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{"name": "Confidence Interval Doctor's Surgery", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"parts": [{"extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "scripts": {}, "showFeedbackIcon": true, "sortAnswers": false, "prompt": "<p>Calculate the mean and standard deviation of the&nbsp;sample of patients visiting on Mondays.</p>\n<p>Enter your answers to&nbsp;<strong><em></em>2 decimal places</strong>.</p>\n<p>Mean: [[0]]</p>\n<p>Standard Deviation: [[1]]</p>", "unitTests": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"mustBeReduced": false, "scripts": {}, "precisionPartialCredit": 0, "variableReplacements": [], "precisionMessage": "You have not given your answer to the correct precision.", "showPrecisionHint": false, "mustBeReducedPC": 0, "allowFractions": false, "customMarkingAlgorithm": "", "marks": 1, "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "correctAnswerStyle": "plain", "showFeedbackIcon": true, "type": "numberentry", "correctAnswerFraction": false, "maxValue": "monday_mean", "precisionType": "dp", "notationStyles": ["plain", "en", "si-en"], "unitTests": [], "minValue": "monday_mean", "precision": "2", "variableReplacementStrategy": "originalfirst", "strictPrecision": false}, {"mustBeReduced": false, "scripts": {}, "precisionPartialCredit": 0, "variableReplacements": [], "precisionMessage": "You have not given your answer to the correct precision.", "showPrecisionHint": false, "mustBeReducedPC": 0, "allowFractions": false, "customMarkingAlgorithm": "", "marks": 1, "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "correctAnswerStyle": "plain", "showFeedbackIcon": true, "type": "numberentry", "correctAnswerFraction": false, "maxValue": "monday_sd", "precisionType": "dp", "notationStyles": ["plain", "en", "si-en"], "unitTests": [], "minValue": "monday_sd", "precision": "2", "variableReplacementStrategy": "originalfirst", "strictPrecision": false}], "variableReplacements": [], "type": "gapfill", "customMarkingAlgorithm": "", "marks": 0}, {"extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "scripts": {}, "showFeedbackIcon": true, "sortAnswers": false, "prompt": "<p style=\"text-align: left;\">Find the 95% confidence interval for the population&nbsp;mean $\\mu_m$ of the age of patients visiting on Monday, using the $t$-distribution with $\\var{num_monday-1}$ degrees of freedom.</p>\n<p style=\"text-align: left;\">Enter your answers to&nbsp;<strong>2 decimal places</strong>.</p>\n<p style=\"text-align: left;\">Lower limit:&nbsp;[[0]]</p>\n<p style=\"text-align: left;\">Upper limit: [[1]]&nbsp;</p>", "unitTests": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"mustBeReduced": false, "scripts": {}, "precisionPartialCredit": 0, "variableReplacements": [{"variable": "monday_mean", "part": "p0g0", "must_go_first": false}, {"variable": "monday_sd", "part": "p0g1", "must_go_first": false}], "precisionMessage": "You have not given your answer to the correct precision.", "showPrecisionHint": false, "mustBeReducedPC": 0, "allowFractions": false, "customMarkingAlgorithm": "", "marks": 1, "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "correctAnswerStyle": "plain", "showFeedbackIcon": true, "type": "numberentry", "correctAnswerFraction": false, "maxValue": "monday_low", "precisionType": "dp", "notationStyles": ["plain", "en", "si-en"], "unitTests": [], "minValue": "monday_low", "precision": "2", "variableReplacementStrategy": "originalfirst", "strictPrecision": false}, {"mustBeReduced": false, "scripts": {}, "precisionPartialCredit": 0, "variableReplacements": [], "precisionMessage": "You have not given your answer to the correct precision.", "showPrecisionHint": false, "mustBeReducedPC": 0, "allowFractions": false, "customMarkingAlgorithm": "", "marks": 1, "extendBaseMarkingAlgorithm": true, "showCorrectAnswer": true, "correctAnswerStyle": "plain", "showFeedbackIcon": true, "type": "numberentry", "correctAnswerFraction": false, "maxValue": "monday_high", "precisionType": "dp", "notationStyles": ["plain", "en", "si-en"], "unitTests": [], "minValue": "monday_high", "precision": "2", "variableReplacementStrategy": "originalfirst", "strictPrecision": false}], "variableReplacements": [], "type": "gapfill", "customMarkingAlgorithm": "", "marks": 0}], "variable_groups": [], "name": "Confidence Interval Doctor's Surgery", "statement": "<p>The ages of a random sample of patients coming into a doctor's surgery during the school holidays are noted down on each Monday and Friday thoughout August.</p>\n<p>{table([['Monday']+mon,['Friday']+fri],[])}</p>", "advice": "<h4>a)</h4>\n<p>The mean of the sample of patients visiting on Mondays is&nbsp;</p>\n<p>\\[ \\bar{x} = \\frac{1}{n}(\\sum x_i) = \\var{precround(monday_mean,2)} \\text{ (to 2 d.p.)} \\]</p>\n<p>The standard deviation is</p>\n<p>\\[ s = \\sqrt{\\frac{1}{n-1}\\sum \\left(x_i-\\bar{x}\\right)^2} = \\var{precround(monday_sd,2)} \\text{ (to 2 d.p.)} \\]</p>\n<h4>b)</h4>\n<p>The sample mean $\\mu_m$&nbsp;follows a $t$-distribution with $n-1 = \\var{num_monday-1}$ degrees of freedom&nbsp;centred on&nbsp;the population mean, with a standard error of $\\frac{s}{\\sqrt{n}} = \\simplify{{precround(monday_sd,2)}/sqrt({num_monday})} = \\var{precround(monday_se,2)}$.