View of /trunk/NationalProblemLibrary/UVA-Stat/setStat212-Homework12/stat212-HW12-16.pg

    1 ## 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 ## TitleText1('Mathematical Statistics')
   15 ## 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.