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<!DOCTYPE html>
<html data-require="math graphie word-problems stat">
<head>
<meta http-equiv="Content-Type" content="text/html; charset=UTF-8">
<title>Z-scores 2: Reading a z-table</title>
<script src="../khan-exercise.js"></script>
<style>
.reading span {
width: 32px;
font-size: 11px;
padding: 5px;
}
.reading span:first-child {
font-weight: bold;
width: 22px;
border-right: 2px solid #CCCCCC;
}
</style>
</head>
<body>
<div class="exercise">
<div class="problems">
<div>
<div class="vars">
<var id="COL_INDEX">randRange(0, 9)</var>
<var id="ROW_INDEX">randRange(0, 9)</var>
<var id="ZSCORE">roundTo(2, randRange(0, 2) + ROW_INDEX / 10 + COL_INDEX / 100)</var>
<var id="GRADE">randRange(70, 90)</var>
<var id="STDDEV">randRange(2, 6)</var>
<var id="MEAN">roundTo(2, GRADE - ZSCORE * STDDEV)</var>
<var id="ROWS">
(function() {
var rowNames = [];
for(var i = floorTo(0, ZSCORE); i &lt; (floorTo(0, ZSCORE) + 1); i += 0.1) {
rowNames.push(i.toFixed(1));
}
return rowNames;
})()
</var>
<var id="COLUMNS">[".00", ".01", ".02", ".03", ".04", ".05", ".06", ".07", ".08", ".09"]</var>
<var id="ZGRID">
(function() {
var zGrid = [];
for (var i = 0; i &lt; ROWS.length; i++) {
var zRow = [];
for (var j = 0; j &lt; COLUMNS.length; j++) {
zRow.push(zScores(roundTo(2,
(floorTo(0, ZSCORE) + i / 10 + j / 100) * 100)).toFixed(4));
}
zGrid.push(zRow);
}
return zGrid;
})()
</var>
<var id="ANSWER">ZGRID[ROW_INDEX][COL_INDEX]</var>
<var id="HEADER">"z"</var>
</div>
<div class="problem">
The scores on a statewide <var>course(1)</var> exam were normally distributed with
<code>\mu = <var>MEAN</var></code> and <code>\sigma = <var>STDDEV</var></code>.
<br /><var>person(1)</var> earned a<span data-if="floorTo(0, GRADE / 10) === 8">n</span>
<code><var>GRADE</var></code> on the exam.
</div>
<p class="question"><var>person(1)</var>'s exam grade was higher than what percentage
of test-takers? Use the cumulative z-table provided below. Round to two decimal places.</p>
<div class="fake_header reading">
<span><var>HEADER</var></span><span data-each="COLUMNS as i, column"><var>column</var></span>
</div>
<div class="fake_row reading" data-each="ROWS as i, row">
<span><var>row</var></span><span data-each="ZGRID[i] as j, zgrid"><var>zgrid</var></span>
</div>
<div class="solution" data-inexact data-max-error="0.01" data-forms="percent">
<span><var>roundTo(2, ANSWER * 100)</var></span>
</div>
<div class="hints">
<p>
A cumulative z-table shows the probability that a standard normal variable will be
<b>less than</b> a certain value <b>(z)</b>.
</p>
<p>
In order to use the z-table, we first need to determine the
<span class="hint_blue">z-score</span> of <var>person( 1 )</var>'s exam grade.
</p>
<div>
<p>
Recall that we can calculate <var>his( 1 )</var> z-score by subtracting the
<span class="hint_pink">mean <code>(\mu)</code></span> from <var>his(1)</var>
grade and then dividing by the <span class="hint_green">standard deviation
<code>(\sigma)</code></span>.
</p>
<p>
<code>
\large{\quad z \quad = \quad
\dfrac{x - \pink{\mu}}{\green{\sigma}}
\quad = \quad \dfrac{<var>GRADE</var> - \pink{<var>MEAN</var>}}{\green{<var>STDDEV</var>}}
\quad = \quad <var>ZSCORE</var>}
</code>
</p>
</div>
<div>
<p>
Look up <code><var>ZSCORE</var></code> on the z-table. This value,
<code class="hint_blue"><var>ANSWER</var></code>, represents
the portion of the population that scored lower than
<code><var>GRADE</var></code> on the exam.
</p>
<div class="graphie">
var nth = ":nth-child(" + (COL_INDEX + 2) + ")";
$(".fake_row").eq(ROW_INDEX).find("span").css("background", ORANGE);
$(".fake_row span" + nth).css("background", ORANGE);
$(".fake_header span" + nth).css("background", ORANGE);
$(".fake_row").eq(ROW_INDEX).find("span" + nth).css("background", BLUE);
</div>
</div>
<p>
<var>person( 1 )</var> scored higher than
<code class="hint_blue"><var>roundTo(2, ANSWER * 100)</var>\%</code> of the
test-takers on the <var>course( 1 )</var> exam.
</p>
</div>
</div>
</div>
</div>
</body>
</html>
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