This repository has been archived by the owner on May 11, 2021. It is now read-only.
-
Notifications
You must be signed in to change notification settings - Fork 863
/
z_scores_3.html
313 lines (290 loc) · 15.4 KB
/
z_scores_3.html
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
191
192
193
194
195
196
197
198
199
200
201
202
203
204
205
206
207
208
209
210
211
212
213
214
215
216
217
218
219
220
221
222
223
224
225
226
227
228
229
230
231
232
233
234
235
236
237
238
239
240
241
242
243
244
245
246
247
248
249
250
251
252
253
254
255
256
257
258
259
260
261
262
263
264
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
<!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 3: Finding areas under the curve with 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="INDEX">randRange(0, 3)</var>
<var id="STDDEV">[4, 5, 8, 10][INDEX]</var>
<var id="ZSCORES">
(function() {
var zScores = [];
if (STDDEV === 8) {
var zDecimal1 = randRange(0, 2) * 2;
var zDecimal2 = randRange(zDecimal1 / 2 + 1, 3) * 2;
} else {
var zDecimal1 = randRange(0, STDDEV - 2);
var zDecimal2 = randRange(zDecimal1 + 1, STDDEV - 1);
}
zScores.push(roundTo(2, randRange(0, 2) + zDecimal1 / STDDEV) * randRangeNonZero(-1, 1));
zScores.push(roundTo(2, floorTo(0,
abs(zScores[0])) + zDecimal2 / STDDEV) * randRangeNonZero(-1, 1));
return sortNumbers(zScores);
})()
</var>
<var id="ROW_INDEX_1">floorTo(0, abs(ZSCORES[0]) * 10) % 10</var>
<var id="COL_INDEX_1">(roundTo(0, abs(ZSCORES[0]) * 100)) % 10</var>
<var id="ROW_INDEX_2">floorTo(0, abs(ZSCORES[1]) * 10) % 10</var>
<var id="COL_INDEX_2">(roundTo(0, abs(ZSCORES[1]) * 100)) % 10</var>
<var id="MEAN">randRange(95, 99) - STDDEV * 3</var>
<var id="GRADES">[MEAN + ZSCORES[0] * STDDEV, MEAN + ZSCORES[1] * STDDEV]</var>
<var id="ROWS">
(function() {
var rowNames = [];
for (var i = floorTo(0, abs(ZSCORES[0])); i < (floorTo(0, abs(ZSCORES[0])) + 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 < ROWS.length; i++) {
var zRow = [];
for (var j = 0; j < COLUMNS.length; j++) {
zRow.push(zScores(roundTo(2,
(floorTo(0, abs(ZSCORES[0])) + i / 10 + j / 100) * 100)).toFixed(4));
}
zGrid.push(zRow);
}
return zGrid;
})()
</var>
<var id="ANSWER">
(function() {
var a, b;
if (ZSCORES[0] >= 0) {
a = ZGRID[ROW_INDEX_1][COL_INDEX_1];
} else {
a = 1 - ZGRID[ROW_INDEX_1][COL_INDEX_1];
}
if (ZSCORES[1] >= 0) {
b = ZGRID[ROW_INDEX_2][COL_INDEX_2];
} else {
b = 1 - ZGRID[ROW_INDEX_2][COL_INDEX_2];
}
return roundTo(4, b - a);
})()
</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>.
</div>
<p class="question">
What percentage of test-takers had a grade between <code><var>GRADES[0]</var></code>
and <code><var>GRADES[1]</var></code> on the exam?
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-scores</span> of the two grades which we were given.
</p>
<div>
<p>
Recall that we can calculate the z-scores by subtracting the
<span class="hint_pink">mean <code>(\mu)</code></span> from
each exam grade and then dividing by the <span class="hint_green">standard
deviation <code>(\sigma)</code></span>.
