You have not given your answer to the correct precision.
", "showPrecisionHint": false, "minValue": "{p}*100", "precisionType": "dp", "type": "numberentry", "precision": "1", "correctAnswerStyle": "plain", "strictPrecision": false, "showFeedbackIcon": true, "precisionPartialCredit": 0, "showCorrectAnswer": true, "marks": 1}], "showFeedbackIcon": true, "scripts": {}, "prompt": "Calculate, correct to one decimal place, the percentage of capacitors that will be destroyed. [[0]]
"}, {"variableReplacements": [], "marks": 0, "variableReplacementStrategy": "originalfirst", "type": "gapfill", "showCorrectAnswer": true, "gaps": [{"variableReplacements": [], "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "allowFractions": false, "correctAnswerFraction": false, "maxValue": "{m1}-{sd1}*{z1}", "mustBeReducedPC": 0, "variableReplacementStrategy": "originalfirst", "scripts": {}, "precisionMessage": "You have not given your answer to the correct precision.", "showPrecisionHint": false, "minValue": "{m1}-{sd1}*{z1}", "precisionType": "dp", "type": "numberentry", "precision": "0", "correctAnswerStyle": "plain", "strictPrecision": false, "showFeedbackIcon": true, "precisionPartialCredit": 0, "showCorrectAnswer": true, "marks": 1}, {"variableReplacements": [], "mustBeReduced": false, "notationStyles": ["plain", "en", "si-en"], "allowFractions": false, "correctAnswerFraction": false, "maxValue": "{m1}+{sd1}*{z1}", "mustBeReducedPC": 0, "variableReplacementStrategy": "originalfirst", "scripts": {}, "precisionMessage": "You have not given your answer to the correct precision.", "showPrecisionHint": false, "minValue": "{m1}+{sd1}*{z1}", "precisionType": "dp", "type": "numberentry", "precision": "0", "correctAnswerStyle": "plain", "strictPrecision": false, "showFeedbackIcon": true, "precisionPartialCredit": 0, "showCorrectAnswer": true, "marks": 1}], "showFeedbackIcon": true, "scripts": {}, "prompt": "If the manufacturer wants to destroy no more than \\(\\var{pr}\\)% of production, calculate the new acceptance limits that must apply.
\nLower limit = [[0]] Upper limit = [[1]]
"}], "variablesTest": {"maxRuns": 100, "condition": ""}, "variable_groups": [], "metadata": {"licence": "Creative Commons Attribution-NonCommercial 4.0 International", "description": ""}, "name": "normal_1", "statement": "The production of capacitors is normally distributed about a mean of \\(\\var{m1}\\) μF with a standard deviation of \\(\\var{sd1}\\) μF.
\nA capacitor is considered to be faulty and must be destroyed unless its capacitance lies in the range [\\(\\var{z_l}\\) μF, \\(\\var{z_u}\\)μF].
", "variables": {"inv_pr": {"name": "inv_pr", "definition": "(100-{pr}/2)/100", "group": "Ungrouped variables", "description": "If we want to find upper z-value where the prob that z is between lower z-value and upper z-value is pr then 1-pr/2 is the total prob less than upper z-value
", "templateType": "anything"}, "sd1": {"name": "sd1", "definition": "random(2..5#1)", "group": "Ungrouped variables", "description": "", "templateType": "randrange"}, "z_u": {"name": "z_u", "definition": "{m1}+{n}*{sd1}", "group": "Ungrouped variables", "description": "", "templateType": "anything"}, "pr": {"name": "pr", "definition": "random(0.5..2#0.1)", "group": "Ungrouped variables", "description": "", "templateType": "randrange"}, "m1": {"name": "m1", "definition": "random(50..250#25)", "group": "Ungrouped variables", "description": "", "templateType": "randrange"}, "p": {"name": "p", "definition": "2-2*{p1}", "group": "Ungrouped variables", "description": "", "templateType": "anything"}, "z_l": {"name": "z_l", "definition": "{m1}-{n}*{sd1}", "group": "Ungrouped variables", "description": "", "templateType": "anything"}, "p1": {"name": "p1", "definition": "normalcdf(n,0,1)", "group": "Ungrouped variables", "description": "", "templateType": "anything"}, "n": {"name": "n", "definition": "random(1.65..2.1#0.01)", "group": "Ungrouped variables", "description": "", "templateType": "randrange"}, "z1": {"name": "z1", "definition": "normalinv({inv_pr},0,1)", "group": "Ungrouped variables", "description": "", "templateType": "anything"}}, "functions": {}, "preamble": {"css": "", "js": ""}, "advice": "\\(X\\) has a mean of \\(\\var{m1}\\) μF with a standard deviation of \\(\\var{sd1}\\) μF.
\nResistor is faulty if its capacitance lies in the range [\\(\\var{z_l}\\) μF, \\(\\var{z_u}\\)μF]
\n\n(a) \\(P(faulty)=P(\\var{z_l}<X<\\var{z_u})\\)
\n=\\(P\\left(\\frac{\\var{z_l}-\\var{m1}}{\\var{sd1}}<Z<\\frac{\\var{z_u}-\\var{m1}}{\\var{sd1}}\\right)\\)
\n=\\(P(Z<\\var{n})-P(Z<-\\var{n})\\)
\n=\\(P(Z<\\var{n})-(1-P(Z<\\var{n}))\\)
\n=\\(\\var{p1}-(1-\\var{p1})\\)
\n=\\(\\var{p}\\)
\n\n(b) Find the values \\({k_1}\\) and \\({k_2}\\) such that \\(P({k_1}<X<{k_2})=\\simplify{1-{pr}*0.0100}\\)
\n\\(P\\left(\\frac{{k_1}-\\var{m1}}{\\var{sd1}}<Z<\\frac{{k_2}-\\var{m1}}{\\var{sd1}}\\right)=\\simplify{1-{pr}*0.0100}\\)
\nBut \\(P(-\\var{z1}<Z<\\var{z1})=\\simplify{1-{pr}*0.0100}\\)
\nSo \\(\\frac{{k_1}-\\var{m1}}{\\var{sd1}}=-\\var{z1}\\) and \\(\\frac{{k_2}-\\var{m1}}{\\var{sd1}}=\\var{z1}\\)
\n\\({k_1}-\\var{m1}=-\\simplify{{sd1}*{z1}}\\) and \\({k_2}-\\var{m1}=\\simplify{{sd1}*{z1}}\\)
\n\\({k_1}=\\simplify{{m1}-{sd1}*{z1}}\\) and \\({k_2}=\\simplify{{m1}+{sd1}*{z1}}\\)
\n", "tags": [], "ungrouped_variables": ["m1", "sd1", "n", "z_l", "z_u", "p1", "p", "pr", "z1", "inv_pr"], "extensions": ["stats"], "type": "question", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}