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| 1 : | sh002i | 1050 | #!/usr/local/bin/webwork-perl |
| 2 : | |||
| 3 : | sub _PGstatisticsmacros_init { | ||
| 4 : | foreach my $t (@Distributions::EXPORT_OK) { | ||
| 5 : | *{$t} = *{"Distributions::$t"} | ||
| 6 : | } | ||
| 7 : | foreach my $t (@Regression::EXPORT_OK) { | ||
| 8 : | *{$t} = *{"Regression::$t"} | ||
| 9 : | } | ||
| 10 : | } | ||
| 11 : | |||
| 12 : | =head1 Statistics Macros | ||
| 13 : | |||
| 14 : | =head3 Normal distribution | ||
| 15 : | |||
| 16 : | =pod | ||
| 17 : | |||
| 18 : | Usage: normal_prob(a, b, mean=>0, deviation=>1); | ||
| 19 : | |||
| 20 : | Computes the probability of x being in the interval (a,b) for normal distribution. | ||
| 21 : | The first two arguments are required. Use '-infty' for negative infinity, and 'infty' or '+infty' for positive infinity. | ||
| 22 : | The mean and deviation are optional, and are 0 and 1 respectively by default. | ||
| 23 : | Load PGnumericalmacros.pl in your problem if you use this method. | ||
| 24 : | |||
| 25 : | =cut | ||
| 26 : | |||
| 27 : | sub normal_prob { | ||
| 28 : | warn 'You must also load PGnumericalmacros to use PGstatisticsmacros' unless defined(&_PGnumericalmacros_init); | ||
| 29 : | |||
| 30 : | my $a = shift; | ||
| 31 : | my $b = shift; | ||
| 32 : | my %options=@_; | ||
| 33 : | |||
| 34 : | my $mean = $options{'mean'} if defined ($options{'mean'}); | ||
| 35 : | $mean = 0 unless defined $mean; | ||
| 36 : | |||
| 37 : | my $deviation = $options{'deviation'} if defined ($options{'deviation'}); | ||
| 38 : | $deviation = 1 unless defined $deviation; | ||
| 39 : | |||
| 40 : | if ($deviation <= 0) { | ||
| 41 : | warn 'Deviation must be a positive number.'; | ||
| 42 : | return; | ||
| 43 : | } | ||
| 44 : | |||
| 45 : | my $z_score_of_a; | ||
| 46 : | my $z_score_of_b; | ||
| 47 : | |||
| 48 : | if ( $a eq '-infty' ) { | ||
| 49 : | $z_score_of_a = -6; | ||
| 50 : | } else { | ||
| 51 : | $z_score_of_a = ($a - $mean)/$deviation; | ||
| 52 : | } | ||
| 53 : | |||
| 54 : | if (($b eq 'infty') or ($b eq '+infty')) { | ||
| 55 : | $z_score_of_b = 6; | ||
| 56 : | } else { | ||
| 57 : | $z_score_of_b = ($b - $mean)/$deviation; | ||
| 58 : | } | ||
| 59 : | |||
| 60 : | my $function = sub { my $x=shift; | ||
| 61 : | $E**(-$x**2/2)/sqrt(2*$PI); | ||
| 62 : | }; | ||
| 63 : | |||
| 64 : | my $prob = romberg($function, $z_score_of_a, $z_score_of_b, 8); | ||
| 65 : | $prob; | ||
| 66 : | } | ||
| 67 : | |||
| 68 : | =head3 "Inverse" of normal distribution | ||
| 69 : | |||
| 70 : | =pod | ||
| 71 : | |||
| 72 : | Usage: normal_distr(prob, mean=>0, deviation=>1); | ||
| 73 : | |||
| 74 : | Computes the positive number b such that the probability of x being in the interval (0,b) | ||
| 75 : | is equal to the given probability (first argument). The mean and deviation are | ||
| 76 : | optional, and are 0 and 1 respectively by default. | ||
| 77 : | Caution: since students may use tables, they may only be able to provide the answer correct to 2 or 3 | ||
| 78 : | decimal places. Use tolerance when evaluating answers. | ||
| 79 : | Load PGnumericalmacros.pl if you use this method. | ||
| 80 : | |||
| 81 : | =cut | ||
| 82 : | |||
| 83 : | sub normal_distr { | ||
| 84 : | warn 'You must also load PGnumericalmacros to use PGstatisticsmacros' unless defined(&_PGnumericalmacros_init); | ||
| 85 : | |||
| 86 : | my $prob = shift; | ||
| 87 : | my %options=@_; | ||
| 88 : | |||
| 89 : | my $mean = $options{'mean'} if defined ($options{'mean'}); | ||
| 90 : | $mean = 0 unless defined $mean; | ||
| 91 : | |||
| 92 : | my $deviation = $options{'deviation'} if defined ($options{'deviation'}); | ||
| 93 : | $deviation = 1 unless defined $deviation; | ||
| 94 : | |||
| 95 : | if ($deviation <= 0) { | ||
| 96 : | warn 'Deviation must be a positive number.'; | ||
| 97 : | return; | ||
| 98 : | } | ||
| 99 : | |||
| 100 : | my $function = sub { my $x=shift; | ||
| 101 : | $E**(-$x**2/2)/sqrt(2*$PI); | ||
| 102 : | }; | ||
| 103 : | |||
| 104 : | my $z_score_of_b = inv_romberg($function, 0, $prob); | ||
| 105 : | |||
| 106 : | my $b = $z_score_of_b * $deviation + $mean; | ||
| 107 : | $b; | ||
| 108 : | } | ||
| 109 : | |||
| 110 : | ########################################## | ||
| 111 : | |||
| 112 : | 1; | ||
| 113 : |
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