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| 1 : | jjholt | 451 | ## DESCRIPTION |
| 2 : | ## Statistics: Sampling Distributions | ||
| 3 : | ## ENDDESCRIPTION | ||
| 4 : | ##DESCRIPTION | ||
| 5 : | ##KEYWORDS('statistics','hypothesis testing') | ||
| 6 : | ## | ||
| 7 : | ## naw tagged this problem | ||
| 8 : | |||
| 9 : | ## DBchapter('Hypothesis Testing') | ||
| 10 : | ## DBsection('Common Large-Sample Tests') | ||
| 11 : | ## Date('7/8/2005') | ||
| 12 : | ## Author('Nolan A. Wages') | ||
| 13 : | ## Institution('University of Virgnia') | ||
| 14 : | jjholt | 453 | ## TitleText1('Mathematical Statistics') |
| 15 : | jjholt | 451 | ## EditionText1('4') |
| 16 : | ## AuthorText1('Wackerly, Mendenhall, Scheaffer') | ||
| 17 : | ## Section1('10.3') | ||
| 18 : | ## Problem1('20') | ||
| 19 : | |||
| 20 : | |||
| 21 : | ##ENDDESCRIPTION | ||
| 22 : | |||
| 23 : | DOCUMENT(); # This should be the first executable line in the problem. | ||
| 24 : | |||
| 25 : | loadMacros( | ||
| 26 : | "PG.pl", | ||
| 27 : | "PGbasicmacros.pl", | ||
| 28 : | "PGchoicemacros.pl", | ||
| 29 : | "PGanswermacros.pl", | ||
| 30 : | "PGgraphmacros.pl", | ||
| 31 : | "PGnumericalmacros.pl", | ||
| 32 : | "PGstatisticsmacros.pl", | ||
| 33 : | "PGauxiliaryFunctions.pl", | ||
| 34 : | "extraAnswerEvaluators.pl" | ||
| 35 : | ); | ||
| 36 : | |||
| 37 : | |||
| 38 : | |||
| 39 : | TEXT(beginproblem()); | ||
| 40 : | $showPartialCorrectAnswers = 0; | ||
| 41 : | install_problem_grader(~~&std_problem_grader); | ||
| 42 : | |||
| 43 : | $x[1] = random(-3,-1,1); | ||
| 44 : | $x[2] = random(0,3,1); | ||
| 45 : | $x[3] = random(4,6,1); | ||
| 46 : | $x[4] = random(7,9,1); | ||
| 47 : | $x[5] = random(9,12,1); | ||
| 48 : | |||
| 49 : | $y[1] = 3 + random(-3,-1,1); | ||
| 50 : | $y[2] = 6 + random(0,3,1); | ||
| 51 : | $y[3] = 9 + random(4,6,1); | ||
| 52 : | $y[4] = 12 + random(6,8,1); | ||
| 53 : | $y[5] = 15 + random(8,12,1); | ||
| 54 : | |||
| 55 : | $n = 5; | ||
| 56 : | |||
| 57 : | $xbar = ($x[1]+$x[2]+$x[3]+$x[4]+$x[5])/$n; | ||
| 58 : | $ybar = ($y[1]+$y[2]+$y[3]+$y[4]+$y[5])/$n; | ||
| 59 : | |||
| 60 : | $sx = sqrt( (($x[1]-$xbar)**2 + ($x[2]-$xbar)**2 + ($x[3]-$xbar)**2 + ($x[4]-$xbar)**2 + ($x[5]-$xbar)**2)/($n-1) ); | ||
| 61 : | $sy = sqrt( (($y[1]-$ybar)**2 + ($y[2]-$ybar)**2 + ($y[3]-$ybar)**2 + ($y[4]-$ybar)**2 + ($y[5]-$ybar)**2)/($n-1) ); | ||
| 62 : | |||
| 63 : | $cov = ( ($x[1]-$xbar)*($y[1]-$ybar) + ($x[2]-$xbar)*($y[2]-$ybar) + ($x[3]-$xbar)*($y[3]-$ybar) | ||
| 64 : | + ($x[4]-$xbar)*($y[4]-$ybar) + ($x[5]-$xbar)*($y[5]-$ybar) )/($n-1); | ||
| 65 : | |||
| 66 : | $b1 = $cov/$sx**2; | ||
| 67 : | $b0 = $ybar - $b1*$xbar; | ||
| 68 : | |||
| 69 : | $sse = ($n-1)*($sy**2 - ($cov/$sx)**2); | ||
| 70 : | $se = sqrt($sse/($n-2)); | ||
| 71 : | $r2 = ($cov/($sx*$sy))**2; | ||
| 72 : | |||
| 73 : | $x1 = random(1,5,1); | ||
| 74 : | $x2 = random(6,10,1); | ||
| 75 : | |||
| 76 : | $yhat1 = $b0 + $b1*$x1; | ||
| 77 : | $yhat2 = $b0 + $b1*$x2; | ||
| 78 : | |||
| 79 : | |||
| 80 : | BEGIN_TEXT | ||
| 81 : | |||
| 82 : | Find the least-squares regression line | ||
| 83 : | \(\hat{y} = b_0 + b_1 x\) through the points | ||
| 84 : | |||
| 85 : | \[ | ||
| 86 : | ($x[1],$y[1]), ($x[2],$y[2]), ($x[3],$y[3]), ($x[4],$y[4]), ($x[5],$y[5]), | ||
| 87 : | \] | ||
| 88 : | and then use it to find point estimates \(\hat{y}\) corresponding to | ||
| 89 : | \(x = $x1\) and \(x = $x2\). | ||
| 90 : | $PAR | ||
| 91 : | |||
| 92 : | For \(x = $x1\), \(\hat{y}\) = \{ ans_rule(15) \} | ||
| 93 : | |||
| 94 : | $PAR | ||
| 95 : | |||
| 96 : | For \(x = $x2\), \(\hat{y}\) = \{ ans_rule(15) \} | ||
| 97 : | |||
| 98 : | $PAR | ||
| 99 : | |||
| 100 : | END_TEXT | ||
| 101 : | |||
| 102 : | ANS(num_cmp($yhat1)); | ||
| 103 : | ANS(num_cmp($yhat2)); | ||
| 104 : | |||
| 105 : | ENDDOCUMENT(); # This should be the last executable line in the problem. |
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