</p>
<p data-each="GRADES as index, grade">
<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>ZSCORES[index]</var>}
</code>
</p>
</div>
<div>
<p>
We can find the percentage of test-takers that earned between
<code><var>GRADES[0]</var></code> and <code><var>GRADES[1]</var></code>
by finding the area between <code><var>ZSCORES[0]</var></code> and
<code><var>ZSCORES[1]</var></code> under the standard normal curve. After
looking up these two z-scores in our z-table, subtracting the two table
values will provide us with the total area.
</p>
<div class="graphie" id="grid">
init({
range: [[-4, 4], [-1.5, 4.7]],
scale: [430 / 8, 30]
});
style({ stroke: "#bbb" }, function() {
line([-3.5, 0], [3.5, 0]);
});
graph.pdf = function(x) {
return gaussianPDF(0, 1, x) * 4 / gaussianPDF(0, 1, 0) + 0.2;
};
style({ stroke: BLUE }, function() {
plot(graph.pdf, [-3.5, 3.5]);
});
</div>
</div>
<div>
<div data-if="ZSCORES[1] < 0"><p>
Since the normal curve is symmetrical, the area <b>less than</b>
<code><var>ZSCORES[1]</var></code> is equal to the area
<b>greater than</b> <code><var>abs( ZSCORES[1] )</var></code>, which can
be found by looking up <code><var>abs( ZSCORES[1] )</var></code>
on the z-table and subtracting the table value from <code>1</code>, the
total area under the curve.
</p></div>
<p data-if="ZSCORES[1] < 0">
<code>1 - <var>ZGRID[ ROW_INDEX_2 ][ COL_INDEX_2 ]</var> =
<var>roundTo( 4, 1 - ZGRID[ ROW_INDEX_2 ][ COL_INDEX_2 ] )</var></code>.
This value, <code class="hint_blue">
<var>roundTo( 4, 1 - ZGRID[ ROW_INDEX_2 ][ COL_INDEX_2 ] )</var></code>,
represents the portion of the population that scored lower than
<code><var>GRADES[1]</var></code> on the exam.
</p>
<p data-else>
Look up <code><var>ZSCORES[1]</var></code> on the z-table. This value,
<code class="hint_blue"> <var>ZGRID[ ROW_INDEX_2 ][ COL_INDEX_2 ]</var></code>,
represents the portion of the population that scored lower than
<code><var>GRADES[1]</var></code> on the exam.
</p>
<div class="graphie">
var nth = ":nth-child(" + ( COL_INDEX_2 + 2 ) + ")";
$( ".fake_row" ).eq( ROW_INDEX_2 ).find( "span" + nth )
.css( "background", KhanUtil.BLUE );
</div>
<div class="graphie" data-update="grid">
style({ stroke: "#bbb" }, function() {
graph.zLine1 = line([ZSCORES[1], 0], [ZSCORES[1], graph.pdf(ZSCORES[1])]).toBack();
});
graph.zLabel = label([ZSCORES[1], 0], roundTo(2, ZSCORES[1]), "below", { color: "#bbb" });
var step = 1 / 50;
blueShape = [];
blueShape.push([-3.5, 0]);
for (var i = -3.5; i <= ZSCORES[1]; i += step) {
blueShape.push([i, graph.pdf(i)]);
}
blueShape.push([ZSCORES[1], graph.pdf(ZSCORES[1])]);
blueShape.push([ZSCORES[1], 0]);
blueShape.push([-3.5, 0]);
style({ stroke: null, fill: BLUE, opacity: 0.3 }, function() {
graph.blueShape = path(blueShape);
});
</div>
</div>
<div>
<div data-if="ZSCORES[0] < 0"><p>
Since the normal curve is symmetrical, the area <b>less than</b>
<code><var>ZSCORES[0]</var></code> is equal to the area
<b>greater than</b> <code><var>abs(ZSCORES[0])</var></code>, which can
be found by looking up <code><var>abs(ZSCORES[0])</var></code>
on the z-table and subtracting the table value from <code>1</code>, the
total area under the curve.
