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New exercise: Z-scores 2: Reading a z-table
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| <!DOCTYPE html> | ||
| <html data-require="math graphie word-problems"> | ||
| <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: 50px; | ||
| } | ||
| .reading span:first-child { | ||
| font-weight: bold; | ||
| width: 60px; | ||
| 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">GRADE - ZSCORE * STDDEV</var> | ||
| <var id="ROWS"> | ||
| (function() { | ||
| var rowNames = []; | ||
| for( var i = floorTo( 0, ZSCORE ); i < ( 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="ZSCORES"> | ||
| { | ||
| "0": 0.5, "1": 0.504, "2": 0.508, "3": 0.512, "4": 0.516, "5": 0.5199, "6": 0.5239, "7": 0.5279, "8": 0.5319, "9": 0.5359, | ||
| "10": 0.5398, "11": 0.5438, "12": 0.5478, "13": 0.5517, "14": 0.5557, "15": 0.5596, "16": 0.5636, "17": 0.5675, "18": 0.5714, | ||
| "19": 0.5753, "20": 0.5793, "21": 0.5832, "22": 0.5871, "23": 0.591, "24": 0.5948, "25": 0.5987, "26": 0.6026, "27": 0.6064, | ||
| "28": 0.6103, "29": 0.6141, "30": 0.6179, "31": 0.6217, "32": 0.6255, "33": 0.6293, "34": 0.6331, "35": 0.6368, "36": 0.6406, | ||
| "37": 0.6443, "38": 0.648, "39": 0.6517, "40": 0.6554, "41": 0.6591, "42": 0.6628, "43": 0.6664, "44": 0.67, "45": 0.6736, | ||
| "46": 0.6772, "47": 0.6808, "48": 0.6844, "49": 0.6879, "50": 0.6915, "51": 0.695, "52": 0.6985, "53": 0.7019, "54": 0.7054, | ||
| "55": 0.7088, "56": 0.7123, "57": 0.7157, "58": 0.719, "59": 0.7224, "60": 0.7257, "61": 0.7291, "62": 0.7324, "63": 0.7357, | ||
| "64": 0.7389, "65": 0.7422, "66": 0.7454, "67": 0.7486, "68": 0.7517, "69": 0.7549, "70": 0.758, "71": 0.7611, "72": 0.7642, | ||
| "73": 0.7673, "74": 0.7704, "75": 0.7734, "76": 0.7764, "77": 0.7794, "78": 0.7823, "79": 0.7852, "80": 0.7881, "81": 0.791, | ||
| "82": 0.7939, "83": 0.7967, "84": 0.7995, "85": 0.8023, "86": 0.8051, "87": 0.8078, "88": 0.8106, "89": 0.8133, "90": 0.8159, | ||
| "91": 0.8186, "92": 0.8212, "93": 0.8238, "94": 0.8264, "95": 0.8289, "96": 0.8315, "97": 0.834, "98": 0.8365, "99": 0.8389, | ||
| "100": 0.8413, "101": 0.8438, "102": 0.8461, "103": 0.8485, "104": 0.8508, "105": 0.8531, "106": 0.8554, "107": 0.8577, | ||
| "108": 0.8599, "109": 0.8621, "110": 0.8643, "111": 0.8665, "112": 0.8686, "113": 0.8708, "114": 0.8729, "115": 0.8749, | ||
| "116": 0.877, "117": 0.879, "118": 0.881, "119": 0.883, "120": 0.8849, "121": 0.8869, "122": 0.8888, "123": 0.8907, "124": 0.8925, | ||
| "125": 0.8944, "126": 0.8962, "127": 0.898, "128": 0.8997, "129": 0.9015, "130": 0.9032, "131": 0.9049, "132": 0.9066, | ||
| "133": 0.9082, "134": 0.9099, "135": 0.9115, "136": 0.9131, "137": 0.9147, "138": 0.9162, "139": 0.9177, "140": 0.9192, | ||
| "141": 0.9207, "142": 0.9222, "143": 0.9236, "144": 0.9251, "145": 0.9265, "146": 0.9279, "147": 0.9292, "148": 0.9306, | ||
| "149": 0.9319, "150": 0.9332, "151": 0.9345, "152": 0.9357, "153": 0.937, "154": 0.9382, "155": 0.9394, "156": 0.9406, | ||
| "157": 0.9418, "158": 0.9429, "159": 0.9441, "160": 0.9452, "161": 0.9463, "162": 0.9474, "163": 0.9484, "164": 0.9495, | ||
