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<!DOCTYPE html>
<html data-require="math graphie stat word-problems">
<head>
<meta charset="UTF-8">
<title>Empirical rule</title>
<script data-main="../local-only/main.js" src="../local-only/require.js"></script>
</head>
<body>
<div class="exercise">
<div class="problems">
<div id="longer">
<div class="vars">
<var id="ANIMAL">animal( 1 )</var>
<var id="ANIMALS">plural( animal( 1 ) )</var>
<div data-ensure="MEAN - STDDEV * 4 &gt; 0">
<var id="MEAN">roundTo( 1, animalAvgLifespan( 1 ) * ( randRange( 80, 120 ) / 100 ) )</var>
<var id="STDDEV">roundTo( 1, animalStddevLifespan( 1 ) * ( randRange( 20, 120 ) / 100 ) )</var>
</div>
<var id="Z">randRangeNonZero( -3, 3 )</var>
<var id="EMPIRICAL">
{
"-3": 99.7,
"-2": 95,
"-1": 68,
"1": 68,
"2": 95,
"3": 99.7
}[ Z ]
</var>
<var id="ANSWER">
{
"-3": 99.85,
"-2": 97.5,
"-1": 84,
"1": 16,
"2": 2.5,
"3": 0.15
}[ Z ]
</var>
</div>
<div class="problem">
The lifespans of <var>ANIMALS</var> in a particular zoo are normally distributed.
The average <var>ANIMAL</var> lives <code><var>MEAN</var></code> years; the
standard deviation is <code><var>STDDEV</var></code> years.
</div>
<p class="question">
Use the empirical rule <code>(68 - 95 - 99.7\%)</code> to estimate the probability of a <var>ANIMAL</var> living longer than
<code><var>roundTo( 1, MEAN + STDDEV * Z )</var></code> years.
</p>
<div class="solution" data-forms="integer, proper, improper, mixed, decimal, percent"><var>ANSWER / 100</var></div>
<div class="hints">
<div>
<div class="graphie" id="normaldist">
init({
range: [ [ MEAN - STDDEV * 3.5, MEAN + STDDEV * 3.5 ], [ -1.5, 4.3 ] ],
scale: [ 475 / ( STDDEV * 7 ), 40 ]
});
style({ stroke: "#bbb" }, function() {
line( [ MEAN - STDDEV * 4, 0 ], [ MEAN + STDDEV * 4, 0 ] );
});
graph.pdf = function( x ) {
return gaussianPDF( MEAN, STDDEV, x ) * 4 / gaussianPDF( MEAN, STDDEV, MEAN ) + 0.2;
};
style({ stroke: BLUE }, function() {
plot( graph.pdf, [ MEAN - STDDEV * 3.5, MEAN + STDDEV * 3.5 ]);
});
style({ stroke: PINK }, function() {
graph.meanLine = line( [ MEAN, 0 ], [ MEAN, graph.pdf( MEAN ) ] ).toBack();
});
graph.meanLabel = label( [ MEAN, 0 ], MEAN, "below", { color: PINK } );
graph.zLine = [];
graph.zLabel = [];
</div>
<p>
We know the lifespans are normally distributed with an average lifespan of
<code class="hint_pink" id="meanHint"><var>MEAN</var></code> years.
</p>
</div>
<div>
<div class="graphie" data-update="normaldist">
graph.meanLine.attr({ stroke: "#bbb" });
graph.meanLabel.css({ color: "#bbb" });
$( "#meanHint" ).removeClass("hint_pink");
style({ stroke: PINK }, function() {
graph.zLine[ -1 ] = line( [ MEAN - STDDEV, 0 ], [ MEAN - STDDEV, graph.pdf( MEAN - STDDEV ) ] ).toBack();
graph.zLine[ 1 ] = line( [ MEAN + STDDEV, 0 ], [ MEAN + STDDEV, graph.pdf( MEAN + STDDEV ) ] ).toBack();
});
graph.zLabel[ -1 ] = label( [ MEAN - STDDEV, 0 ], roundTo( 1, MEAN - STDDEV ), "below", { color: PINK } );
graph.zLabel[ 1 ] = label( [ MEAN + STDDEV, 0 ], roundTo( 1, MEAN + STDDEV ), "below", { color: PINK } );
</div>
<p>
We know the standard deviation is <code><var>STDDEV</var></code> years, so one
standard deviation below the mean is
<code class="hint_pink" id="zm1Hint"><var>roundTo( 1, MEAN - STDDEV )</var></code> years
and one standard deviation above the mean is
<code class="hint_pink" id="zp1Hint"><var>roundTo( 1, MEAN + STDDEV )</var></code> years.
