// Numbas version: exam_results_page_options
{"name": "Calculate expectation and a probability from a frequency table, , , ", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [{"variables": ["idef", "thing", "episodes", "period", "activity"], "name": "Strings"}, {"variables": ["p0", "p1", "p2", "p3", "p4", "p5", "p6", "p7", "p8", "probabilities", "values"], "name": "Probabilities"}, {"variables": ["r", "s", "t", "t1", "t2", "u1", "u2", "u3", "d"], "name": "Stuff to generate probabilities"}], "variables": {"p4": {"templateType": "anything", "group": "Probabilities", "definition": "t-p8-p7-p6-p5", "description": "", "name": "p4"}, "expected_number": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sum(map(x*y,[x,y],zip(probabilities,values)))", "description": "", "name": "expected_number"}, "u2": {"templateType": "anything", "group": "Stuff to generate probabilities", "definition": "u1", "description": "", "name": "u2"}, "p1": {"templateType": "anything", "group": "Probabilities", "definition": "p0+t1", "description": "", "name": "p1"}, "p3": {"templateType": "anything", "group": "Probabilities", "definition": "r-p0-p1-p2", "description": "", "name": "p3"}, "t": {"templateType": "anything", "group": "Stuff to generate probabilities", "definition": "100-r", "description": "", "name": "t"}, "probexceed": {"templateType": "anything", "group": "Ungrouped variables", "definition": "sum(map(if(j>expected_number,probabilities[j],0),j,0..8))", "description": "", "name": "probexceed"}, "values": {"templateType": "anything", "group": "Probabilities", "definition": "list(0..8)", "description": "", "name": "values"}, "thing": {"templateType": "string", "group": "Strings", "definition": "\"airline\"", "description": "", "name": "thing"}, "u3": {"templateType": "anything", "group": "Stuff to generate probabilities", "definition": "u1", "description": "", "name": "u3"}, "u1": {"templateType": "anything", "group": "Stuff to generate probabilities", "definition": "round(d*random(70..100)/100)", "description": "", "name": "u1"}, "expect_int": {"templateType": "anything", "group": "Ungrouped variables", "definition": "floor(expected_number)", "description": "", "name": "expect_int"}, "activity": {"templateType": "string", "group": "Strings", "definition": "\"luggage handling\"", "description": "", "name": "activity"}, "probabilities": {"templateType": "anything", "group": "Probabilities", "definition": "map(x/100,x,[p0,p1,p2,p3,p4,p5,p6,p7,p8])", "description": "<p>Probability of there being $i$ episodes</p>", "name": "probabilities"}, "d": {"templateType": "anything", "group": "Stuff to generate probabilities", "definition": "round(t/15)", "description": "", "name": "d"}, "episodes": {"templateType": "string", "group": "Strings", "definition": "\"complaints\"", "description": "", "name": "episodes"}, "t2": {"templateType": "anything", "group": "Stuff to generate probabilities", "definition": "t1", "description": "", "name": "t2"}, "p8": {"templateType": "anything", "group": "Probabilities", "definition": "d", "description": "", "name": "p8"}, "p7": {"templateType": "anything", "group": "Probabilities", "definition": "p8+u1", "description": "", "name": "p7"}, "p5": {"templateType": "anything", "group": "Probabilities", "definition": "p6+u3", "description": "", "name": "p5"}, "idef": {"templateType": "string", "group": "Strings", "definition": "\"an\"", "description": "", "name": "idef"}, "p2": {"templateType": "anything", "group": "Probabilities", "definition": "p1+t2", "description": "", "name": "p2"}, "t1": {"templateType": "anything", "group": "Stuff to generate probabilities", "definition": "round(s*random(70..100)/100)", "description": "", "name": "t1"}, "r": {"templateType": "anything", "group": "Stuff to generate probabilities", "definition": "random(45..65)", "description": "", "name": "r"}, "s": {"templateType": "anything", "group": "Stuff to generate probabilities", "definition": "round(r/10)", "description": "", "name": "s"}, "p0": {"templateType": "anything", "group": "Probabilities", "definition": "s", "description": "", "name": "p0"}, "p6": {"templateType": "anything", "group": "Probabilities", "definition": "p7+u2", "description": "", "name": "p6"}, "period": {"templateType": "string", "group": "Strings", "definition": "\"day\"", "description": "", "name": "period"}}, "ungrouped_variables": ["expected_number", "expect_int", "probexceed"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Calculate expectation and a probability from a frequency table, , , ", "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"precisionPartialCredit": 0, "allowFractions": false, "correctAnswerFraction": false, "minValue": "expected_number", "maxValue": "expected_number", "precision": "2", "type": "numberentry", "precisionType": "dp", "showPrecisionHint": false, "strictPrecision": false, "scripts": {}, "precisionMessage": "You have not given your answer to the correct precision.", "showCorrectAnswer": true, "marks": 2}], "type": "gapfill", "prompt": "<p>Find the expected number of {episodes} per {period}.