// Numbas version: exam_results_page_options
{"name": "Confidence intervals for the mean and standard deviation of a protein in infant formula 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 intervals for the mean and standard deviation of a protein in infant formula adaptive marking", "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>The amount of protein in a particular brand of infant formula is to be determined. {n} replicate samples weighing 100 grams were selected at random and the amount of protein (grams) measured for each replicate is shown below.</p>\n<p>&nbsp;&nbsp;</p>\n<p>&nbsp; &nbsp;{data}</p>\n<p>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;</p>\n<p>(i) Produce a {percentage}% confidence interval for the mean amount of protein per 100 grams&nbsp;</p>\n<p>(ii) Produce a {percentage}% confidence interval for the variance of the amount of protein per 100 grams. &nbsp;</p>\n<p></p>", "advice": "<p>The sample mean $\\bar{x}=\\var{mean}$</p>\n<p>The sample&nbsp;standard deviation $s=\\var{s}$</p>\n<p>The sample variance is $s^2=\\var{sample_var}$</p>\n<p></p>\n<p>The sample&nbsp;mean $\\mu_m$&nbsp;follows a $t$-distribution with $n-1 = \\var{n-1}$ degrees of freedom&nbsp;centred on&nbsp;the population mean, with a standard error of $\\frac{s}{\\sqrt{n}} = \\simplify{{precround(s,2)}/sqrt({n})} = \\var{precround(standard_error,2)}$</p>\n<p>That means&nbsp;that the {percentage}% confidence interval for the population mean&nbsp;is as follows:</p>\n<p>$a=\\bar{x}-t_{\\frac{\\alpha}{2},n-1}\\times\\frac{s}{\\sqrt{n}}$</p>\n<p>$a=\\var{mean}-t_{\\frac{\\var{alpha}}{2},\\var{n-1}}\\times\\frac{\\var{s}}{\\sqrt{\\var{n}}}=\\var{a}$</p>\n<p>$a=\\bar{x}+t_{\\frac{\\alpha}{2},n-1}\\times\\frac{s}{\\sqrt{n}}$</p>\n<p>$b=\\var{mean}+t_{\\frac{\\var{alpha}}{2},\\var{n-1}}\\times\\frac{\\var{s}}{\\sqrt{\\var{n}}}=\\var{b}$</p>\n<p></p>\n<p>The {percentage}% confidence interval for the population variance $\\sigma^2$ is:</p>\n<p></p>\n<p>$a=\\frac{(n-1)\\times s^2}{\\chi^2_{upper,n-1}}=\\frac{(\\var{n}-1)\\times \\var{s}^2}{\\chi^2_{\\simplify{1-{alpha/2}},\\var{n-1}}}=\\var{lower}$</p>\n<p></p>\n<p>$b=\\frac{(n-1)\\times s^2}{\\chi^2_{lower,n-1}}=\\frac{(\\var{n}-1)\\times \\var{s}^2}{\\chi^2_{\\simplify{{alpha/2}},\\var{n-1}}}=\\var{upper}$</p>\n<p></p>\n<p>The {percentage}% confidence interval for the population&nbsp;standard deviation $\\sigma$ is:</p>\n<p></p>\n<p>$a=\\sqrt{\\var{lower}}=\\var{lower_sd}$</p>\n<p>$a=\\sqrt{\\var{upper}}=\\var{upper_sd}$</p>\n<p></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(alpha,(n-1)),2)", "description": "", "templateType": "anything", "can_override": false}, "chi_upper": {"name": "chi_upper", "group": "Ungrouped variables", "definition": "precround(chisquareinv(1-alpha,(n-1)),2)", "description": "", "templateType": "anything", "can_override": false}, "lower": {"name": "lower", "group": "Ungrouped variables", "definition": "precround(s^2*(n-1)/chi_upper,4)", "description": "", "templateType": "anything", "can_override": false}, "upper": {"name": "upper", "group": "Ungrouped variables", "definition": "precround(s^2*(n-1)/chi_lower,4)", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(6..9#1)", "description": "", "templateType": "anything", "can_override": false}, "data": {"name": "data", "group": "Ungrouped variables", "definition": "shuffle(13.6..14.7#0.1)[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}, "percentage": {"name": "percentage", "group": "Ungrouped variables", "definition": "perc[Index]", "description": "", "templateType": "anything", "can_override": false}, "perc": {"name": "perc", "group": "Ungrouped variables", "definition": "[95,99]", "description": "", "templateType": "anything", "can_override": false}, "Index": {"name": "Index", "group": "Ungrouped variables", "definition": "random(0,1)", "description": "", "templateType": "anything", "can_override": false}, "alpha": {"name": "alpha", "group": "Ungrouped variables", "definition": "(100-percentage)/200", "description": "", "templateType": "anything", "can_override": false}, "mean": {"name": "mean", "group": "Ungrouped variables", "definition": "precround(mean(data),3)", "description": "", "templateType": "anything", "can_override": false}, "standard_error": {"name": "standard_error", "group": "Ungrouped variables", "definition": "precround(s/sqrt(n),3)", "description": "", "templateType": "anything", "can_override": false}, "units": {"name": "units", "group": "Ungrouped variables", "definition": "'grams'", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "precround((mean-t*standard_error),3)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "precround((mean+t*standard_error),3)", "description": "", "templateType": "anything", "can_override": false}, "t": {"name": "t", "group": "Ungrouped variables", "definition": "precround(studenttinv((1-alpha),(n-1)),3)", "description": "", "templateType": "anything", "can_override": false}, "alpha_comp": {"name": "alpha_comp", "group": "Ungrouped variables", "definition": "1-alpha", "description": "", "templateType": "anything", "can_override": false}, "s": {"name": "s", "group": "Ungrouped variables", "definition": "precround(sqrt(sample_var),3)", "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", "percentage", "perc", "Index", "alpha", "mean", "standard_error", "units", "a", "b", "t", "alpha_comp", "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 answers to three decimal places.