</p>
</div>
<div>
<div class="graphie" data-update="normaldist">
graph.zLine[ -1 ].attr({ stroke: "#bbb" });
graph.zLine[ 1 ].attr({ stroke: "#bbb" });
graph.zLabel[ -1 ].css({ color: "#bbb" });
graph.zLabel[ 1 ].css({ color: "#bbb" });
$( "#zm1Hint" ).removeClass("hint_pink");
$( "#zp1Hint" ).removeClass("hint_pink");
style({ stroke: PINK }, function() {
graph.zLine[ -2 ] = line( [ MEAN - STDDEV * 2, 0 ], [ MEAN - STDDEV * 2, graph.pdf( MEAN - STDDEV * 2 ) ] ).toBack();
graph.zLine[ 2 ] = line( [ MEAN + STDDEV * 2, 0 ], [ MEAN + STDDEV * 2, graph.pdf( MEAN + STDDEV * 2 ) ] ).toBack();
});
graph.zLabel[ -2 ] = label( [ MEAN - STDDEV * 2, 0 ], roundTo( 1, MEAN - STDDEV * 2 ), "below", { color: PINK } );
graph.zLabel[ 2 ] = label( [ MEAN + STDDEV * 2, 0 ], roundTo( 1, MEAN + STDDEV * 2 ), "below", { color: PINK } );
</div>
<p>
Two standard deviations below the mean is
<code class="hint_pink" id="zm2Hint"><var>roundTo( 1, MEAN - STDDEV * 2 )</var></code> years
and two standard deviations above the mean is
<code class="hint_pink" id="zp2Hint"><var>roundTo( 1, MEAN + STDDEV * 2 )</var></code> years.
</p>
</div>
<div>
<div class="graphie" data-update="normaldist">
graph.zLine[ -2 ].attr({ stroke: "#bbb" });
graph.zLine[ 2 ].attr({ stroke: "#bbb" });
graph.zLabel[ -2 ].css({ color: "#bbb" });
graph.zLabel[ 2 ].css({ color: "#bbb" });
$( "#zm2Hint" ).removeClass("hint_pink");
$( "#zp2Hint" ).removeClass("hint_pink");
style({ stroke: PINK }, function() {
graph.zLine[ -3 ] = line( [ MEAN - STDDEV * 3, 0 ], [ MEAN - STDDEV * 3, graph.pdf( MEAN - STDDEV * 3 ) ] ).toBack();
graph.zLine[ 3 ] = line( [ MEAN + STDDEV * 3, 0 ], [ MEAN + STDDEV * 3, graph.pdf( MEAN + STDDEV * 3 ) ] ).toBack();
});
graph.zLabel[ -3 ] = label( [ MEAN - STDDEV * 3, 0 ], roundTo( 1, MEAN - STDDEV * 3 ), "below", { color: PINK } );
graph.zLabel[ 3 ] = label( [ MEAN + STDDEV * 3, 0 ], roundTo( 1, MEAN + STDDEV * 3 ), "below", { color: PINK } );
</div>
<p>
Three standard deviations below the mean is
<code class="hint_pink" id="zm3Hint"><var>roundTo( 1, MEAN - STDDEV * 3 )</var></code> years
and three standard deviations above the mean is
<code class="hint_pink" id="zp3Hint"><var>roundTo( 1, MEAN + STDDEV * 3 )</var></code> years.