</p>\n<p>Expected number = [[0]]</p>", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"precisionPartialCredit": 0, "allowFractions": false, "correctAnswerFraction": false, "minValue": "probexceed", "maxValue": "probexceed", "precision": "2", "type": "numberentry", "precisionType": "dp", "showPrecisionHint": false, "strictPrecision": false, "scripts": {}, "precisionMessage": "You have not given your answer to the correct precision.", "showCorrectAnswer": true, "marks": 2}], "type": "gapfill", "prompt": "<p>What is the probability that the number of {episodes} will exceed the expected number?</p>\n<p>Probability = [[0]]</p>", "showCorrectAnswer": true, "marks": 0}], "statement": "<p>The probabilities that {idef} {thing} will receive {episodes} per {period} about its {activity} are given by the following table:</p>\n<table border=\"1\">\n<tbody>\n<tr><th align=\"left\">Complaints</th><th>{values[0]}</th><th>{values[1]}</th><th>{values[2]}</th><th>{values[3]}</th><th>{values[4]}</th><th>{values[5]}</th><th>{values[6]}</th><th>{values[7]}</th><th>{values[8]}</th></tr>\n<tr><th align=\"left\">Probability</th>\n<td>{probabilities[0]}</td>\n<td>{probabilities[1]}</td>\n<td>{probabilities[2]}</td>\n<td>{probabilities[3]}</td>\n<td>{probabilities[4]}</td>\n<td>{probabilities[5]}</td>\n<td>{probabilities[6]}</td>\n<td>{probabilities[7]}</td>\n<td>{probabilities[8]}</td>\n</tr>\n</tbody>\n</table>\n<p>Answer the following two parts, giving your answers to $2$ decimal&nbsp;places.</p>", "tags": ["checked2015", "discrete distribution", "expectation", "expected value", "MAS1604", "MAS2304", "MAS8380", "MAS8401", "mass function", "pmf", "PMF", "Probability", "probability", "probability mass function", "query", "sc", "statistics", "tested1"], "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>7/07/2012:</strong></p>\n                <p>Added tags.</p>\n                <p>Checked calculation.</p>\n                <p><strong>22/07/2012:</strong></p>\n                <p>Added description.</p>\n                <p>Ticked stats extension box.</p>\n                <p><strong>31/07/2012:</strong></p>\n                <p>Added tags.</p>\n                <p>Question appears to be working correctly.</p>\n                <p><strong>20/12/2012:</strong></p>\n                <p>Could increase the number of scenarios by using random string variables. Query tag added for that.</p>\n                <p>Also very cumbersome use of variables. But no change proposed for now.</p>\n                <p>Checked calculation, OK. Added tested1 tag.</p>\n                <p><strong>21/12/2012:</strong></p>\n                <p>Although asks for solution to 2 dps, there is no rounding as the raw values are to 2 dps. Added sc tag for possible scenarios.</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Given a probability mass function $P(X=i)$ with outcomes $i \\in \\{0,1,2,\\ldots 8\\}$, find the expectation $E$ and $P(X \\gt E)$.</p>"}, "variablesTest": {"condition": "", "maxRuns": "100"}, "advice": "<h4>a)</h4>\n<p>The expected number of {episodes} is given by:</p>\n<p>\\[ \\simplify[]{{probabilities[0]}*{values[0]} +&nbsp;{probabilities[1]}*{values[1]} +&nbsp;{probabilities[2]}*{values[2]} +&nbsp;{probabilities[3]}*{values[3]} +&nbsp;{probabilities[4]}*{values[4]} +&nbsp;{probabilities[5]}*{values[5]} +&nbsp;{probabilities[6]}*{values[6]} +&nbsp;{probabilities[7]}*{values[7]} +&nbsp;{probabilities[8]}*{values[8]}} = \\var{expected_number} \\]</p>\n<h4>b)</h4>\n<p>We want the probability that the number of {episodes} exceeds $\\var{expected_number}$.</p>\n<p>Since the number of {episodes} is a whole number, this is the same as the probability that the number is&nbsp;$\\var{expect_int+1}$ or more and is</p>\n<p>\\[\\sum_{i=\\var{expect_int+1}}^{i=8} \\left( \\text{Probability}(\\var{episodes} = i ) \\right)= \\simplify[zeroTerm]{ {if(expect_int&lt;1,probabilities[1],0)} + {if(expect_int&lt;2,probabilities[2],0)} + {if(expect_int&lt;3,probabilities[3],0)} + {if(expect_int&lt;4,probabilities[4],0)} + {if(expect_int&lt;5,probabilities[5],0)} + {if(expect_int&lt;6,probabilities[6],0)} + {if(expect_int&lt;7,probabilities[7],0)} + {if(expect_int&lt;8,probabilities[8],0)}} = \\var{probexceed}\\]</p>", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}