// Numbas version: exam_results_page_options
{"name": "Calculate probabilities from a normal distribution and use inverse tables ", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Calculate probabilities from a normal distribution and use inverse tables ", "tags": [], "metadata": {"description": "<p></p>\n<p>Given a random variable $X$ &nbsp;normally distributed as $\\operatorname{N}(m,\\sigma^2)$ find probabilities $P(X \\gt a),\\; a \\gt m;\\;\\;P(X \\lt b),\\;b \\lt m$.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Let $X$ represent the {chosen_amount} of {chosen_stuff}. It is known that $X$ follows a Normal distribution with a mean of {chosen_m}{chosen_units} and a standard deviation of {chosen_s}{chosen_units}.</p>\n<p>&nbsp;</p>", "advice": "<p>a)</p>\n<p>1. Converting to $\\operatorname{N}(0,1)$</p>\n<p>$\\simplify[all,!collectNumbers]{P(X &lt; {lower}) = P(Z &lt; ({lower} -{m}) / {s})} = P(Z&lt;-\\var{zlower})= \\var{prob1}$ to 2 decimal places.</p>\n<p>2. Converting to $\\operatorname{N}(0,1)$</p>\n<p>$\\simplify[all,!collectNumbers]{P(X &gt; {upper}) = P(Z &gt; ({upper} -{m}) / {s})} = P(Z&gt;\\var{zupper}) = 1-P(Z&lt;\\var{zupper})=1-\\var{p1} = \\var{prob2}$ to 2 decimal places.</p>\n<p>3.</p>\n<p>$\\simplify[all,!collectNumbers]{P({lower} &lt; X &lt; {upper}) = P(X &lt; {upper})-P(X &lt; {lower})}=P(Z&lt;\\var{zupper})-P(Z&lt;-\\var{zlower}) =\\var{p1}-\\var{p2} = \\var{prob3}$ to 2 decimal places.</p>\n<p></p>\n<p>b)</p>\n<p>If {percentage}% of {chosen_item}s contain less than $x$ {chosen_units} then we can write $P(X &lt; x ) = \\var{prob}$</p>\n<p>Using the Standard Normal tables we can write $P(Z &lt; \\var{z1}) = \\var{prob}$</p>\n<p>Now $x = \\mu + z \\times \\sigma$</p>\n<p>so that $x = \\var{chosen_m}&nbsp;+ (\\var{z1}\\times \\var{chosen_s})= \\var{x1}$</p>\n<p></p>", "rulesets": {}, "extensions": ["stats"], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"chosen_amount": {"name": "chosen_amount", "group": "Ungrouped variables", "definition": "amount[Index]", "description": "", "templateType": "anything", "can_override": false}, "prob": {"name": "prob", "group": "Ungrouped variables", "definition": "percentage/100", "description": "", "templateType": "anything", "can_override": false}, "zlower": {"name": "zlower", "group": "Ungrouped variables", "definition": "precround((chosen_m-lower)/chosen_s,2)", "description": "", "templateType": "anything", "can_override": false}, "percentage": {"name": "percentage", "group": "Ungrouped variables", "definition": "random(1..10#1)", "description": "", "templateType": "anything", "can_override": false}, "chosen_s": {"name": "chosen_s", "group": "Ungrouped variables", "definition": "s[Index]", "description": "", "templateType": "anything", "can_override": false}, "x1": {"name": "x1", "group": "Ungrouped variables", "definition": "precround(normalinv(percentage/100, chosen_m,chosen_s),1)", "description": "", "templateType": "anything", "can_override": false}, "chosen_item": {"name": "chosen_item", "group": "Ungrouped variables", "definition": "item[Index]", "description": "", "templateType": "anything", "can_override": false}, "prob2": {"name": "prob2", "group": "Ungrouped variables", "definition": "precround(1-p1,2)", "description": "", "templateType": "anything", "can_override": false}, "upper": {"name": "upper", "group": "Ungrouped variables", "definition": "random(chosen_m+0.5*chosen_s..chosen_m+1.5*chosen_s#5)", "description": "", "templateType": "anything", "can_override": false}, "tol": {"name": "tol", "group": "Ungrouped variables", "definition": "0.01", "description": "", "templateType": "anything", "can_override": false}, "chosen_units": {"name": "chosen_units", "group": "Ungrouped variables", "definition": "units1[Index]", "description": "", "templateType": "anything", "can_override": false}, "item": {"name": "item", "group": "Ungrouped variables", "definition": "[\"packet\",\"bottle\"]", "description": "", "templateType": "anything", "can_override": false}, "prob3": {"name": "prob3", "group": "Ungrouped variables", "definition": "precround(p1-p2,2)", "description": "", "templateType": "anything", "can_override": false}, "lower": {"name": "lower", "group": "Ungrouped