</p>
</div>
<div id="hintGoal">
<div class="graphie" data-update="normaldist">
graph.zLine[ -3 ].attr({ stroke: "#bbb" });
graph.zLine[ 3 ].attr({ stroke: "#bbb" });
graph.zLabel[ -3 ].css({ color: "#bbb" });
graph.zLabel[ 3 ].css({ color: "#bbb" });
$( "#zm3Hint" ).removeClass("hint_pink");
$( "#zp3Hint" ).removeClass("hint_pink");
graph.zLine[ Z ].attr({ stroke: PINK });
graph.zLabel[ Z ].css({ color: PINK });
style({ stroke: PINK, fill: PINK, arrows: "-&gt;" }, function() {
line( [ MEAN + STDDEV * Z, -1 ], [ MEAN + STDDEV * 3.5, -1 ] );
ellipse( [ MEAN + STDDEV * Z, -1 ], [ 3 / ( 600 / ( STDDEV * 7 ) ), 3 / 40 ] );
});
</div>
<p>
We are interested in the probability of a <var>ANIMAL</var> living longer than
<code class="hint_pink"><var>roundTo( 1, MEAN + STDDEV * Z )</var></code> years.
</p>
</div>
<div id="graph1">
<div class="graphie" data-update="normaldist">
var shape = [];
shape.push([ MEAN - STDDEV * abs( Z ), 0 ]);
var step = STDDEV / 50;
for ( var x = MEAN - STDDEV * abs( Z ); x &lt;= MEAN + STDDEV * abs( Z ); x += step ) {
shape.push([ x, graph.pdf( x ) ]);
}
shape.push([ MEAN + STDDEV * abs( Z ), graph.pdf( MEAN + STDDEV * abs( Z ) ) ]);
shape.push([ MEAN + STDDEV * abs( Z ), 0 ]);
shape.push([ MEAN - STDDEV * abs( Z ), 0 ]);
style({ stroke: null, fill: BLUE, opacity: 0.3 }, function() {
path( shape );
});
label([ MEAN, graph.pdf( MEAN - STDDEV * Z ) -0.3 ], EMPIRICAL + "\\%", "above",
{ color: GREEN } );
style({ arrows: "-&gt;", stroke: GREEN }, function() {
line([ MEAN, graph.pdf( MEAN - STDDEV * Z ) -0.1 ],
[ MEAN + STDDEV * Z, graph.pdf( MEAN + STDDEV * Z ) -0.1 ]);
line([ MEAN, graph.pdf( MEAN + STDDEV * Z ) -0.1 ],
[ MEAN - STDDEV * Z, graph.pdf( MEAN - STDDEV * Z ) -0.1 ]);
});
</div>
<p data-if="isSingular(abs( Z ))">
The empirical rule (or the <code>68-95-99.7\%</code> rule)
tells us that <code>\green{<var>EMPIRICAL</var>\%}</code>
of the <var>ANIMALS</var> will have lifespans within
<var>abs( Z )</var> standard deviation of the average lifespan.
</p><p data-else="">
The empirical rule (or the <code>68-95-99.7\%</code> rule)
tells us that <code>\green{<var>EMPIRICAL</var>\%}</code>
of the <var>ANIMALS</var> will have lifespans within
<var>abs( Z )</var> standard deviations of the average lifespan.
</p>
</div>
<div id="graph2">
<div class="graphie" data-update="normaldist">
style({ arrows: "-&gt;", stroke: GREEN }, function() {
path([
[ MEAN - STDDEV * abs( Z ), graph.pdf( MEAN - STDDEV * Z ) + 0.2 ],
[ MEAN - STDDEV * abs( Z ), graph.pdf( MEAN - STDDEV * Z ) + 0.4 ],
[ MEAN - STDDEV * 3.5, graph.pdf( MEAN - STDDEV * Z ) + 0.4 ]
]);
});
label( [ MEAN - STDDEV * ( ( abs( Z ) + 3.5 ) / 2 ), graph.pdf( MEAN - STDDEV * Z ) + 0.3 ],
roundTo( 2, ( 100 - EMPIRICAL ) / 2 ) + "\\%", "above", { color: GREEN } );
style({ arrows: "-&gt;", stroke: GREEN }, function() {
path([
[ MEAN + STDDEV * abs( Z ), graph.pdf( MEAN + STDDEV * Z ) + 0.2 ],
[ MEAN + STDDEV * abs( Z ), graph.pdf( MEAN + STDDEV * Z ) + 0.4 ],
[ MEAN + STDDEV * 3.5, graph.pdf( MEAN + STDDEV * Z ) + 0.4 ]
]);
});
label( [ MEAN + STDDEV * ( ( abs( Z ) + 3.5 ) / 2 ), graph.pdf( MEAN + STDDEV * Z ) + 0.3 ],
roundTo( 2, ( 100 - EMPIRICAL ) / 2 ) + "\\%", "above", { color: GREEN } );
</div>
<p>
The remaining <code><var>roundTo( 2, 100 - EMPIRICAL )</var>\%</code>
of the <var>ANIMALS</var> will have lifespans that fall outside the shaded area.
