// Numbas version: finer_feedback_settings
{"name": "SM11 Linear Interpolation", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "SM11 Linear Interpolation", "tags": [], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>This frequence table stores data about the lengths of some plants:</p>\n<table style=\"width: 285px;\">\n<tbody>\n<tr>\n<td style=\"width: 163.625px; text-align: left;\"><strong>Length ($x$ cm)</strong></td>\n<td style=\"width: 88.375px; text-align: left;\"><strong>Frequency</strong></td>\n</tr>\n<tr>\n<td style=\"width: 163.625px; text-align: left;\">10 &lt; $x$ $\\leq$ 20</td>\n<td style=\"width: 88.375px; text-align: left;\">{f1}</td>\n</tr>\n<tr>\n<td style=\"width: 163.625px; text-align: left;\">20 &lt; $x$ $\\leq$ 30</td>\n<td style=\"width: 88.375px; text-align: left;\">{f2}</td>\n</tr>\n<tr>\n<td style=\"width: 163.625px; text-align: left;\">30 &lt; $x$ $\\leq$ 40</td>\n<td style=\"width: 88.375px; text-align: left;\">{f3}</td>\n</tr>\n<tr>\n<td style=\"width: 163.625px; text-align: left;\">40 &lt; $x$ $\\leq$ 50</td>\n<td style=\"width: 88.375px; text-align: left;\">{f4}</td>\n</tr>\n</tbody>\n</table>", "advice": "<p><strong>a)</strong></p>\n<p>To find the median we must find the middle value. To do this we sum the frequencies and divide the result by 2:</p>\n<p>$\\frac{\\var{f1}+\\var{f2}+\\var{f3}+\\var{f4}}{2} = \\frac{\\var{total}}{2} = \\var{middle1}$.</p>\n<p>Hence we must find value number $\\var{middle1}$, we can do this by totalling the cumulative frequency in the table below.</p>\n<table style=\"height: 235px;\" width=\"966\">\n<tbody>\n<tr style=\"height: 46px;\">\n<td style=\"width: 176.25px; height: 14px; text-align: left;\">\n<p><strong>Length $x$ cm</strong></p>\n</td>\n<td style=\"width: 165.375px; height: 14px; text-align: left;\"><strong>Frequency</strong></td>\n<td style=\"width: 575.375px; height: 14px; text-align: left;\"><strong>Cumulative Frequency</strong></td>\n</tr>\n<tr style=\"height: 18px;\">\n<td style=\"width: 176.25px; text-align: left; height: 18px;\"><strong>1.</strong> 10 &lt; $x$ $\\leq$ 20</td>\n<td style=\"width: 165.375px; height: 18px; text-align: left;\">{f1}</td>\n<td style=\"width: 575.375px; height: 18px; text-align: left;\">$\\var{f1}$</td>\n</tr>\n<tr style=\"height: 18px;\">\n<td style=\"width: 176.25px; text-align: left; height: 18px;\"><strong>2.</strong> 20 &lt; $x$ $\\leq$ 30</td>\n<td style=\"width: 165.375px; height: 18px; text-align: left;\">{f2}</td>\n<td style=\"width: 575.375px; height: 18px; text-align: left;\">$\\var{f1}+\\var{f2}=\\var{cf2}$</td>\n</tr>\n<tr style=\"height: 18px;\">\n<td style=\"width: 176.25px; text-align: left; height: 18px;\"><strong>3.</strong> 30 &lt; $x$ $\\leq$ 40&nbsp;</td>\n<td style=\"width: 165.375px; height: 18px; text-align: left;\">{f3}</td>\n<td style=\"width: 575.375px; height: 18px; text-align: left;\">$\\var{f1}+\\var{f2} +\\var{f3} =\\var{cf3}$</td>\n</tr>\n<tr style=\"height: 10px;\">\n<td style=\"width: 176.25px; text-align: left; height: 10px;\"><strong>4.</strong> 40 &lt; $x$ $\\leq$ 50</td>\n<td style=\"width: 165.375px; height: 10px; text-align: left;\">{f4}</td>\n<td style=\"width: 575.375px; height: 10px; text-align: left;\">$\\var{f1}+\\var{f2} +\\var{f3} + \\var{f4} =\\var{total}$</td>\n</tr>\n</tbody>\n</table>\n<p>You can see value number $\\var{middle1}$ lies in interval number $\\var{interval1}$ hence this is the median class interval for the time taken.