// Numbas version: exam_results_page_options
{"name": "normal_2", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "normal_2", "statement": "<p></p>\n<p>The width of a line etched on an integrated circuit chip is normally distributed with mean \\(\\var{m1}\\)nm and standard deviation \\(\\var{sd1}\\)nm.</p>\n<p>A chip is rejected if its line width is found to be greater than \\(\\var{z_u}\\)nm or less than \\(\\var{z_l}\\)nm.</p>\n<p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;</p>", "ungrouped_variables": ["m1", "sd1", "n", "z_l", "z_u", "p1", "p", "pr", "z1", "inv_pr"], "extensions": ["stats"], "type": "question", "metadata": {"licence": "Creative Commons Attribution-NonCommercial 4.0 International", "description": ""}, "variable_groups": [], "preamble": {"js": "", "css": ""}, "variables": {"z1": {"templateType": "anything", "name": "z1", "group": "Ungrouped variables", "description": "", "definition": "normalinv({inv_pr},0,1)"}, "p": {"templateType": "anything", "name": "p", "group": "Ungrouped variables", "description": "", "definition": "2-2*{p1}"}, "m1": {"templateType": "randrange", "name": "m1", "group": "Ungrouped variables", "description": "", "definition": "random(80..250#25)"}, "z_l": {"templateType": "anything", "name": "z_l", "group": "Ungrouped variables", "description": "", "definition": "{m1}-{n}*{sd1}"}, "n": {"templateType": "randrange", "name": "n", "group": "Ungrouped variables", "description": "", "definition": "random(1.65..2.1#0.01)"}, "z_u": {"templateType": "anything", "name": "z_u", "group": "Ungrouped variables", "description": "", "definition": "{m1}+{n}*{sd1}"}, "pr": {"templateType": "randrange", "name": "pr", "group": "Ungrouped variables", "description": "", "definition": "random(90..99#1)"}, "inv_pr": {"templateType": "anything", "name": "inv_pr", "group": "Ungrouped variables", "description": "<p>If we want to find upper z-value where that prob that z is between lower z-value and upper z-value is pr% then (pr/2+50)/100 is the total prob less than upper z-value</p>", "definition": "({pr}/2+50)/100"}, "p1": {"templateType": "anything", "name": "p1", "group": "Ungrouped variables", "description": "", "definition": "normalcdf(n,0,1)"}, "sd1": {"templateType": "randrange", "name": "sd1", "group": "Ungrouped variables", "description": "", "definition": "random(5..10#1)"}}, "tags": [], "parts": [{"variableReplacements": [], "showFeedbackIcon": true, "prompt": "<p>Calculate, correct to three decimal places, the probability that a resistor chosen at random will be rejected.&nbsp;[[0]]</p>\n<p></p>", "scripts": {}, "showCorrectAnswer": true, "type": "gapfill", "gaps": [{"scripts": {}, "variableReplacements": [], "precisionPartialCredit": 0, "minValue": "{p}", "type": "numberentry", "maxValue": "{p}", "precision": "3", "notationStyles": ["plain", "en", "si-en"], "showPrecisionHint": false, "correctAnswerStyle": "plain", "correctAnswerFraction": false, "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "mustBeReduced": false, "allowFractions": false, "showCorrectAnswer": true, "showFeedbackIcon": true, "marks": 1, "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false}], "variableReplacementStrategy": "originalfirst", "marks": 0}, {"variableReplacements": [], "showFeedbackIcon": true, "prompt": "<p>Determine the line widths within which \\(\\var{pr}\\)% of production will lie. Give your answers as integer values.