Because the normal distribution is symmetrical, half
<code>(\green{<var>roundTo( 2, ( 100 - EMPIRICAL ) / 2 )</var>\%})</code>
will live less than <code><var>roundTo( 2, MEAN - STDDEV * abs( Z ) )</var></code> years
and the other half
<code>(\green{<var>roundTo( 2, ( 100 - EMPIRICAL ) / 2 )</var>\%})</code>
will live longer than <code><var>roundTo( 2, MEAN + STDDEV * abs( Z ) )</var></code> years.
</p>
</div>
<p id="finalHint1" data-if="Z < 0">
The probability of a particular <var>ANIMAL</var> living longer than
<code class="hint_pink"><var>roundTo( 1, MEAN + STDDEV * Z )</var></code> years is
<code>\green{<var>EMPIRICAL</var>\%} +
\green{<var>roundTo( 2, ( 100 - EMPIRICAL ) / 2 )</var>\%}</code>, or
<code><var>ANSWER</var>\%</code>.
</p>
<p id="finalHint2" data-else="">
The probability of a particular <var>ANIMAL</var> living longer than
<code class="hint_pink"><var>roundTo( 1, MEAN + STDDEV * Z )</var></code> years is
<code>\green{<var>ANSWER</var>\%}</code>.
</p>
</div>
</div>
<div id="shorter" data-type="longer">
<div class="vars" data-apply="appendVars">
<var id="ANSWER">
{
"3": 99.85,
"2": 97.5,
"1": 84,
"-1": 16,
"-2": 2.5,
"-3": 0.15
}[ Z ]
</var>
</div>
<p class="question">
Use the empirical rule <code>(68-95-99.7\%)</code> to estimate the probability of a <var>ANIMAL</var>
living less than <code><var>roundTo( 1, MEAN + STDDEV * Z )</var></code> years.
</p>
<div class="hints" data-apply="appendContents">
<div id="hintGoal">
<div class="graphie" data-update="normaldist">
graph.zLine[ -3 ].attr({ stroke: "#bbb" });
graph.zLine[ 3 ].attr({ stroke: "#bbb" });
graph.zLabel[ -3 ].css({ color: "#bbb" });
graph.zLabel[ 3 ].css({ color: "#bbb" });
$( "#zm3Hint" ).removeClass("hint_pink");
$( "#zp3Hint" ).removeClass("hint_pink");
graph.zLine[ Z ].attr({ stroke: PINK });
graph.zLabel[ Z ].css({ color: PINK });
style({ stroke: PINK, fill: PINK, arrows: "-&gt;" }, function() {
line( [ MEAN + STDDEV * Z, -1 ], [ MEAN + STDDEV * -3.5, -1 ] );
ellipse( [ MEAN + STDDEV * Z, -1 ], [ 3 / ( 600 / ( STDDEV * 7 ) ), 3 / 40 ] );
});
</div>
<p>
We are interested in the probability of a <var>ANIMAL</var> living less than
<code class="hint_pink"><var>roundTo( 1, MEAN + STDDEV * Z )</var></code> years.
</p>
</div>
<p id="finalHint1" data-if="Z > 0">
The probability of a particular <var>ANIMAL</var> living less than
<code class="hint_pink"><var>roundTo( 1, MEAN + STDDEV * Z )</var></code> years is
<code>\green{<var>EMPIRICAL</var>\%} +
\green{<var>roundTo( 2, ( 100 - EMPIRICAL ) / 2 )</var>\%}</code>, or
<code><var>ANSWER</var>\%</code>.
