// Numbas version: exam_results_page_options
{"name": "Find a confidence interval for the population mean with variance unknown. Sample mean and sample standard deviation given. ", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Find a confidence interval for the population mean with variance unknown. Sample mean and sample standard deviation given. ", "tags": [], "metadata": {"description": "<p>Finding the confidence interval at either 90%, 95% or 99% for the mean given the mean and standard deviation of a sample. The population variance is not given and so the t test has to be used. Various scenarios are included.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>The management of {sc[s]} wants to {dothis[s]}.</p>\n<p>A random sample of {spec} $\\var{n}$ {t[s]} gave a mean and standard deviation of {p}$\\var{m[s]}$ {units} and {p}$\\var{sd[s]}${units} respectively.</p>", "advice": "<p>1.</p>\n<p>The population variance is unknown. So we have to use the t tables to find the confidence interval.</p>\n<p>2.</p>\n<p>We now calculate the $\\var{confl}$ confidence interval:</p>\n<p>As we have $\\var{n}-1=\\var{n-1}$ degrees of freedom, the interval is given by:</p>\n<p>\\[ \\var{m[s]} \\pm t_{\\var{n-1}}\\sqrt{\\frac{\\var{sd[s]}^2}{\\var{n}}}\\]</p>\n<p>Looking up the t tables for $\\var{confl}$% we see that $t_{\\var{n-1}}=\\var{invt}$ to 3 decimal places.</p>\n<p>Hence:</p>\n<p>Lower value of the confidence interval $=\\;\\displaystyle \\var{m[s]} -\\var{invt} \\sqrt{\\frac{\\var{sd[s]} ^ 2} {\\var{n}}} =\\var{p} \\var{lci}${units} to 2 decimal places.</p>\n<p>Upper value of the confidence interval $=\\;\\displaystyle \\var{m[s]} +\\var{invt} \\sqrt{\\frac{\\var{sd[s]} ^ 2} {\\var{n}}} = \\var{p}\\var{uci}${units} to 2 decimal places.</p>", "rulesets": {}, "extensions": ["stats"], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"tlci": {"name": "tlci", "group": "Ungrouped variables", "definition": "m[s]-invt*standard_error", "description": "", "templateType": "anything", "can_override": false}, "dothis": {"name": "dothis", "group": "Ungrouped variables", "definition": "\n    [\"estimate the mean cost per room of repairing damage caused by its customers during a bank holiday weekend\",\n      \"estimate the mean monthly sales of all of its outlets\",\n      \"estimate the mean hours worked per week of all its employees\",\n      \"estimate the mean cost of a ticket on its most popular route\",\n      \"estimate the mean weight of \"+sc5ch+\" inside bars of its most popular product\",\n      \"estimate the mean amount of saturated fat in its \"+ sc6ch]\n      \n      \n      \n    ", "description": "", "templateType": "anything", "can_override": false}, "invt": {"name": "invt", "group": "Ungrouped variables", "definition": "precround(tinvt,3)", "description": "", "templateType": "anything", "can_override": false}, "sc2ch": {"name": "sc2ch", "group": "Ungrouped variables", "definition": "random(\"local\",\"national\")", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(10..30)", "description": "", "templateType": "anything", "can_override": false}, "sc1ch": {"name": "sc1ch", "group": "Ungrouped variables", "definition": "random(\"hotels\",\"motels\", \"Bed and Breakfasts\",\"budget hotels\")", "description": "", "templateType": "anything", "can_override": false}, "s": {"name": "s", "group": "Ungrouped variables", "definition": "random(0..abs(sc)-1)", "description": "", "templateType": "anything", "can_override": false}, "sc6ch": {"name": "sc6ch", "group": "Ungrouped variables", "definition": "random(\"Cornish pasties\",\"sausage rolls\",\"chicken pies\",\"minced beef pasties\")", "description": "", "templateType": "anything", "can_override": false}, "t": {"name": "t", "group": "Ungrouped variables", "definition": "\n    [\"vacated rooms was inspected by the management and this\",\n      \"clothing outlets \",\n      \"workers\",\n      \"tickets \",\n      \"chocolate bars \",\n      sc6ch+\" \"]\n      \n    ", "description": "", "templateType": "anything", "can_override": false}, "sc": {"name": "sc", "group": "Ungrouped variables", "definition": "\n    [\"a national