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{"name": "2012 2013 CBA3_1", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"tol": {"templateType": "anything", "group": "Ungrouped variables", "definition": "0.01", "name": "tol", "description": ""}, "ulim": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(tulim,2)", "name": "ulim", "description": ""}, "z": {"templateType": "anything", "group": "Ungrouped variables", "definition": "switch(per=95,1.96,per=99,2.58,3.29)", "name": "z", "description": ""}, "llim": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(tllim,2)", "name": "llim", "description": ""}, "sd": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(5..12)", "name": "sd", "description": ""}, "tllim": {"templateType": "anything", "group": "Ungrouped variables", "definition": "r11-z*sqrt(sd^2/n)", "name": "tllim", "description": ""}, "mu": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(150..250)", "name": "mu", "description": ""}, "tulim": {"templateType": "anything", "group": "Ungrouped variables", "definition": "r11+z*sqrt(sd^2/n)", "name": "tulim", "description": ""}, "n": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(25..100#5)", "name": "n", "description": ""}, "inf": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(n/sd^2,2)", "name": "inf", "description": ""}, "per": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(95,99,99.9)", "name": "per", "description": ""}, "r11": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(mean(repeat(normalsample(mu,sd),n)),1)", "name": "r11", "description": ""}}, "ungrouped_variables": ["r11", "tulim", "per", "n", "mu", "tol", "sd", "ulim", "inf", "z", "llim", "tllim"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "2012 2013 CBA3_1", "showQuestionGroupNames": false, "functions": {}, "parts": [{"scripts": {}, "gaps": [{"showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "showCorrectAnswer": true, "minValue": "r11", "correctAnswerFraction": false, "marks": 1, "maxValue": "r11"}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "<p>Enter the m.l.e. $\\hat{\\mu}$ for $\\mu$ here:</p>\n      <p>$\\hat{\\mu}=\\;$?[[0]]</p>", "marks": 0}, {"scripts": {}, "gaps": [{"showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "showCorrectAnswer": true, "minValue": "inf-tol", "correctAnswerFraction": false, "marks": 1, "maxValue": "inf+tol"}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "<p>Calculate the expected information $I(\\mu)$:</p>\n      <p>$I(\\mu)=\\;$[[0]] (to 2 decimal places).</p>", "marks": 0}, {"scripts": {}, "gaps": [{"showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "showCorrectAnswer": true, "minValue": "llim-tol", "correctAnswerFraction": false, "marks": 1, "maxValue": "llim+tol"}, {"showPrecisionHint": false, "scripts": {}, "allowFractions": false, "type": "numberentry", "showCorrectAnswer": true, "minValue": "ulim-tol", "correctAnswerFraction": false, "marks": 1, "maxValue": "ulim+tol"}], "type": "gapfill", "showCorrectAnswer": true, "prompt": "<p>Calculate a $\\var{per}$% confidence $(a,b)$ interval for $\\mu$:</p>\n      <p>$a=\\;$[[0]]</p>\n      <p>$b=\\;$[[1]]</p>", "marks": 0}], "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>$X_1,\\;X_2,\\;\\dots, X_n$ is a random sample from $\\operatorname{N}(\\mu,\\var{sd}^2)$.</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"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "<p><strong>27/01/2013:</strong></p>\n    <p>First draft completed.</p>", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>For a sample of size n from a normal distribution and given the mean of the sample and the standard deviation of the distribution, find the MLE for the mean. Also the expected information and a confidence interval for the mean.</p>"}, "advice": "<p>a)</p>\n  <p>The maximum likelihood estimator (m.l.e) is, in this case, the sample mean i.e. $\\hat{\\mu}=\\var{r11}$.</p>\n  <p>b)</p>\n  <p>The expected information in this case is $\\displaystyle \\frac{n}{\\sigma^2}$ where $\\sigma=\\var{sd}$ is the standard deviation of the sampled normal distribution.</p>\n  <p>Hence $\\displaystyle I(\\mu)=\\frac{\\var{n}}{\\var{sd^2}}=\\var{inf}$ to 2 decimal places.</p>\n  <p>c)</p>\n  <p>The $\\var{per}$% confidence interval in this case is given by $(a,b)$ where:</p>\n  <p>\\[a=\\bar{x}-z\\sqrt{\\frac{\\sigma^2}{n}},\\;\\;b=\\bar{x}+z\\sqrt{\\frac{\\sigma^2}{n}},\\;\\;\\;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 the mean 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/"}]}