</p>
<p id="finalHint2" data-else="">
The probability of a particular <var>ANIMAL</var> living less than
<code class="hint_pink"><var>roundTo( 1, MEAN + STDDEV * Z )</var></code> years is
<code>\green{<var>ANSWER</var>\%}</code>.
</p>
</div>
</div>
<div id="between" data-type="shorter">
<div class="vars" data-apply="appendVars">
<var id="Z1">randRangeNonZero( -3, 2 )</var>
<var id="Z2">randRangeNonZero( Z1 + 1, 3 )</var>
<var id="Z_MAX">max( abs( Z1 ), abs( Z2 ) )</var>
<var id="Z_MIN">min( abs( Z1 ), abs( Z2 ) )</var>
<var id="EMPIRICAL1">
{
"-3": 99.7,
"-2": 95,
"-1": 68,
"1": 68,
"2": 95,
"3": 99.7
}[ Z_MAX ]
</var>
<var id="EMPIRICAL2">
{
"-3": 99.7,
"-2": 95,
"-1": 68,
"1": 68,
"2": 95,
"3": 99.7
}[ Z_MIN ]
</var>
<var id="TOTAL1">
{
"3": 99.85,
"2": 97.5,
"1": 84,
"-1": 16,
"-2": 2.5,
"-3": 0.15
}[ Z1 ]
</var>
<var id="TOTAL2">
{
"3": 99.85,
"2": 97.5,
"1": 84,
"-1": 16,
"-2": 2.5,
"-3": 0.15
}[ Z2 ]
</var>
<var id="AREA">TOTAL2 - TOTAL1</var>
</div>
<p class="question">
Use the empirical rule <code>(68-95-99.7\%)</code> to estimate the probability of a <var>ANIMAL</var> living between
<code><var>roundTo( 1, MEAN + STDDEV * Z1 )</var></code> and <code><var>roundTo( 1, MEAN + STDDEV * Z2 )</var></code> years.
</p>
<div class="solution" data-forms="integer, proper, improper, mixed, decimal, percent"><var>AREA / 100</var></div>
<div class="hints" data-apply="appendContents">
<div id="hintGoal">
<div class="graphie" data-update="normaldist">
graph.zLine[ -3 ].attr({ stroke: "#bbb" });
graph.zLine[ 3 ].attr({ stroke: "#bbb" });
graph.zLabel[ -3 ].css({ color: "#bbb" });
graph.zLabel[ 3 ].css({ color: "#bbb" });
$( "#zm3Hint" ).removeClass("hint_pink");
$( "#zp3Hint" ).removeClass("hint_pink");
graph.zLine[ Z1 ].attr({ stroke: PINK });
graph.zLabel[ Z1 ].css({ color: PINK });
graph.zLine[ Z2 ].attr({ stroke: PINK });
graph.zLabel[ Z2 ].css({ color: PINK });
style({ stroke: PINK, fill: PINK, arrows: "" }, function() {
line( [ MEAN + STDDEV * Z2, -1 ], [ MEAN + STDDEV * Z1, -1 ] );
ellipse( [ MEAN + STDDEV * Z2, -1 ], [ 3 / ( 600 / ( STDDEV * 7 ) ), 3 / 40 ] );
ellipse( [ MEAN + STDDEV * Z1, -1 ], [ 3 / ( 600 / ( STDDEV * 7 ) ), 3 / 40 ] );
});
</div>
<p>
We are interested in the probability of a <var>ANIMAL</var> living between
<code class="hint_pink"><var>roundTo( 1, MEAN + STDDEV * Z1 )</var></code> and
<code class="hint_pink"><var>roundTo( 1, MEAN + STDDEV * Z2 )</var></code> years.