</p>\n<p></p>\n<p style=\"padding-left: 30px;\">Lower limit = &nbsp;[[0]]nm&nbsp; &nbsp; &nbsp; &nbsp; &nbsp;Upper limit =&nbsp;[[1]]nm</p>", "scripts": {}, "showCorrectAnswer": true, "type": "gapfill", "gaps": [{"scripts": {}, "variableReplacements": [], "precisionPartialCredit": 0, "minValue": "{m1}-{sd1}*{z1}", "type": "numberentry", "maxValue": "{m1}-{sd1}*{z1}", "precision": "0", "notationStyles": ["plain", "en", "si-en"], "showPrecisionHint": false, "correctAnswerStyle": "plain", "correctAnswerFraction": false, "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "mustBeReduced": false, "allowFractions": false, "showCorrectAnswer": true, "showFeedbackIcon": true, "marks": 1, "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false}, {"scripts": {}, "variableReplacements": [], "precisionPartialCredit": 0, "minValue": "{m1}+{sd1}*{z1}", "type": "numberentry", "maxValue": "{m1}+{sd1}*{z1}", "precision": "0", "notationStyles": ["plain", "en", "si-en"], "showPrecisionHint": false, "correctAnswerStyle": "plain", "correctAnswerFraction": false, "variableReplacementStrategy": "originalfirst", "mustBeReducedPC": 0, "mustBeReduced": false, "allowFractions": false, "showCorrectAnswer": true, "showFeedbackIcon": true, "marks": 1, "precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false}], "variableReplacementStrategy": "originalfirst", "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>\\(X\\)&nbsp;is normally distributed with mean \\(\\var{m1}\\) nm and standard deviation \\(\\var{sd1}\\)&nbsp;nm.</p>\n<p>A chip is rejected if its line width is found to be greater than \\(\\var{z_u}\\)&nbsp;nm or less than \\(\\var{z_l}\\) nm.</p>\n<p></p>\n<p>(a) &nbsp; \\(P(Reject)=P(\\var{z_l}&gt;X\\,\\,or\\,\\,X&gt;\\var{z_u})\\)</p>\n<p style=\"padding-left: 120px;\">\\(=1 - P(Acceptable)\\)</p>\n<p style=\"padding-left: 60px;\">\\(P(Acceptable)=P(\\var{z_l}&lt;X&lt;\\var{z_u})\\)</p>\n<p style=\"padding-left: 120px;\">=\\(P\\left(\\frac{\\var{z_l}-\\var{m1}}{\\var{sd1}}&lt;Z&lt;\\frac{\\var{z_u}-\\var{m1}}{\\var{sd1}}\\right)\\)</p>\n<p style=\"padding-left: 120px;\">=\\(P(Z&lt;\\var{n})-P(Z&lt;-\\var{n})\\)</p>\n<p style=\"padding-left: 120px;\">=\\(P(Z&lt;\\var{n})-(1-P(Z&lt;\\var{n}))\\)</p>\n<p style=\"padding-left: 120px;\">=\\(\\var{p1}-(1-\\var{p1})\\)</p>\n<p style=\"padding-left: 120px;\">=\\(\\simplify{2*{p1}-1}\\)</p>\n<p style=\"padding-left: 60px;\">\\(P(Reject)=1-\\simplify{2*{p1}-1}\\)</p>\n<p style=\"padding-left: 60px;\">\\(P(Reject)=\\simplify{2-2*{p1}}\\)</p>\n<p style=\"padding-left: 120px;\"></p>\n<p></p>\n<p>(b) &nbsp; &nbsp; Find the values \\({k_1}\\) and \\({k_2}\\) such that \\(P({k_1}&lt;X&lt;{k_2})=\\simplify{{pr}*0.0100}\\)</p>\n<p style=\"padding-left: 120px;\">\\(P\\left(\\frac{{k_1}-\\var{m1}}{\\var{sd1}}&lt;Z&lt;\\frac{{k_2}-\\var{m1}}{\\var{sd1}}\\right)=\\simplify{{pr}*0.0100}\\)</p>\n<p style=\"padding-left: 120px;\">But &nbsp; \\(P(-\\var{z1}&lt;Z&lt;\\var{z1})=\\simplify{{pr}*0.0100}\\)</p>\n<p style=\"padding-left: 120px;\">So &nbsp; \\(\\frac{{k_1}-\\var{m1}}{\\var{sd1}}=-\\var{z1}\\) &nbsp; and &nbsp; \\(\\frac{{k_2}-\\var{m1}}{\\var{sd1}}=\\var{z1}\\)</p>\n<p style=\"padding-left: 120px;\">\\({k_1}-\\var{m1}=-\\simplify{{sd1}*{z1}}\\) &nbsp; and &nbsp;&nbsp;\\({k_2}-\\var{m1}=\\simplify{{sd1}*{z1}}\\)</p>\n<p style=\"padding-left: 120px;\">\\({k_1}=\\simplify{{m1}-{sd1}*{z1}}\\) &nbsp; and &nbsp;&nbsp;\\({k_2}=\\simplify{{m1}+{sd1}*{z1}}\\)</p>\n<p style=\"padding-left: 120px;\"></p>", "rulesets": {}, "functions": {}, "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}