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| 1 : | jjholt | 451 | ##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 : | jjholt | 453 | ## TitleText1('Statistics for Management and Economics') |
| 12 : | jjholt | 451 | ## EditionText1('6') |
| 13 : | jjholt | 454 | ## AuthorText1('Keller, Warrack') |
| 14 : | jjholt | 451 | ## 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. |
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