</p>\n<p>We can estimate the median by using interpolation:</p>\n<p>$$<br/>\\text{Estimate of median} = \\text{class start value} + \\frac{\\text{position in class}}{\\text{frequency in class}}\\times \\text{class width}<br/>$$</p>\n<p>In this case this is $\\var{lower1}+\\frac{\\var{middle1}-\\var{cf11}}{\\var{fint1}}\\times 10 =\\var{lower1}+\\var{interpolation1}=\\var{median1}.$</p>\n<p>(Note: that sometimes it is convenient to use $(\\var{total} + 1)/2 = \\var{middle2}$ as the way to work out the median position. The distinction does not really matter as this is an estimated value and often the decision is made based on which is the most convenient to calculate). This question has been written so that both answers will be marked correctly.</p>\n<p><strong>b)</strong></p>\n<p>To calculate the $i^{th}$ quartile ($Q_i$) we must use the following formula:</p>\n<p>$Q_i = l + \\frac{\\frac{iN}{4}-F}{f}\\times h,$</p>\n<p>where:</p>\n<p>$l$ = lower limit of the interval in which $Q_i$ lies;</p>\n<p>$N$ = Total number of observations;</p>\n<p>$F$ = Cumulative frequency of class previous to the $i^{th}$ quartile class;<br/><br/>$f$ = Frequency of $i^{th}$ quartile class.</p>\n<p>Since we want to calculate the lower quartile, $i=1$.</p>\n<p>If we calculate $\\frac{iN}{4} = \\frac{\\var{total}}{4} = \\var{quarter}$, we can see in the table from part <strong>a)</strong> that the lower quartile will be in interval number $\\var{intervallq}$. This means that $F = \\var{fprev}$ and $f = \\var{frequency}$.</p>\n<p>Hence,&nbsp;</p>\n<p>$Q_1 = \\var{lowerlq} + \\frac{\\var{quarter} - \\var{Fprev}}{\\var{frequency}} \\times 10 = \\var{lowerlq} + \\frac{\\var{quarter-Fprev}}{\\var{frequency}} \\times 10 = \\var{lowerlq} + \\var{((quarter-Fprev)/frequency)*10} = \\var{lq}.$&nbsp;<br/><br/>So, The lower quartile is $\\var{lqround}$ to two decimal places.</p>\n<p></p>\n<p><strong>c)&nbsp;</strong></p>\n<p>To calculate the upper quartile we use the same formula as in part <strong>b) </strong>but this time $i=3$. So, $\\frac{iN}{4} = \\frac{\\var{3*total}}{4} = \\var{3*quarter}$, we can see in the table from part <strong>a)</strong> that the lower quartile will be in interval number $\\var{intervaluq}$. This means that $F = \\var{ufprev}$ and $f = \\var{ufrequency}$.</p>\n<p>Hence,&nbsp;</p>\n<p>$Q_3 = \\var{upperlq} + \\frac{\\var{3*quarter} - \\var{uFprev}}{\\var{ufrequency}} \\times 10 = \\var{upperlq} + \\frac{\\var{3*quarter-uFprev}}{\\var{ufrequency}} \\times 10 = \\var{upperlq} + \\var{((3*quarter-uFprev)/ufrequency)*10} = \\var{uq}.$&nbsp;<br/><br/>So, The lower quartile is $\\var{uqround}$ to two decimal places.</p>\n<p><strong>d)</strong></p>\n<p>To calculate the interquartile range we subtract the lower quartile from the upper quartile.</p>\n<p>Hence,&nbsp;</p>\n<p>$IQR = Q_3 - Q_1 = \\var{uq}-\\var{lq}=\\var{iqr}$.</p>\n<p>So the interquartile range rounded to two decimal places is $\\var{iqrround}$.</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": 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false}, "total": {"name": "total", "group": "Ungrouped variables", "definition": "f1+f2+f3+f4", "description": "<p>Sum of frequencies&nbsp;</p>", "templateType": "anything", "can_override": false}, "middle1": {"name": "middle1", "group": "Part a", "definition": "total/2", "description": "<p>Middle Value</p>", "templateType": "anything", "can_override": true}, "cf2": {"name": "cf2", "group": "Ungrouped variables", "definition": "f1+f2", "description": "<p>Cumulative Frequency - first and second frequencies</p>", "templateType": "anything", "can_override": false}, "cf3": {"name": "cf3", "group": "Ungrouped variables", "definition": "f1+f2+f3", "description": "<p>Cumulative