| "165": 0.9505, "166": 0.9515, "167": 0.9525, "168": 0.9535, "169": 0.9545, "170": 0.9554, "171": 0.9564, "172": 0.9573, | ||
| "173": 0.9582, "174": 0.9591, "175": 0.9599, "176": 0.9608, "177": 0.9616, "178": 0.9625, "179": 0.9633, "180": 0.9641, | ||
| "181": 0.9649, "182": 0.9656, "183": 0.9664, "184": 0.9671, "185": 0.9678, "186": 0.9686, "187": 0.9693, "188": 0.9699, | ||
| "189": 0.9706, "190": 0.9713, "191": 0.9719, "192": 0.9726, "193": 0.9732, "194": 0.9738, "195": 0.9744, "196": 0.975, | ||
| "197": 0.9756, "198": 0.9761, "199": 0.9767, "200": 0.9772, "201": 0.9778, "202": 0.9783, "203": 0.9788, "204": 0.9793, | ||
| "205": 0.9798, "206": 0.9803, "207": 0.9808, "208": 0.9812, "209": 0.9817, "210": 0.9821, "211": 0.9826, "212": 0.983, | ||
| "213": 0.9834, "214": 0.9838, "215": 0.9842, "216": 0.9846, "217": 0.985, "218": 0.9854, "219": 0.9857, "220": 0.9861, | ||
| "221": 0.9864, "222": 0.9868, "223": 0.9871, "224": 0.9875, "225": 0.9878, "226": 0.9881, "227": 0.9884, "228": 0.9887, | ||
| "229": 0.989, "230": 0.9893, "231": 0.9896, "232": 0.9898, "233": 0.9901, "234": 0.9904, "235": 0.9906, "236": 0.9909, | ||
| "237": 0.9911, "238": 0.9913, "239": 0.9916, "240": 0.9918, "241": 0.992, "242": 0.9922, "243": 0.9925, "244": 0.9927, | ||
| "245": 0.9929, "246": 0.9931, "247": 0.9932, "248": 0.9934, "249": 0.9936, "250": 0.9938, "251": 0.994, "252": 0.9941, | ||
| "253": 0.9943, "254": 0.9945, "255": 0.9946, "256": 0.9948, "257": 0.9949, "258": 0.9951, "259": 0.9952, "260": 0.9953, | ||
| "261": 0.9955, "262": 0.9956, "263": 0.9957, "264": 0.9959, "265": 0.996, "266": 0.9961, "267": 0.9962, "268": 0.9963, | ||
| "269": 0.9964, "270": 0.9965, "271": 0.9966, "272": 0.9967, "273": 0.9968, "274": 0.9969, "275": 0.997, "276": 0.9971, | ||
| "277": 0.9972, "278": 0.9973, "279": 0.9974, "280": 0.9974, "281": 0.9975, "282": 0.9976, "283": 0.9977, "284": 0.9977, | ||
| "285": 0.9978, "286": 0.9979, "287": 0.9979, "288": 0.998, "289": 0.9981, "290": 0.9981, "291": 0.9982, "292": 0.9982, | ||
| "293": 0.9983, "294": 0.9984, "295": 0.9984, "296": 0.9985, "297": 0.9985, "298": 0.9986, "299": 0.9986, "300": 0.9987, | ||
| "301": 0.9987, "302": 0.9987, "303": 0.9988, "304": 0.9988, "305": 0.9989, "306": 0.9989, "307": 0.9989, "308": 0.999, | ||
| "309": 0.999 | ||
| } | ||
| </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( 0, ( 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-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 - \color{<var>PINK</var>}{\mu}}{\color{<var>GREEN</var>}{\sigma}} | ||
| \quad = \quad \dfrac{<var>GRADE</var> - \color{<var>PINK</var>}{<var>MEAN</var>}}{\color{<var>GREEN</var>}{<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. | ||
| <div class="graphie"> | ||
| var nth = ":nth-child(" + ( COL_INDEX + 2 ) + ")"; | ||
| jQuery( ".fake_row" ).eq( ROW_INDEX ).find( "span" ) | ||
| .css( "background", KhanUtil.ORANGE ); | ||
| jQuery( ".fake_row span" + nth ) | ||
| .css( "background", KhanUtil.ORANGE ); | ||
| jQuery( ".fake_header span" + nth ) | ||
| .css( "background", KhanUtil.ORANGE ); | ||
| jQuery( ".fake_row" ).eq( ROW_INDEX ).find( "span" + nth ) | ||
| .css( "background", KhanUtil.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> |