// Numbas version: exam_results_page_options
{"name": "2012 2013 CBA3_3", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variablesTest": {"condition": "", "maxRuns": 100}, "variables": {"tol": {"group": "Ungrouped variables", "templateType": "anything", "definition": "0.01", "name": "tol", "description": ""}, "ulim": {"group": "Ungrouped variables", "templateType": "anything", "definition": "precround(tulim,2)", "name": "ulim", "description": ""}, "z": {"group": "Ungrouped variables", "templateType": "anything", "definition": "switch(per=95,1.96,per=99,2.58,3.29)", "name": "z", "description": ""}, "llim": {"group": "Ungrouped variables", "templateType": "anything", "definition": "precround(tllim,2)", "name": "llim", "description": ""}, "tllim": {"group": "Ungrouped variables", "templateType": "anything", "definition": "1/r11-z*sqrt(1/(n*r11^2))", "name": "tllim", "description": ""}, "mu": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(4..10)", "name": "mu", "description": ""}, "tulim": {"group": "Ungrouped variables", "templateType": "anything", "definition": "1/r11+z*sqrt(1/(n*r11^2))", "name": "tulim", "description": ""}, "n": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(25..100#5)", "name": "n", "description": ""}, "inf": {"group": "Ungrouped variables", "templateType": "anything", "definition": "precround(n*r11^2,2)", "name": "inf", "description": ""}, "per": {"group": "Ungrouped variables", "templateType": "anything", "definition": "random(95,99,99.9)", "name": "per", "description": ""}, "r11": {"group": "Ungrouped variables", "templateType": "anything", "definition": "precround(mean(repeat(exponentialsample(mu),n)),1)", "name": "r11", "description": ""}, "mle": {"group": "Ungrouped variables", "templateType": "anything", "definition": "precround(1/r11,2)", "name": "mle", "description": ""}}, "ungrouped_variables": ["mle", "r11", "tulim", "per", "n", "mu", "tol", "ulim", "inf", "z", "llim", "tllim"], "rulesets": {}, "name": "2012 2013 CBA3_3", "functions": {}, "variable_groups": [], "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"showPrecisionHint": false, "scripts": {}, "showCorrectAnswer": true, "allowFractions": false, "type": "numberentry", "correctAnswerFraction": false, "minValue": "mle-tol", "maxValue": "mle+tol", "marks": 1}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "<p>Enter the m.l.e. $\\hat{\\lambda}$ for $\\lambda$ here:</p>\n      <p>$\\hat{\\lambda}=\\;$?[[0]]</p>", "marks": 0}, {"scripts": {}, "gaps": [{"showPrecisionHint": false, "scripts": {}, "showCorrectAnswer": true, "allowFractions": false, "type": "numberentry", "correctAnswerFraction": false, "minValue": "inf-tol", "maxValue": "inf+tol", "marks": 1}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "<p>Calculate the expected information $I(\\lambda)$:</p>\n      <p>$I(\\lambda)=\\;$[[0]] (to 2 decimal places).</p>", "marks": 0}, {"scripts": {}, "gaps": [{"showPrecisionHint": false, "scripts": {}, "showCorrectAnswer": true, "allowFractions": false, "type": "numberentry", "correctAnswerFraction": false, "minValue": "llim-tol", "maxValue": "llim+tol", "marks": 1}, {"showPrecisionHint": false, "scripts": {}, "showCorrectAnswer": true, "allowFractions": false, "type": "numberentry", "correctAnswerFraction": false, "minValue": "ulim-tol", "maxValue": "ulim+tol", "marks": 1}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "<p>Calculate a $\\var{per}$% confidence $(a,b)$ interval for $\\lambda$:</p>\n      <p>$a=\\;$[[0]]</p>\n      <p>$b=\\;$[[1]]</p>", "marks": 0}], "statement": "<p>$X_1,\\;X_2,\\;\\dots, X_n$ is a random sample from $\\operatorname{Exponential}(\\lambda)$.</p>\n  <p>&nbsp;Let $\\bar{x}$ denote the mean of this sample and for this exercise we have $n=\\var{n},\\; \\bar{x}=\\var{r11}$</p>", "tags": ["MAS2302", "MLE", "Normal distribution", "checked2015", "confidence interval", "expected information", "maximum likelihood estimator", "mean ", "mle", "normal distribution", "random sample", "sample mean"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "\n    \t\t    \t\t<p><strong>27/01/2013:</strong></p>\n    \t\t    \t\t    <p>First draft completed.</p>\n    \t\t    \n    \t\t", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>For a sample of size n from an Exponential distribution $\\operatorname{Exponential}(\\lambda)$ &nbsp;and given the mean of the sample, find the MLE for $\\lambda$. Also find the expected information and a confidence interval for $\\lambda$.</p>"}, "advice": "<p>a)</p>\n  <p>The maximum likelihood estimator (MLE) for $\\lambda$ is given by</p>\n  <p>\\[\\operatorname{MLE} = \\frac{1}{\\bar{x}}=\\frac{1}{\\var{r11}}=\\var{mle}\\] to 2 decimal places.</p>\n  <p>b)</p>\n  <p>The expected information in this case is:</p>\n  <p>$\\displaystyle \\operatorname{I}(\\lambda)=\\frac{n}{\\hat{\\lambda}^2}=n\\bar{x}^2=\\var{n}\\times\\var{r11}^2=\\var{inf}$ to 2 decimal places.</p>\n  <p>c)</p>\n  <p>The $\\var{per}$% confidence interval for $\\lambda$ in this case is given by $(a,b)$ where:</p>\n  <p>\\[a=\\frac{1}{\\bar{x}}-z\\sqrt{\\frac{1}{n\\bar{x}^2}},\\;\\;b=\\frac{1}{\\bar{x}}+z\\sqrt{\\frac{1}{n\\bar{x}^2}},\\;\\;\\;z=\\var{z}\\]&nbsp;</p>\n  <p>Calculating to 2 decimal places gives:</p>\n  <p>$a=\\var{llim},\\;\\;\\;b=\\var{ulim}$.</p>\n  <p>Hence a $\\var{per}$% confidence interval for $\\lambda$ is given by $(\\var{llim},\\var{ulim})$.</p>", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}