freqeuncy - sum of first, second, third frequencies</p>", "templateType": "anything", "can_override": false}, "lower1": {"name": "lower1", "group": "Part a", "definition": "if(interval1=1,10,if(interval1=2,20,if(interval1=3,30,if(interval1=4,40,0))))", "description": "<p>Lower bound of interval which median sits</p>", "templateType": "anything", "can_override": false}, "interval1": {"name": "interval1", "group": "Part a", "definition": "if(middle1<f1,1,if(middle1<cf2,2,if(middle1<cf3,3,if(middle1<=total,4,0))))", "description": "<p>Interval which median sits in</p>", "templateType": "anything", "can_override": false}, "lq": {"name": "lq", "group": "Part b", "definition": "rational(lowerlq + ((quarter - Fprev)/frequency) * 10)", "description": "<p>Formula to find lower quartile</p>", "templateType": "anything", "can_override": false}, "intervallq": {"name": "intervallq", "group": "Part b", "definition": "if(quarter<=f1,1,if(quarter<=cf2,2,if(quarter<=cf3,3,if(quarter<=total,4,0))))", "description": "<p>Interval which lower quartile sits in</p>", "templateType": "anything", "can_override": false}, "quarter": {"name": "quarter", "group": "Part b", "definition": "total/4", "description": "<p>Quarter value</p>", "templateType": "anything", "can_override": false}, "fprev": {"name": "fprev", "group": "Part b", "definition": "if(intervallq=1,0,if(intervallq=2,f1,if(intervallq=3,cf2,if(intervallq=4,cf3,0))))", "description": "<p>Cumulative frequency of class previous to one which lower quartile sits in</p>", "templateType": "anything", "can_override": false}, "frequency": {"name": "frequency", "group": "Part b", "definition": "if(intervallq=1,f1,if(intervallq=2,f2,if(intervallq=3,f3,if(intervallq=4,f4,0))))", "description": "<p>Frequency of class which lower quartile sits in</p>", "templateType": "anything", "can_override": false}, "lqround": {"name": "lqround", "group": "Part b", "definition": "precround(lq,2)", "description": "<p>Lower quartile rounded to 2 decimal places</p>", "templateType": "anything", "can_override": false}, "uqround": {"name": "uqround", "group": "Part c", "definition": "precround(uq,2)", "description": "<p>Upper quartile rounded to 2 decimal places</p>", "templateType": "anything", "can_override": false}, "uq": {"name": "uq", "group": "Part c", "definition": "rational(upperlq + (((total*0.75) - UFprev)/Ufrequency) * 10)", "description": "<p>Upper quartile</p>", "templateType": "anything", "can_override": false}, "ufprev": {"name": "ufprev", "group": "Part c", "definition": "if(intervaluq=1,0,if(intervaluq=2,f1,if(intervaluq=3,cf2,if(intervaluq=4,cf3,0))))", "description": "<p>cumulative frequency of class previous to the one which the upper quartile sits in</p>", "templateType": "anything", "can_override": false}, "ufrequency": {"name": "ufrequency", "group": "Part c", "definition": "if(intervaluq=1,f1,if(intervaluq=2,f2,if(intervaluq=3,f3,if(intervaluq=4,f4,0))))", "description": "<p>Frequency of class upper quartile sits in</p>", "templateType": "anything", "can_override": false}, "intervaluq": {"name": "intervaluq", "group": "Part c", "definition": "if(3*quarter<=f1,1,if(3*quarter<=cf2,2,if(3*quarter<=cf3,3,if(3*quarter<=total,4,0))))", "description": "<p>class which upper quartile sits in</p>", "templateType": "anything", "can_override": false}, "iqr": {"name": "iqr", "group": "Part d", "definition": "rational(uq-lq)", "description": "<p>Interquartile range</p>", "templateType": "anything", "can_override": false}, "lowerlq": {"name": "lowerlq", "group": "Part