// Numbas version: finer_feedback_settings
{"name": "SM10 Estimated Mean", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "SM10 Estimated Mean", "tags": [], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Tina needs to take a train to work.</p>\n<p>The table below shows information about how long the train is delayed for on each day.</p>\n<p>The policy of the train company is that if your train is delayed for longer than 30 minutes, you will recieve a 50% discount on your train ticket.</p>\n<table width=\"294\" height=\"161\" style=\"height: 108px;\">\n<tbody>\n<tr style=\"height: 18px;\">\n<td style=\"width: 130.5px; height: 18px;\"><strong>Delay Time</strong></td>\n<td style=\"width: 130.5px; height: 18px;\"><strong>Frequency&nbsp;</strong></td>\n</tr>\n<tr style=\"height: 18px;\">\n<td style=\"width: 130.5px; height: 18px;\">0 &lt; t $\\leq$ 10</td>\n<td style=\"width: 130.5px; height: 18px;\">{f1}</td>\n</tr>\n<tr style=\"height: 18px;\">\n<td style=\"width: 130.5px; height: 18px;\">10 &lt; t $\\leq$ 20</td>\n<td style=\"width: 130.5px; height: 18px;\">{f2}</td>\n</tr>\n<tr style=\"height: 18px;\">\n<td style=\"width: 130.5px; height: 18px;\">20 &lt; t $\\leq$ 30</td>\n<td style=\"width: 130.5px; height: 18px;\">{f3}</td>\n</tr>\n<tr style=\"height: 18px;\">\n<td style=\"width: 130.5px; height: 18px;\">30 &lt; t $\\leq$ 40</td>\n<td style=\"width: 130.5px; height: 18px;\">{f4}</td>\n</tr>\n<tr style=\"height: 18px;\">\n<td style=\"width: 130.5px; height: 18px;\">40 &lt; t $\\leq$ 50</td>\n<td style=\"width: 130.5px; height: 18px;\">{f5}</td>\n</tr>\n</tbody>\n</table>", "advice": "<p><strong>a)</strong></p>\n<p>To find the estimated mean we must work out the midpoint of the delay time intervals and multiply this by the frequency to create the $fx$ column on the table.</p>\n<table width=\"294\" height=\"361\" style=\"width: 747px;\">\n<tbody>\n<tr>\n<td style=\"width: 154.688px;\"><strong>Delay Time</strong></td>\n<td style=\"width: 167px;\"><strong>Frequency&nbsp;</strong></td>\n<td style=\"width: 119.891px;\"><strong>Midpoint</strong></td>\n<td style=\"width: 240.422px;\"><strong>Frequency $\\times$ Midpoint, $fx$</strong></td>\n</tr>\n<tr>\n<td style=\"width: 154.688px;\">0 &lt; t $\\leq$ 10</td>\n<td style=\"width: 167px;\">{f1}</td>\n<td style=\"width: 119.891px;\">5</td>\n<td style=\"width: 240.422px;\">= 5$\\times\\var{f1} = \\var{f1m}$</td>\n</tr>\n<tr>\n<td style=\"width: 154.688px;\">10 &lt; t $\\leq$ 20</td>\n<td style=\"width: 167px;\">{f2}</td>\n<td style=\"width: 119.891px;\">15</td>\n<td style=\"width: 240.422px;\">= 15$\\times\\var{f2} = \\var{f2m}$</td>\n</tr>\n<tr>\n<td style=\"width: 154.688px;\">20 &lt; t $\\leq$ 30</td>\n<td style=\"width: 167px;\">{f3}</td>\n<td style=\"width: 119.891px;\">25</td>\n<td style=\"width: 240.422px;\">= 25$\\times\\var{f3} = \\var{f3m}$</td>\n</tr>\n<tr>\n<td style=\"width: 154.688px;\">30 &lt; t $\\leq$ 40</td>\n<td style=\"width: 167px;\">{f4}</td>\n<td style=\"width: 119.891px;\">35</td>\n<td style=\"width: 240.422px;\">= 35$\\times\\var{f4} = \\var{f4m}$</td>\n</tr>\n<tr>\n<td style=\"width: 154.688px;\">40 &lt; t $\\leq$ 50</td>\n<td style=\"width: 167px;\">{f5}</td>\n<td style=\"width: 119.891px;\">45</td>\n<td style=\"width: 240.422px;\">= 45$\\times\\var{f5} = \\var{f5m}$</td>\n</tr>\n</tbody>\n</table>\n<p>Now to find the mean we calculate the sum of the $fx$ column and divide it by the sum of the frequencies:</p>\n<p>Estimated Mean = $\\frac{\\var{f1m}+\\var{f2m}+\\var{f3m}+\\var{f4m}+\\var{f5m}}{5+15+25+35+45} = \\frac{\\var{freqmid}}{\\var{sum}} = \\var{meana}.