// Numbas version: finer_feedback_settings
{"name": "Normal distribution 1", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Normal distribution 1", "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>A machine mass-produces steel bolts with a target diameter of {size}mm, but records show that the diameters are normally distributed with mean {size}mm and standard deviation {sd}mm.</p>\n<p>The bolts can only be sold if they have a diameter in the range {x1}mm and {x2}mm.</p>\n<p>What proportion of the bolts produced are unacceptable for sale?</p>", "advice": "<p>Let $X$ be the diameter of a bolt. Since $X$ is normally distributed, $X\\sim N(\\var{size},\\var{sd}^2).$</p>\n<p>If we want to make a transformation to the standard normal distribution $Z \\sim N(0,1^2)$, we can use the transformation</p>\n<p>\\[ Z = \\frac{X-\\mu}{\\sigma} = \\frac{X-\\var{size}}{\\var{sd}} \\]</p>\n<p>The lower level of acceptability, $x_1=\\var{x1}$, such that $z_1=\\tfrac{\\var{x1}-\\var{size}}{\\var{sd}}=\\var{z1}$.</p>\n<p>The upper level of acceptability, $x_2=\\var{x2}$, such that $z_2=\\tfrac{\\var{x2}-\\var{size}}{\\var{sd}}=\\var{z2}$.</p>\n<p>The range of acceptable $Z$ values is therefore $-1$ and $1$.</p>\n<p>Using the symmetrical property of the normal distribution,</p>\n<p>\\[ \\begin{split} P(-1&lt;Z&lt;1) &amp;\\,= 2 P(0&lt;Z&lt;1) \\\\ &amp;\\,= 2 \\times 0.34134 \\\\ &amp;\\,=0.68268. \\end{split} \\]</p>\n<p>Therefore, the proportion of bolts produced which are unacceptable for sale are&nbsp;</p>\n<p>\\[ 1 - 0.68268 = 0.31732, \\quad \\text{or} \\quad 31.73 \\% \\text{ (2 d.p.)} .\\]</p>", "rulesets": {}, "extensions": ["stats"], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"size": {"name": "size", "group": "Ungrouped variables", "definition": "random(20..50 #10)", "description": "", "templateType": "anything", "can_override": false}, "sd": {"name": "sd", "group": "Ungrouped variables", "definition": "0.05", "description": "", "templateType": "anything", "can_override": false}, "x1": {"name": "x1", "group": "Ungrouped variables", "definition": "size-sd", "description": "", "templateType": "anything", "can_override": false}, "x2": {"name": "x2", "group": "Ungrouped variables", "definition": "size+sd", "description": "", "templateType": "anything", "can_override": false}, "z1": {"name": "z1", "group": "Ungrouped variables", "definition": "(x1-size)/sd", "description": "", "templateType": "anything", "can_override": false}, "z2": {"name": "z2", "group": "Ungrouped variables", "definition": "(x2-size)/sd", "description": "", "templateType": "anything", "can_override": false}, "ans": {"name": "ans", "group": "Ungrouped variables", "definition": "31.73", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["size", "sd", "x1", "x2", "z1", "z2", "ans"], "variable_groups": [], "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>[[0]]%</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{ans}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": "0.01", "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question", "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}]}], "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}