variables", "definition": "random(chosen_m-1.5*chosen_s..chosen_m-0.5*chosen_s#5)", "description": "", "templateType": "anything", "can_override": false}, "s": {"name": "s", "group": "Ungrouped variables", "definition": "[random(10..40#5), random(6..15#2)]", "description": "", "templateType": "anything", "can_override": false}, "z1": {"name": "z1", "group": "Ungrouped variables", "definition": "precround(normalinv(percentage/100,0,1),2)", "description": "", "templateType": "anything", "can_override": false}, "units1": {"name": "units1", "group": "Ungrouped variables", "definition": "[\"g\", \"ml\"]", "description": "", "templateType": "anything", "can_override": false}, "p1": {"name": "p1", "group": "Ungrouped variables", "definition": "precround(normalcdf(zupper,0,1),4)", "description": "", "templateType": "anything", "can_override": false}, "stuff": {"name": "stuff", "group": "Ungrouped variables", "definition": "[\"flour in a packet\", \"soft drink in a bottle\"]", "description": "", "templateType": "anything", "can_override": false}, "prob1": {"name": "prob1", "group": "Ungrouped variables", "definition": "precround(1-p,2)", "description": "", "templateType": "anything", "can_override": false}, "amount": {"name": "amount", "group": "Ungrouped variables", "definition": "[\"number of grams\", \"volume (in ml)\"]", "description": "", "templateType": "anything", "can_override": false}, "chosen_m": {"name": "chosen_m", "group": "Ungrouped variables", "definition": "m[Index]", "description": "", "templateType": "anything", "can_override": false}, "zupper": {"name": "zupper", "group": "Ungrouped variables", "definition": "precround((upper-chosen_m)/chosen_s,2)", "description": "", "templateType": "anything", "can_override": false}, "Index": {"name": "Index", "group": "Ungrouped variables", "definition": "random(0,1)", "description": "", "templateType": "anything", "can_override": false}, "p": {"name": "p", "group": "Ungrouped variables", "definition": "precround(normalcdf(zlower,0,1),4)", "description": "", "templateType": "anything", "can_override": false}, "chosen_stuff": {"name": "chosen_stuff", "group": "Ungrouped variables", "definition": "stuff[index]", "description": "", "templateType": "anything", "can_override": false}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "[random(950..1100#10),random(300..1000#50)]", "description": "", "templateType": "anything", "can_override": false}, "p2": {"name": "p2", "group": "Ungrouped variables", "definition": "1-p", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["units1", "upper", "lower", "p1", "m", "s", "zupper", "p", "amount", "p2", "tol", "zlower", "stuff", "prob2", "prob3", "prob1", "Index", "chosen_amount", "chosen_stuff", "chosen_units", "chosen_m", "chosen_s", "chosen_item", "item", "percentage", "x1", "z1", "prob"], "variable_groups": [{"name": "Unnamed group", "variables": []}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Suppose a {chosen_item} is chosen at random, find the probability that the {chosen_amount} is less than {lower} {chosen_units}:</p>\n<p>Probability = ?[[0]](to 2 &nbsp;decimal places)</p>\n<p>Find the probability that the {chosen_amount} is greater than {upper} {chosen_units}:</p>\n<p>Probability = ?[[1]](to 2 &nbsp;decimal places)</p>\n<p>Find the probability that the {chosen_amount} is between {lower}{chosen_units} and {upper} {chosen_units}:</p>\n<p>Probability = ?[[2]](to 2 &nbsp;decimal places)</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "prob1-tol", "maxValue": "prob1+tol", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "prob2-tol", "maxValue": "prob2+tol", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "prob3-tol", "maxValue": "prob3+tol", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>{percentage}% of {chosen_item}s contain less than $x$ {chosen_units}.<br/>Calculate the value of $x$.</p>\n<p>$x$ = [[0]]&nbsp; &nbsp;(rounded to the nearest {chosen_units})</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "x1-0.6", "maxValue": "x1+0.6", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Clodagh Carroll", "profile_url": 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