</p></div>
<p data-if="ZSCORES[0] < 0">
<code>1 - <var>ZGRID[ROW_INDEX_1][COL_INDEX_1]</var> =
<var>roundTo(4, 1 - ZGRID[ROW_INDEX_1][COL_INDEX_1])</var></code>. This value,
<code class="hint_orange"><var>roundTo(4, 1 - ZGRID[ROW_INDEX_1][COL_INDEX_1])</var></code>,
represents the portion of the population that scored lower than
<code><var>GRADES[0]</var></code> on the exam.
</p>
<p data-else>
Look up <code><var>ZSCORES[0]</var></code> on the z-table. This value,
<code class="hint_orange"><var>ZGRID[ ROW_INDEX_1 ][ COL_INDEX_1 ]</var></code>,
represents the portion of the population that scored lower than
<code><var>GRADES[0]</var></code> on the exam.
</p>
<div class="graphie">
var nth = ":nth-child(" + (COL_INDEX_1 + 2) + ")";
$(".fake_row").eq(ROW_INDEX_1).find("span" + nth).css("background", ORANGE);
</div>
<div class="graphie" data-update="grid">
style({ stroke: "#bbb" }, function() {
graph.zLine2 = line([ZSCORES[0], 0], [ZSCORES[0], graph.pdf(ZSCORES[0])]).toBack();
});
graph.zLabel.remove();
graph.zLabel = label([ZSCORES[0], 0], roundTo(2, ZSCORES[0]), "below", { color: "#bbb" });
var step = 1 / 50;
graph.blueShape.remove();
orangeShape = [];
orangeShape.push([-3.5, 0]);
for (var i = -3.5; i <= ZSCORES[0]; i += step) {
orangeShape.push([i, graph.pdf(i)]);
}
orangeShape.push([ZSCORES[0], graph.pdf(ZSCORES[0])]);
orangeShape.push([ZSCORES[0], 0]);
orangeShape.push([-3.5, 0]);
style({ stroke: null, fill: ORANGE, opacity: 0.7 }, function() {
graph.orangeShape = path(orangeShape);
});
blueShape = [];
blueShape.push([ZSCORES[0], 0]);
for (var i = ZSCORES[0]; i <= ZSCORES[1]; i += step) {
blueShape.push([i, graph.pdf(i)]);
}
blueShape.push([ZSCORES[1], graph.pdf(ZSCORES[1])]);
blueShape.push([ZSCORES[1], 0]);
blueShape.push([ZSCORES[0], 0]);
style({ stroke: null, fill: BLUE, opacity: 0.3 }, function() {
graph.blueShape = path(blueShape);
});
</div>
</div>
<div>
<p>
Finally, subtract the two cumulative areas to arrive at our final answer:
<code>\blue{<span data-if="ZSCORES[1] < 0">
<var>roundTo(4, 1 - ZGRID[ROW_INDEX_2][COL_INDEX_2])</var>
</span><span data-else>
<var>roundTo(4, ZGRID[ROW_INDEX_2][COL_INDEX_2])</var>
</span>} - \orange{<span data-if="ZSCORES[0] < 0">
<var>roundTo(4, 1 - ZGRID[ROW_INDEX_1][COL_INDEX_1])</var>
</span><span data-else>
<var>roundTo(4, ZGRID[ROW_INDEX_1][COL_INDEX_1])</var>
</span>} = <var>ANSWER</var></code><br />
Thus, <code class="hint_pink"><var>roundTo( 2, ANSWER * 100 )</var>\%</code>
of the test-takers scored between <code><var>GRADES[0]</var></code> and
<code><var>GRADES[1]</var></code> on the <var>course( 1 )</var> exam.
</p>
<div class="graphie" data-update="grid">
graph.orangeShape.remove();
graph.zLabel.remove();
</div>
</div>
</div>
</div>
</div>
</div>
</body>
</html>