// Numbas version: finer_feedback_settings
{"name": "Confidence interval for variance", "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 variance", "tags": [], "metadata": {"description": "<p>Finding the confidence interval at either 90%, 95% or 99% for the population&nbsp;variance&nbsp;of a sample. $\\chi^2$ tables are used. A single scenario is given.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>A smartphone manufacturer claims that their new smartphone will last 24 hours on a full charge under normal usage with a standard deviation of 0.8 hours.&nbsp;</p>\n<div class=\"page\" title=\"Page 29\">\n<div class=\"section\">\n<div class=\"layoutArea\">\n<div class=\"column\">\n<p>Testing&nbsp;6 of these smartphones&nbsp;we see the following usage (in hours) until the battery dies:</p>\n<p>{x}&nbsp;hours.</p>\n<p>Construct a {confl}% confidence interval for the standard deviation.</p>\n</div>\n</div>\n</div>\n</div>", "advice": "<p>We use the $\\chi^2$&nbsp;tables to find the confidence interval for&nbsp;the population variance and then we can square root our lower and upper points of the interval to give us the confidence interval for the standard deviation.&nbsp;</p>\n<p>The formula for the confidence interval for the population variance is:</p>\n<p>\\[\\frac{(n-1)s^2}{\\chi^2_{n-1;\\alpha/2}}\\le\\sigma^2\\le\\frac{(n-1)s^2}{\\chi^2_{n-1;1-\\alpha/2}}\\]</p>\n<p></p>\n<p>We first calculate the sample variance using our formula:</p>\n<p>\\[\\textrm{Sample Variance} = \\frac{1}{n-1}\\left(\\sum_{j=1}^{n}x_j^2 -n\\bar{x}^2\\right)=\\var{var}\\]</p>\n<p></p>\n<p></p>\n<p>We now need to find the critical values of the $\\chi^2$ distribution.&nbsp;Our degrees of freedom here are $n-1=\\var{n}-1=\\var{n-1}$.&nbsp;</p>\n<p>Our confidence interval is $\\var{confl}\\%$ so our significance level is $\\alpha=\\var{alpha}$. Our critical values are then</p>\n<p>\\[\\chi^2_{n-1;\\alpha/2}=\\chi^2_{\\var{n-1};\\var{alpha/2}}=\\var{chi_lower}\\]</p>\n<p>and</p>\n<p>\\[\\chi^2_{n-1;1-\\alpha/2}=\\chi^2_{\\var{n-1};\\var{1-alpha/2}}=\\var{chi_upper}\\]</p>\n<p>Hence, inputting all of these values into our formula for the confidence interval</p>\n<p>\\[\\frac{(n-1)s^2}{\\chi^2_{n-1;\\alpha/2}}\\le\\sigma^2\\le\\frac{(n-1)s^2}{\\chi^2_{n-1;1-\\alpha/2}}\\]</p>\n<p>gives the confidence intervals for the population variance $\\sigma^2$</p>\n<p>Lower value of the confidence interval $=\\;\\displaystyle \\var{var_lower}$.</p>\n<p>Upper value of the confidence interval $=\\;\\displaystyle \\var{var_upper}$.</p>\n<p>Square rooting these values gives us the confidence interval for the population standard deviation $\\sigma$ (to 3 d.p.):</p>\n<p>\\[(\\var{lower_round},\\var{upper_round})\\]</p>", "rulesets": {}, "extensions": ["stats"], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"n": {"name": "n", "group": "Ungrouped variables", "definition": "6", "description": "", "templateType": "anything", "can_override": false}, "xbar": {"name": "xbar", "group": "Ungrouped variables", "definition": "sum(x)/n", "description": "", "templateType": "anything", "can_override": false}, "sd": {"name": "sd", "group": "Ungrouped variables", "definition": "stdev(x,true)", "description": "", "templateType": "anything", "can_override": false}, "chi_lower": {"name": "chi_lower", "group": "Ungrouped variables", "definition": "if(confl=90,11.07,if(confl=95,12.83,16.75))", "description": "", "templateType": "anything", "can_override": false}, "chi_upper": {"name": "chi_upper", "group": "Ungrouped variables", "definition": "if(confl=90,1.145,if(confl=95,0.8312,0.4117))", "description": "", "templateType": "anything", "can_override": false}, "var_lower": {"name": "var_lower", "group": "Ungrouped variables", "definition": "((n-1)*var)/chi_lower", "description": "", "templateType": "anything", "can_override": false}, "var": {"name": "var", "group": "Ungrouped variables", "definition": "sd^2", "description": "", "templateType": "anything", "can_override": false}, "var_upper": {"name": "var_upper", "group": "Ungrouped variables", "definition": "((n-1)*var)/chi_upper", "description": "", "templateType": "anything", "can_override": false}, "sd_lower": {"name": "sd_lower", "group": "Ungrouped variables", "definition": "sqrt(var_lower)", "description": "", "templateType": "anything", "can_override": false}, "sd_upper": {"name": "sd_upper", "group": "Ungrouped variables", "definition": "sqrt(var_upper)", "description": "", "templateType": "anything", "can_override": false}, "alpha": {"name": "alpha", "group": "Ungrouped variables", "definition": "if(confl=90,0.1,if(confl=95,0.05,0.01))", "description": "", "templateType": "anything", "can_override": false}, "confl": {"name": "confl", "group": "Ungrouped variables", "definition": "random(90,95,99)", "description": "", "templateType": "anything", "can_override": false}, "x": {"name": "x", "group": "Ungrouped variables", "definition": "repeat(random(230..240)*0.1,n)", "description": "", "templateType": "anything", "can_override": false}, "lower_round": {"name": "lower_round", "group": "Ungrouped variables", "definition": "precround(sd_lower,3)", "description": "", "templateType": "anything", "can_override": false}, "upper_round": {"name": "upper_round", "group": "Ungrouped variables", "definition": "precround(sd_upper,3)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["n", "xbar", "sd", "chi_lower", "chi_upper", "var_lower", "var", "var_upper", "sd_lower", "sd_upper", "alpha", "confl", "x", "lower_round", "upper_round"], "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>Calculate a {confl}% confidence interval $(a,b)$ for the population standard deviation:</p>\n<p>$a=\\;$[[0]] hours&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; $b=\\;$[[1]]&nbsp;hours</p>\n<p>Enter both to&nbsp;1 decimal place.</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": "sd_lower-0.1", "maxValue": "sd_lower+0.1", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "dp", "precision": "3", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true, "showPrecisionHint": 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": "sd_upper-0.1", "maxValue": "sd_upper+0.1", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "dp", "precision": "3", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question", "contributors": [{"name": "Aamir Khan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4537/"}, {"name": "Laura Midgley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18287/"}]}]}], "contributors": [{"name": "Aamir Khan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4537/"}, {"name": "Laura Midgley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18287/"}]}