// Numbas version: exam_results_page_options
{"name": "Confidence interval for the standard deviation of a sodium solution adaptive marking", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Confidence interval for the standard deviation of a sodium solution adaptive marking", "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>To determine the average sodium concentration of a solution, {n} replicate samples were taken and the sodium content of each sample (mg/L) was recorded:</p>\n<p>&nbsp;</p>\n<p>&nbsp; &nbsp;{data}</p>\n<p>The variance of the sodium concentration in the samples was calculated to be $\\var{sample_var}$.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</p>\n<p>Produce a 99% confidence interval for the standard deviation of the sodium concentration. &nbsp;</p>\n<p></p>", "advice": "<p>The 99% confidence interval for the population variance $\\sigma^2$ is:</p>\n<p></p>\n<p>$lower=\\frac{(n-1)\\times s^2}{\\chi^2_{upper,n-1}}=\\frac{(\\var{n}-1)\\times \\var{s}^2}{\\chi^2_{0.995,\\var{n-1}}}=\\var{lower}$</p>\n<p></p>\n<p>$upper=\\frac{(n-1)\\times s^2}{\\chi^2_{lower,n-1}}=\\frac{(\\var{n}-1)\\times \\var{s}^2}{\\chi^2_{0.005,\\var{n-1}}}=\\var{upper}$</p>\n<p></p>\n<p>The 99% confidence interval for the population standard deviation $\\sigma$ is:</p>\n<p></p>\n<p>$lower=\\sqrt{\\var{lower}}=\\var{lower_sd}$</p>\n<p>$upper=\\sqrt{\\var{upper}}=\\var{upper_sd}$</p>", "rulesets": {}, "extensions": ["stats"], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"sample_var": {"name": "sample_var", "group": "Ungrouped variables", "definition": "precround(variance(data,true),4)", "description": "", "templateType": "anything", "can_override": false}, "chi_lower": {"name": "chi_lower", "group": "Ungrouped variables", "definition": " precround(chisquareinv(0.005,(n-1)),3)", "description": "", "templateType": "anything", "can_override": false}, "chi_upper": {"name": "chi_upper", "group": "Ungrouped variables", "definition": "precround(chisquareinv(0.995,(n-1)),3)", "description": "", "templateType": "anything", "can_override": false}, "lower": {"name": "lower", "group": "Ungrouped variables", "definition": "precround(sample_var*(n-1)/chi_upper,4)", "description": "", "templateType": "anything", "can_override": false}, "upper": {"name": "upper", "group": "Ungrouped variables", "definition": "precround(sample_var*(n-1)/chi_lower,4)", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(6..10#1)", "description": "", "templateType": "anything", "can_override": false}, "data": {"name": "data", "group": "Ungrouped variables", "definition": "shuffle(5.3..5.8#0.01)[0..n]\n", "description": "", "templateType": "anything", "can_override": false}, "lower_sd": {"name": "lower_sd", "group": "Ungrouped variables", "definition": "precround(sqrt(lower),3)", "description": "", "templateType": "anything", "can_override": false}, "upper_sd": {"name": "upper_sd", "group": "Ungrouped variables", "definition": "precround(sqrt(upper),3)", "description": "", "templateType": "anything", "can_override": false}, "s": {"name": "s", "group": "Ungrouped variables", "definition": "precround(sqrt(sample_var),5)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["sample_var", "chi_lower", "chi_upper", "lower", "n", "data", "lower_sd", "upper", "upper_sd", "s"], "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>Please give your answer to two decimal places.</p>\n<p>What are the values of&nbsp; $\\chi^2_{{\\frac{\\alpha}{2},}{n-1}}$ and $\\chi^2_{1-{\\frac{\\alpha}{2},}{n-1}}$?</p>\n<p>$\\chi^2_{{\\frac{\\alpha}{2},}{n-1}}$ = [[0]]</p>\n<p>$\\chi^2_{1-{\\frac{\\alpha}{2},}{n-1}}$ = [[1]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "chi_lower-0.01", "maxValue": "chi_lower+0.01", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "chi_upper-0.001", "maxValue": "chi_upper+0.001", "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>Please give your answer to three decimal places.</p>\n<p>The&nbsp;lower limit is&nbsp;[[0]]</p>\n<p>The upper limit is&nbsp;[[1]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": false, "variableReplacements": [{"variable": "chi_lower", "part": "p0g0", "must_go_first": false}, {"variable": "chi_upper", "part": "p0g1", "must_go_first": false}], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "alternatives": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "<p>The question asked for a confidence interval for the standard deviation which is the square root of the variance.</p>", "useAlternativeFeedback": false, "minValue": "precround(lower,3)-0.001", "maxValue": "precround(lower,3)+0.001", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "minValue": "lower_sd-0.001", "maxValue": "lower_sd+0.001", "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": [{"variable": "chi_lower", "part": "p0g0", "must_go_first": false}, {"variable": "chi_upper", "part": "p0g1", "must_go_first": false}], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "alternatives": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "<p>The question asked for a confidence interval for the standard deviation which is the square root of the variance.</p>", "useAlternativeFeedback": false, "minValue": "precround(upper,3)-0.001", "maxValue": "precround(upper,3)+0.001", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "minValue": "upper_sd-0.001", "maxValue": "upper_sd+0.001", "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": "Catherine Palmer", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/423/"}]}]}], "contributors": [{"name": "Catherine Palmer", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/423/"}]}