</p>\n<p>That means&nbsp;that the 95% confidence interval for the population mean&nbsp;is as follows:</p>\n<p>\\begin{array}{lrcl}<br/>&amp;\\left[ \\bar{x} - s \\times t_{\\var{num_monday-1},0.975} \\right. \\kern{-0.5em}&amp;,&amp;\\kern{-0.5em} \\left. \\bar{x} + s \\times t_{\\var{num_monday-1},0.975} \\right] \\\\<br/>= &amp; \\left[ \\simplify[]{{precround(monday_mean,2)} - {precround(monday_se,2)}*{precround(monday_t,2)}} \\right. \\kern{-0.5em}&amp;,&amp;\\kern{-0.5em}&nbsp;\\left.&nbsp;\\simplify[]{{precround(monday_mean,2)} + {precround(monday_se,2)}*{precround(monday_t,2)}} \\right] \\\\<br/>= &amp; \\left[ \\var{precround(monday_low,2)} \\right. \\kern{-0.5em}&amp;,&amp;\\kern{-0.5em}&nbsp;\\left.&nbsp;\\var{precround(monday_high,2)} \\right]<br/>\\end{array}</p>\n<h4></h4>", "tags": [], "ungrouped_variables": ["num_monday", "monday_pop_mean", "monday_pop_range", "mon", "num_friday", "friday_pop_mean", "friday_pop_range", "fri", "monday_mean", "monday_sd", "monday_se", "monday_t", "monday_low", "monday_high", "t", "friday_mean", "friday_se", "friday_t", "friday_low", "friday_high", "no_overlap"], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"friday_high": {"definition": "friday_mean+friday_t*friday_se", "group": "Ungrouped variables", "name": "friday_high", "description": "", "templateType": "anything"}, "friday_se": {"definition": "stdev(fri,true)/sqrt(num_friday)", "group": "Ungrouped variables", "name": "friday_se", "description": "", "templateType": "anything"}, "monday_pop_mean": {"definition": "random(20..70)", "group": "Ungrouped variables", "name": "monday_pop_mean", "description": "", "templateType": "anything"}, "monday_high": {"definition": "monday_mean+monday_t*monday_se", "group": "Ungrouped variables", "name": "monday_high", "description": "", "templateType": "anything"}, "monday_se": {"definition": "monday_sd/sqrt(num_monday)", "group": "Ungrouped variables", "name": "monday_se", "description": "", "templateType": "anything"}, "num_monday": {"definition": "12", "group": "Ungrouped variables", "name": "num_monday", "description": "", "templateType": "anything"}, "monday_t": {"definition": "studenttinv(0.975,num_monday-1)", "group": "Ungrouped variables", "name": "monday_t", "description": "", "templateType": "anything"}, "t": {"definition": "studenttinv(0.975,11)", "group": "Ungrouped variables", "name": "t", "description": "", "templateType": "anything"}, "friday_t": {"definition": "studenttinv(0.975,num_friday-1)", "group": "Ungrouped variables", "name": "friday_t", "description": "", "templateType": "anything"}, "monday_mean": {"definition": "mean(mon)", "group": "Ungrouped variables", "name": "monday_mean", "description": "", "templateType": "anything"}, "num_friday": {"definition": "random(12..16)", "group": "Ungrouped variables", "name": "num_friday", "description": "", "templateType": "anything"}, "monday_low": {"definition": "monday_mean-monday_t*monday_se", "group": "Ungrouped variables", "name": "monday_low", "description": "<p>Lower end of the confidence interval for Monday's mean</p>", "templateType": "anything"}, "friday_mean": {"definition": "mean(fri)", "group": "Ungrouped variables", "name": "friday_mean", "description": "", "templateType": "anything"}, "fri": {"definition": "repeat(friday_pop_mean+random(-friday_pop_range..friday_pop_range),num_monday)", "group": "Ungrouped variables", "name": "fri", "description": "<p>Data for Friday</p>", "templateType": "anything"}, "friday_pop_mean": {"definition": "random(20..70)", "group": "Ungrouped variables", "name": "friday_pop_mean", "description": "", "templateType": "anything"}, "monday_sd": {"definition": "stdev(mon,true)", "group": "Ungrouped variables", "name": "monday_sd", "description": "", "templateType": "anything"}, "mon": {"definition": "repeat(monday_pop_mean+random(-monday_pop_range..monday_pop_range),num_monday)", "group": "Ungrouped variables", "name": "mon", "description": "<p>Data for Monday</p>", "templateType": "anything"}, "friday_low": {"definition": "friday_mean-friday_t*friday_se", "group": "Ungrouped variables", "name": "friday_low", "description": "", "templateType": "anything"}, "monday_pop_range": {"definition": "20", "group": "Ungrouped variables", "name": "monday_pop_range", "description": "", "templateType": "anything"}, "no_overlap": {"definition": "friday_low > monday_high or friday_high < monday_low", "group": "Ungrouped variables", "name": "no_overlap", "description": "", "templateType": "anything"}, "friday_pop_range": {"definition": "20", "group": "Ungrouped variables", "name": "friday_pop_range", "description": "", "templateType": "anything"}}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Compute a 95% confidence interval for the population mean given a small sample, and compare it with a confidence interval for a different population.</p>"}, "extensions": ["stats"], "functions": {}, "rulesets": {}, "type": "question", "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Catherine Palmer", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/423/"}, {"name": "Hayley Moore", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/495/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Catherine Palmer", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/423/"}, {"name": "Hayley Moore", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/495/"}]}