$</p>\n<p>Hence the estimated mean delay time is $\\var{mean}$ to two decimal places.</p>\n<p></p>\n<p><strong>b)</strong></p>\n<p>To work out what percentage of delays that were over 30 minutes we first need to calculate the number of days where the train was delayed for over 30 minutes.&nbsp;</p>\n<p>The number of delays that over 30 minutes is $\\var{f4}+\\var{f5} = \\var{f4+f5}$.</p>\n<p>We divide this by the total number of deliveries to find the fraction of delays that were over 30 minutes and then multiply this number by 100 to find the percentage:</p>\n<p>$\\frac{\\var{f4+f5}}{\\var{f1}+\\var{f2}+\\var{f3}+\\var{f4}+\\var{f5}} \\times 100\\% = \\frac{\\var{f4+f5}}{\\var{sum}} \\times 100\\% = \\var{percentagea} \\times 100\\%$.</p>\n<p>Hence, the percentage of delays that were over 30 minutes is $\\var{percentage}\\%$ to two decimal places.</p>\n<p></p>\n<p><strong>c)</strong></p>\n<p>The percentage of delays that take over 40 minutes is:</p>\n<p>$\\frac{\\var{f5}}{\\var{f1}+\\var{f2}+\\var{f3}+\\var{f4}+\\var{f5}} \\times 100\\% = \\frac{\\var{f5}}{\\var{sum}} \\times 100\\% = \\var{percent40} \\times 100\\%$, to one decimal place.</p>\n<p>So, only $\\var{percent40}\\%$ of the time, delays were longer than 40 minutes so the manager would probably wanto the policy changed to 40 minutes.</p>\n<p></p>\n<p style=\"text-align: center;\"><a href=\"https://www.sheffield.ac.uk/mash/stats-resources/first-steps#mean,-median-and-mode\" target=\"_blank\" rel=\"noopener\">Use this link to find some resources which will help you revise this topic.</a></p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"f1": {"name": "f1", "group": "Ungrouped variables", "definition": "random(1 .. 20#1)", "description": "<p>Frequency 1</p>", "templateType": "randrange", "can_override": false}, "f2": {"name": "f2", "group": "Ungrouped variables", "definition": "random(1 .. 20#1)", "description": "<p>Frequency 2</p>", "templateType": "randrange", 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mins</p>", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["f1", "f2", "f3", "f4", "f5", "meana", "sum", "f1m", "f2m", "f3m", "f4m", "f5m", "freqmid", "mean", "f30", "percentagea", "percentage", "percent40"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "numberentry", "useCustomName": true, "customName": "a)", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Using a calculator, calculate an estimate for the mean delay time. Give your answer to two decimal places.</p>", "minValue": "mean", "maxValue": "mean", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "b)", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>What percentage of delays were over 30 minutes? Give your answer to one decimal place.</p>", "minValue": "percentage", "maxValue": "percentage", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "jme", "useCustomName": true, "customName": "c)", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>The train company propose that they should only offer the 50% discount if the train is delayed for over 40 minutes.</p>\n<p>If you were the manager of the train company would you agree? 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