</p>\n<p>The&nbsp;sample mean is &nbsp;[[0]]</p>\n<p>The&nbsp;sample standard deviation is [[1]]</p>\n<p>The standard error is&nbsp;[[2]]</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": "mean-0.001", "maxValue": "mean+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": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [{"variable": "sample_var", "part": "p0g0", "must_go_first": false}], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "s-0.001", "maxValue": "s+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": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [{"variable": "s", "part": "p0g1", "must_go_first": false}], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "standard_error-0.001", "maxValue": "standard_error+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 answers correct to 3 decimal places.</p>\n<p>What is the critical t-value&nbsp;associated with&nbsp;a sample of size $\\var{n}$ and $\\frac{\\alpha}{2}=\\var{alpha}$?</p>\n<p>t<sub>crit&nbsp;=[[0]]</sub><sub></sub><sub></sub></p>\n<p>Calculate a &nbsp;$\\var{percentage}$% confidence interval $(a,b)$ for the population mean:</p>\n<p></p>\n<p>$a=$ [[1]]</p>\n<p>$b=$ [[2]]</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": "t-0.001", "maxValue": "t+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": "mean", "part": "p0g0", "must_go_first": false}, {"variable": "standard_error", "part": "p0g2", "must_go_first": false}, {"variable": "t", "part": "p1g0", "must_go_first": false}], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "a-0.002", "maxValue": "a+0.002", "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": "mean", "part": "p0g0", "must_go_first": false}, {"variable": "standard_error", "part": "p0g2", "must_go_first": false}, {"variable": "t", "part": "p1g0", "must_go_first": false}], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "b-0.002", "maxValue": "b+0.002", "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 answers correct to two decimal places.</p>\n<p>What is the critical $\\chi^2$-value&nbsp;associated with&nbsp;a sample of size $\\var{n}$ and $1-\\frac{\\alpha}{2}=\\var{alpha_comp}$?</p>\n<p>[[0]]</p>\n<p></p>\n<p>What is the critical $\\chi^2$-value&nbsp;associated with&nbsp;a sample of size $\\var{n}$ and $\\frac{\\alpha}{2}=\\var{alpha}$?</p>\n<p>[[1]]</p>\n<p></p>\n<p>Calculate a &nbsp;$\\var{percentage}$% confidence interval $(a,b)$ for the population variance:</p>\n<p>$a=\\;$&nbsp;&nbsp;[[2]]</p>\n<p>$b=\\;$&nbsp;&nbsp;[[3]]</p>\n<p>Calculate a &nbsp;$\\var{percentage}$% confidence interval $(a,b)$ for the population standard deviation:</p>\n<p>$a=\\;$&nbsp; [[4]]</p>\n<p>$b=\\;$&nbsp; [[5]]</p>\n<p></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_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"}, {"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-0.001", "maxValue": "chi_lower+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": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [{"variable": "s", "part": "p0g1", "must_go_first": false}, {"variable": "chi_upper", "part": "p2g0", "must_go_first": false}], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "lower-0.01", "maxValue": "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": [{"variable": "s", "part": "p0g1", "must_go_first": false}, {"variable": "chi_lower", "part": "p2g1", "must_go_first": false}], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "upper-0.01", "maxValue": "upper+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": [{"variable": "s", "part": "p0g1", "must_go_first": false}, {"variable": "chi_upper", "part": "p2g0", "must_go_first": false}], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "lower_sd-0.01", "maxValue": "lower_sd+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": [{"variable": "s", "part": "p0g1", "must_go_first": false}, {"variable": "chi_lower", "part": "p2g1", "must_go_first": false}], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "upper_sd-0.01", "maxValue": "upper_sd+0.01", "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/"}]}