// Numbas version: finer_feedback_settings
{"name": "Wyznaczanie miar pozycyjnych", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Wyznaczanie miar pozycyjnych", "tags": [], "metadata": {"description": "<p>Zadaniem studenta jest wyznaczenie miar pozycyjnych dla losowego zbioru 32 warto\u015bci.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Na podstawie danych w tabeli odpowiedz na podane poni\u017cej pytania. W odpowiedziach podaj warto\u015bci&nbsp;<strong>bez zaokr\u0105glania</strong>.</p>\n<table border=\"1\">\n<tbody>\n<tr>\n<td>$\\var{r1[0]}$</td>\n<td>$\\var{r1[1]}$</td>\n<td>$\\var{r1[2]}$</td>\n<td>$\\var{r1[3]}$</td>\n<td>$\\var{r1[4]}$</td>\n<td>$\\var{r1[5]}$</td>\n<td>$\\var{r1[6]}$</td>\n<td>$\\var{r1[7]}$</td>\n<td>$\\var{r1[8]}$</td>\n<td>$\\var{r1[9]}$</td>\n<td>$\\var{r1[10]}$</td>\n<td>$\\var{r1[11]}$</td>\n<td>$\\var{r1[12]}$</td>\n<td>$\\var{r1[13]}$</td>\n<td>$\\var{r1[14]}$</td>\n<td>$\\var{r1[15]}$</td>\n</tr>\n<tr>\n<td>$\\var{r1[16]}$</td>\n<td>$\\var{r1[17]}$</td>\n<td>$\\var{r1[18]}$</td>\n<td>$\\var{r1[19]}$</td>\n<td>$\\var{r1[20]}$</td>\n<td>$\\var{r1[21]}$</td>\n<td>$\\var{r1[22]}$</td>\n<td>$\\var{r1[23]}$</td>\n<td>$\\var{r1[24]}$</td>\n<td>$\\var{r1[25]}$</td>\n<td>$\\var{r1[26]}$</td>\n<td>$\\var{r1[27]}$</td>\n<td>$\\var{r1[28]}$</td>\n<td>$\\var{r1[29]}$</td>\n<td>$\\var{r1[30]}$</td>\n<td>$\\var{r1[31]}$</td>\n</tr>\n</tbody>\n</table>", "advice": "<h4>a)</h4>\n<p>If you sort the data in increasing order you get the following table:</p>\n<table border=\"1\">\n<tbody>\n<tr>\n<td>$\\var{r1[0]}$</td>\n<td>$\\var{r1[1]}$</td>\n<td>$\\var{r1[2]}$</td>\n<td>$\\var{r1[3]}$</td>\n<td>$\\var{r1[4]}$</td>\n<td>$\\var{r1[5]}$</td>\n<td>$\\var{r1[6]}$</td>\n<td>$\\var{r1[7]}$</td>\n<td>$\\var{r1[8]}$</td>\n<td>$\\var{r1[9]}$</td>\n<td>$\\var{r1[10]}$</td>\n<td>$\\var{r1[11]}$</td>\n<td>$\\var{r1[12]}$</td>\n<td>$\\var{r1[13]}$</td>\n<td>$\\var{r1[14]}$</td>\n<td>$\\var{r1[15]}$</td>\n</tr>\n<tr>\n<td>$\\var{r1[16]}$</td>\n<td>$\\var{r1[17]}$</td>\n<td>$\\var{r1[18]}$</td>\n<td>$\\var{r1[19]}$</td>\n<td>$\\var{r1[20]}$</td>\n<td>$\\var{r1[21]}$</td>\n<td>$\\var{r1[22]}$</td>\n<td>$\\var{r1[23]}$</td>\n<td>$\\var{r1[24]}$</td>\n<td>$\\var{r1[25]}$</td>\n<td>$\\var{r1[26]}$</td>\n<td>$\\var{r1[27]}$</td>\n<td>$\\var{r1[28]}$</td>\n<td>$\\var{r1[29]}$</td>\n<td>$\\var{r1[30]}$</td>\n<td>$\\var{r1[31]}$</td>\n</tr>\n</tbody>\n</table>\n<p>Denote the ordered data by $x_j$, thus $x_{10}=\\var{r1[9]}$ for example.</p>\n<p><strong>Minimum value:</strong>&nbsp;The minimum value is $x_1=\\var{r1[0]}$.</p>\n<p><strong>Lower Quartile:</strong>&nbsp;As there is an even number of values, the Lower Quartile will lie between two values. Its position is calculated by finding</p>\n<p>\\[\\frac{n+1}{4}=\\frac{\\var{n+1}}{4}=8\\frac{1}{4}\\]</p>\n<p>Hence the Lower Quartile lies between the 8th and 9th entries in the ordered table, so it is:</p>\n<p>\\[0.75\\times x_8+0.25\\times x_9 = 0.75\\times\\var{r1[7]}+0.25\\times \\var{r1[8]}=\\var{lquartile}\\]</p>\n<p><strong>Median:</strong>&nbsp;The position of the median in the table is given by</p>\n<p>\\[ \\frac{2(n+1)}{4} = \\frac{\\var{2*(n+1)}}{4} = 16 \\frac{1}{2}\\]</p>\n<p>The median lies between the 16th and 17th entries in the ordered table and is given by:</p>\n<p>\\[0.5\\times x_{16}+0.5\\times x_{17} = 0.5\\times\\var{r1[15]}+0.5\\times \\var{r1[16]}=\\var{median}\\]</p>\n<p><strong>Upper Quartile:</strong> As there is an even number of values, the Upper Quartile will lie between two values. Its position is calculated by finding</p>\n<p>\\[\\frac{3(n+1)}{4}=\\frac{\\var{3*(n+1)}}{4}=24\\frac{3}{4}\\]</p>\n<p>Hence the Upper Quartile lies between the 24th and 25th entries in the ordered table.</p>\n<p>We find it is \\[0.25\\times x_{24}+0.75\\times x_{25} = 0.25\\times\\var{r1[23]}+0.75\\times \\var{r1[24]}=\\var{uquartile}\\]</p>\n<p><strong>Maximum value:&nbsp;</strong>The maximum value is $x_{32}=\\var{r1[31]}$</p>\n<p><strong>b)</strong></p>\n<p>The range is defined to be<br/><br/>Range = Maximum &ndash; Minimum<br/><br/>and so in this case we have:<br/><br/>Range = $\\var{r1[31]}-\\var{r1[0]}=\\var{range}$.</p>\n<h4>c)</h4>\n<p>The interquartile range is defined to be</p>\n<p>\\[ \\text{Upper Quartile} &ndash; \\text{Lower Quartile} \\]</p>\n<p>and so in this case we have:</p>\n<p>\\[ \\text{Interquartile range} = \\var{uquartile}-\\var{lquartile}=\\var{uquartile-lquartile} \\]</p>\n<h4>d)</h4>\n<p>Most of the data should be spanned by $4s$ where $s$ is the sample standard deviation.