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    1 ##DESCRIPTION
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
23 "PG.pl",
24 "PGbasicmacros.pl",
25 "PGchoicemacros.pl",
27 "PGgraphmacros.pl",
28 "PGnumericalmacros.pl",
29 "PGstatisticsmacros.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));