chain of \"+ sc1ch,\n      \"a \"+sc2ch + \" chain of clothing shops \",\n      \" a large factory \",\n      \" a regional passenger airline in  \"+sc4ch,\n      \"Choclastic!, a company producing a variety of chocolate bars \",\n      \" a large bakery \"]\n    ", "description": "", "templateType": "anything", "can_override": false}, "sc5ch": {"name": "sc5ch", "group": "Ungrouped variables", "definition": "random(\"caramel\",\"Turkish delight\",\"honeycomb\",\"nuts\")", "description": "", "templateType": "anything", "can_override": false}, "units": {"name": "units", "group": "Ungrouped variables", "definition": "switch(s=2,\"hours\",s=4,\"g\",s=5,\"g per 100g\",\" \")", "description": "", "templateType": "anything", "can_override": false}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "\n    [random(30..100#0.01),\n      random(40000..80000#500),\n      random(34..48#0.01),\n      random(100..300#0.01),\n      random(10..20#0.01),\n      random(3.5..6#0.01)]\n      \n    ", "description": "", "templateType": "anything", "can_override": false}, "lci": {"name": "lci", "group": "Ungrouped variables", "definition": "precround(tlci,2)", "description": "", "templateType": "anything", "can_override": false}, "tuci": {"name": "tuci", "group": "Ungrouped variables", "definition": "m[s]+invt*standard_error", "description": "", "templateType": "anything", "can_override": false}, "tinvt": {"name": "tinvt", "group": "Ungrouped variables", "definition": "studenttinv((confl+100)/200,n-1)", "description": "", "templateType": "anything", "can_override": false}, "confl": {"name": "confl", "group": "Ungrouped variables", "definition": "random(90,95,99)", "description": "", "templateType": "anything", "can_override": false}, "spec": {"name": "spec", "group": "Ungrouped variables", "definition": "if(s=2,\"the timecards of \", \" \")", "description": "", "templateType": "anything", "can_override": false}, "sc4ch": {"name": "sc4ch", "group": "Ungrouped variables", "definition": "random(\"the Caribbean\",\"the Mediterranean\",\"North East Ireland\",\"South West Ireand\",\"California\")", "description": "", "templateType": "anything", "can_override": false}, "p": {"name": "p", "group": "Ungrouped variables", "definition": "switch(s=0 or s=1 or s=3,'\u00a3',' ')", "description": "", "templateType": "anything", "can_override": false}, "uci": {"name": "uci", "group": "Ungrouped variables", "definition": "precround(tuci,2)", "description": "", "templateType": "anything", "can_override": false}, "sd": {"name": "sd", "group": "Ungrouped variables", "definition": "\n    [random(3..10#0.01),\n      random(500..4000#0.5),\n      random(2..5#0.01),\n      random(10..40#0.01),\n      random(1..3#0.01),\n      random(0.5..1#0.01)]\n    ", "description": "", "templateType": "anything", "can_override": false}, "standard_error": {"name": "standard_error", "group": "Ungrouped variables", "definition": "precround(sqrt(sd[s]^2/n),3)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["m", "uci", "invt", "units", "lci", "spec", "sc2ch", "sc1ch", "tinvt", "confl", "tuci", "dothis", "sc4ch", "sc6ch", "n", "p", "s", "tlci", "t", "sc", "sc5ch", "sd", "standard_error"], "variable_groups": [], "functions": {"pounds": {"parameters": [["n", "number"]], "type": "number", "language": "javascript", "definition": "return Numbas.util.currency(n,'\u00a3','p');"}}, "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>Is the population variance known or unknown?</p>\n<p>[[0]]</p>\n<p></p>\n<p>Which distribution does the sample mean follow?</p>\n<p>[[1]]</p>\n<p></p>\n<p>The standard error of the mean is estimated to be:</p>\n<p>[[4]]</p>\n<p>Enter to 3 decimal places.</p>\n<p></p>\n<p>For a $\\var{confl}$% confidence interval, the critical t- value is</p>\n<p>[[5]]</p>\n<p>Enter to 3 decimal places.</p>\n<p></p>\n<p>Calculate a $\\var{confl}$% confidence interval $(a,b)$ for the population mean:</p>\n<p>$a=\\;${p}[[2]]{units} &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;$b=\\;${p}[[3]]{units}</p>\n<p>Enter both to 2 decimal places.</p>", "gaps": [{"type": "1_n_2", 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