b", "definition": "if(intervallq=1,10,if(intervallq=2,20,if(intervallq=3,30,if(intervallq=4,40,0))))", "description": "<p>width of class which lower quartile sits in</p>", "templateType": "anything", "can_override": false}, "upperlq": {"name": "upperlq", "group": "Part c", "definition": "if(intervaluq=1,10,if(intervaluq=2,20,if(intervaluq=3,30,if(intervaluq=4,40,0))))", "description": "<p>width of class upper quartile sits in</p>", "templateType": "anything", "can_override": false}, "iqrround": {"name": "iqrround", "group": "Part d", "definition": "precround(iqr,2)", "description": "<p>Round interquartle range to two decimal places</p>", "templateType": "anything", "can_override": false}, "interpolation1": {"name": "interpolation1", "group": "Part a", "definition": "if(middle1<f1,((middle1-0)/f1)*10,if(middle1<cf2,((middle1-f1)/f2)*10,if(middle1<cf3,((middle1-cf2)/f3)*10,if(middle1<=total,((middle1-cf3)/f4)*10,0))))", "description": "", "templateType": "anything", "can_override": false}, "median2": {"name": "median2", "group": "Part a", "definition": "lower2 + interpolation2", "description": "", "templateType": "anything", "can_override": false}, "middle2": {"name": "middle2", "group": "Part a", "definition": "(total+1)/2", "description": "", "templateType": "anything", "can_override": false}, "interval2": {"name": "interval2", "group": "Part a", "definition": "if(middle2<f1,1,if(middle2<cf2,2,if(middle2<cf3,3,if(middle2<=total,4,0))))", "description": "", "templateType": "anything", "can_override": false}, "lower2": {"name": "lower2", "group": "Part a", "definition": "if(interval2=1,10,if(interval2=2,20,if(interval2=3,30,if(interval2=4,40,0))))", "description": "", "templateType": "anything", "can_override": false}, "interpolation2": {"name": "interpolation2", "group": "Part a", "definition": "if(middle2<f1,((middle2-0)/f1)*10,if(middle2<cf2,((middle2-f1)/f2)*10,if(middle2<cf3,((middle2-cf2)/f3)*10,if(middle2<=total,((middle2-cf3)/f4)*10,0))))", "description": "", "templateType": "anything", "can_override": false}, "cf11": {"name": "cf11", "group": "Part a", "definition": "if(middle1<f1,0,if(middle1<cf2,f1,if(middle1<cf3,cf2,if(middle1<=total,cf3,0))))", "description": "", "templateType": "anything", "can_override": false}, "fint1": {"name": "fint1", "group": "Part a", "definition": "if(middle1<f1,f1,if(middle1<cf2,f2,if(middle1<cf3,f3,if(middle1<=total,f4,0))))", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["f1", "f2", "f3", "f4", "total", "cf2", "cf3"], "variable_groups": [{"name": "Part a", "variables": ["median1", "middle1", "interval1", "lower1", "interpolation1", "median2", "middle2", "interval2", "lower2", "interpolation2", "cf11", "fint1"]}, {"name": "Part b", "variables": ["lq", "intervallq", "quarter", "fprev", "frequency", "lqround", "lowerlq"]}, {"name": "Part c", "variables": ["uq", "uqround", "ufprev", "ufrequency", "intervaluq", "upperlq"]}, {"name": "Part d", "variables": ["iqrround", "iqr"]}], "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>Calculate an estimate for the median using linear interpolation.</p>", "minValue": "median1", "maxValue": "median2", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "median1", "precisionType": "dp", "precision": "1", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": 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>Calculate the lower quartile, give your answer to 2 decimal places.</p>", "minValue": "lqround", "maxValue": "lqround", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", 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