</p>
</div>
<div id="graph1">
<div class="graphie" data-update="normaldist">
var shape = [];
shape.push([ MEAN - STDDEV * abs( Z_MAX ), 0 ]);
var step = STDDEV / 50;
for ( var x = MEAN - STDDEV * abs( Z_MAX ); x &lt;= MEAN + STDDEV * abs( Z_MAX ); x += step ) {
shape.push([ x, graph.pdf( x ) ]);
}
shape.push([ MEAN + STDDEV * abs( Z_MAX ), graph.pdf( MEAN + STDDEV * abs( Z_MAX ) ) ]);
shape.push([ MEAN + STDDEV * abs( Z_MAX ), 0 ]);
shape.push([ MEAN - STDDEV * abs( Z_MAX ), 0 ]);
style({ stroke: null, fill: BLUE, opacity: 0.3 }, function() {
path( shape );
});
label([ MEAN, graph.pdf( MEAN - STDDEV * Z_MAX ) -0.3 ], EMPIRICAL1 + "\\%", "above",
{ color: GREEN } );
style({ arrows: "-&gt;", stroke: GREEN }, function() {
line([ MEAN, graph.pdf( MEAN - STDDEV * Z_MAX ) -0.1 ],
[ MEAN + STDDEV * Z_MAX, graph.pdf( MEAN + STDDEV * Z_MAX ) -0.1 ]);
line([ MEAN, graph.pdf( MEAN + STDDEV * Z_MAX ) -0.1 ],
[ MEAN - STDDEV * Z_MAX, graph.pdf( MEAN - STDDEV * Z_MAX ) -0.1 ]);
});
</div>
<p data-if="isSingular(abs( Z_MAX ))">
The empirical rule (or the <code>68-95-99.7\%</code> rule)
tells us that <code>\green{<var>EMPIRICAL1</var>\%}</code>
of the <var>ANIMALS</var> will have lifespans within
<var>abs( Z_MAX )</var> standard deviation of the average lifespan.
</p><p data-else="">
The empirical rule (or the <code>68-95-99.7\%</code> rule)
tells us that <code>\green{<var>EMPIRICAL1</var>\%}</code>
of the <var>ANIMALS</var> will have lifespans within
<var>abs( Z_MAX )</var> standard deviations of the average lifespan.
</p>
</div>
<div id="graph2" data-if="Z_MAX !== Z_MIN">
<div class="graphie" data-update="normaldist">
label([ MEAN, graph.pdf( MEAN - STDDEV * Z_MIN ) -0.3 ], EMPIRICAL2 + "\\%", "above",
{ color: GREEN } );
style({ arrows: "-&gt;", stroke: GREEN }, function() {
line([ MEAN, graph.pdf( MEAN - STDDEV * Z_MIN ) -0.1 ],
[ MEAN + STDDEV * Z_MIN, graph.pdf( MEAN + STDDEV * Z_MIN ) -0.1 ]);
line([ MEAN, graph.pdf( MEAN + STDDEV * Z_MIN ) -0.1 ],
[ MEAN - STDDEV * Z_MIN, graph.pdf( MEAN - STDDEV * Z_MIN ) -0.1 ]);
});
label([ MEAN + STDDEV * ( ( Z_MIN + Z_MAX ) / 2 ), graph.pdf( MEAN - STDDEV * Z_MIN ) -0.3 ],
roundTo( 2, ( ( EMPIRICAL1 - EMPIRICAL2 ) / 2 ) ) + "\\%", "above", { color: RED } );
label([ MEAN - STDDEV * ( ( Z_MIN + Z_MAX ) / 2 ), graph.pdf( MEAN - STDDEV * Z_MIN ) -0.3 ],
roundTo( 2, ( ( EMPIRICAL1 - EMPIRICAL2 ) / 2 ) ) + "\\%", "above", { color: RED } );
style({ arrows: "-&gt;", stroke: RED }, function() {
line([ MEAN + STDDEV * ( ( Z_MIN + Z_MAX ) / 2 ), graph.pdf( MEAN - STDDEV * Z_MIN ) -0.1 ],
[ MEAN + STDDEV * Z_MIN, graph.pdf( MEAN + STDDEV * Z_MIN ) -0.1 ]);
line([ MEAN + STDDEV * ( ( Z_MIN + Z_MAX ) / 2 ), graph.pdf( MEAN - STDDEV * Z_MIN ) -0.1 ],
[ MEAN + STDDEV * Z_MAX, graph.pdf( MEAN + STDDEV * Z_MIN ) -0.1 ]);
line([ MEAN - STDDEV * ( ( Z_MIN + Z_MAX ) / 2 ), graph.pdf( MEAN + STDDEV * Z_MIN ) -0.1 ],
[ MEAN - STDDEV * Z_MIN, graph.pdf( MEAN - STDDEV * Z_MIN ) -0.1 ]);
line([ MEAN - STDDEV * ( ( Z_MIN + Z_MAX ) / 2 ), graph.pdf( MEAN + STDDEV * Z_MIN ) -0.1 ],
[ MEAN - STDDEV * Z_MAX, graph.pdf( MEAN - STDDEV * Z_MIN ) -0.1 ]);
});
</div>
<p>
<span data-if="isSingular(Z_MIN)">
It also tells us that <code>\green{<var>EMPIRICAL2</var>\%}</code> of the <var>ANIMALS</var>
will have lifespans within <code><var>Z_MIN</var></code> standard deviation of the mean.