</p>\n<p>The range of values is $\\var{r1[31]}-\\var{r1[0]}=\\var{r1[31]-r1[0]}$ and so $s$ should be approximately</p>\n<p>\\[ \\simplify[std]{({r1[31]}-{r1[0]}) / 4 = {(r1[31] -r1[0]) / 4}} \\]</p>\n<p>The most likely value for the sample standard deviation&nbsp;of the options presented is $\\var{guess1}$.</p>\n<p>(The actual value is $\\var{stdev}$ to 2 decimal places).</p>", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "extensions": ["stats"], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"sig": {"name": "sig", "group": "Ungrouped variables", "definition": "random(4..9)", "description": "", "templateType": "anything", "can_override": false}, "r1": {"name": "r1", "group": "Ungrouped variables", "definition": "sort(r0)", "description": "", "templateType": "anything", "can_override": false}, "median": {"name": "median", "group": "Ungrouped variables", "definition": 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"variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Oblicz warto\u015bci miar wskazanych poni\u017cej:</p>\n<table border=\"1\" align=\"center\" style=\"width: 543px;\">\n<tbody>\n<tr>\n<th style=\"width: 91px;\">Minimum (x<sub>min</sub>)</th>\n<th style=\"width: 99px;\">Pierwszy kwartyl (Q1)</th>\n<th style=\"width: 94px;\">Mediana (Me)</th>\n<th style=\"width: 86px;\">Trzeci kwartyl (Q3)</th>\n<th style=\"width: 92px;\">Maximum (x<sub>max</sub>)</th>\n</tr>\n<tr>\n<td style=\"width: 91px;\">[[0]]</td>\n<td style=\"width: 99px;\">[[1]]</td>\n<td style=\"width: 94px;\">[[2]]</td>\n<td style=\"width: 86px;\">[[3]]</td>\n<td style=\"width: 92px;\">[[4]]</td>\n</tr>\n</tbody>\n</table>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.25", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{r1[0]}", "maxValue": "{r1[0]}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain-eu"], "correctAnswerStyle": "plain-eu"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{lquartile}", "maxValue": "0.5*r1[7]+0.5*r1[8]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain-eu"], "correctAnswerStyle": "plain-eu"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{median}", "maxValue": "{median}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain-eu"], "correctAnswerStyle": "plain-eu"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "0.5*r1[23]+0.5*r1[24]", "maxValue": "{uquartile}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain-eu"], "correctAnswerStyle": "plain-eu"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.25", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{r1[31]}", "maxValue": "{r1[31]}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain-eu"], "correctAnswerStyle": "plain-eu"}], "sortAnswers": false}, {"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>Oblicz rozst\u0119p:</p>\n<p>R= [[0]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "range", "maxValue": "range", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain-eu"], "correctAnswerStyle": "plain-eu"}], "sortAnswers": false}, {"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>Oblicz rozst\u0119p mi\u0119dzykwartylowy:</p>\n<p>R<sub>Q</sub>= [[0]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "(0.5*r1[23]+0.5*r1[24])-(0.5*r1[7]+0.5*r1[8])", "maxValue": "{uquartile-lquartile}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain-eu"], "correctAnswerStyle": "plain-eu"}], "sortAnswers": false}, {"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>Wyznacz typowy obszar zmienno\u015bci wzgl\u0119dem miar pozycyjnych ($Me{+-}Q$):<br/>[[0]]${&lt;=} {x_{typ}}&lt;=$[[1]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": false, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{median}-{uquartile-lquartile}/2", "maxValue": "{median}-{(0.5*r1[23]+0.5*r1[24])-(0.5*r1[7]+0.5*r1[8])}/2", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain-eu"], "correctAnswerStyle": "plain-eu"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": "0.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": false, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{median}+{(0.5*r1[23]+0.5*r1[24])-(0.5*r1[7]+0.5*r1[8])}/2", "maxValue": "{median}+{uquartile-lquartile}/2", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain-eu"], "correctAnswerStyle": "plain-eu"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Aleksandra Romanowska", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/11231/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}, {"name": "Aleksandra Romanowska", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/11231/"}]}