $p1[7]
  101 
  102 $PAR
  103 
  104 Population 2: $p2[1], $p2[2], $p2[3], $p2[4], $p2[5], $p2[6], $p2[7], $p2[8]
  105 
  106 $PAR
  107 
  108 Is there evidence, at an \(\alpha = $alpha\) level of significance,
  109 to conclude that there those who exercise regularly have lower resting heart
  110 rates?  (Assume that the population variances are equal.)  Carry out an appropriate
  111 hypothesis test, filling in the information requested.
  112 $PAR
  113 
  114 A. The value of the standardized test statistic: \{ ans_rule(25) \}
  115 $PAR
  116 $BBOLD Note:$EBOLD For the next part, your answer should use interval notation.  An
  117 answer of the form \((-\infty, a)\) is expressed (-infty, a), an answer of the
  118 form \((b, \infty)\) is expressed (b, infty), and an answer of the
  119 form \((-\infty, a) \cup (b, \infty)\) is expressed (-infty, a)U(b, infty).
  120 $PAR
  121 B. The rejection region for the standardized test statistic: \{ ans_rule(25) \} $PAR
  122 $PAR
  123 C. The p-value is \{ ans_rule(25) \} $PAR
  124 
  125 \{ $mc ->print_q() \} $BR
  126 \{ $mc ->print_a() \}
  127 
  128 END_TEXT
  129 
  130 ANS(num_cmp($tstat));
  131 ANS(interval_cmp("(-infty,-$tcrit)", sloppy=>'yes'));
  132 ANS(num_cmp($pval,tol=>0.005));
  133 ANS(radio_cmp($mc->correct_ans));
  134 
  135 ENDDOCUMENT();       # This should be the last executable line in the problem.