</span><span data-else="">
It also tells us that <code>\green{<var>EMPIRICAL2</var>\%}</code> of the <var>ANIMALS</var>
will have lifespans within <code><var>Z_MIN</var></code> standard deviations of the mean.
</span>
<span data-if="isSingular(Z_MAX)">
That leaves <code><var>EMPIRICAL1</var>\% - <var>EMPIRICAL2</var>\% =
<var>roundTo(2, EMPIRICAL1 - EMPIRICAL2)</var>\%</code> of <var>ANIMALS</var> between
<code><var>Z_MIN</var></code> and <code><var>Z_MAX</var></code> standard deviation of the mean, or
<code>\red{<var>roundTo(2, (EMPIRICAL1 - EMPIRICAL2) / 2)</var>\%}</code> on either side of the distribution.
</span><span data-else="">
That leaves <code><var>EMPIRICAL1</var>\% - <var>EMPIRICAL2</var>\% =
<var>roundTo(2, EMPIRICAL1 - EMPIRICAL2)</var>\%</code> of <var>ANIMALS</var> between
<code><var>Z_MIN</var></code> and <code><var>Z_MAX</var></code> standard deviations of the mean, or
<code>\red{<var>roundTo(2, (EMPIRICAL1 - EMPIRICAL2) / 2)</var>\%}</code> on either side of the distribution.
</span>
</p>
</div>
<p id="finalHint1" data-if="abs( Z1 ) > abs( Z2 )">
The probability of a particular <var>ANIMAL</var> living between
<code class="hint_pink"><var>roundTo( 1, MEAN + STDDEV * Z1 )</var></code> and
<code class="hint_pink"><var>roundTo( 1, MEAN + STDDEV * Z2 )</var></code> years is
<span data-if="Z1 * Z2 > 0"><code>\red{<var>roundTo( 2, ( EMPIRICAL1 - EMPIRICAL2 ) / 2 )</var>\%}</code>.</span>
<span data-else=""><code>\red{<var>roundTo( 2, ( EMPIRICAL1 - EMPIRICAL2 ) / 2 )</var>\%} +
\green{<var>EMPIRICAL2</var>\%}</code>, or <code><var>roundTo( 2, AREA )</var>\%</code>.</span>
</p>
<p id="finalHint2" data-else-if="abs( Z1 ) &lt; abs( Z2 )">
The probability of a particular <var>ANIMAL</var> living between
<code class="hint_pink"><var>roundTo( 1, MEAN + STDDEV * Z1 )</var></code> and
<code class="hint_pink"><var>roundTo( 1, MEAN + STDDEV * Z2 )</var></code> years is
<span data-if="Z1 * Z2 > 0"><code>\red{<var>roundTo( 2, ( EMPIRICAL1 - EMPIRICAL2 ) / 2 )</var>\%}</code>.</span>
<span data-else=""><code>\green{<var>EMPIRICAL2</var>\%} +
\red{<var>roundTo( 2, ( EMPIRICAL1 - EMPIRICAL2 ) / 2 )</var>\%}</code>, or
<code><var>roundTo( 2, AREA )</var>\%</code>.</span>
</p>
<p data-else="">
The probability of a particular <var>ANIMAL</var> living between
<code class="hint_pink"><var>roundTo( 1, MEAN + STDDEV * Z1 )</var></code> and
<code class="hint_pink"><var>roundTo( 1, MEAN + STDDEV * Z2 )</var></code> years is
<code>\green{<var>roundTo( 2, AREA )</var>\%}</code>.
</p>
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