// Numbas version: exam_results_page_options
{"name": "Zentralma\u00dfe f\u00fcr einen Datensatz bestimmen", "extensions": ["stats"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Zentralma\u00dfe f\u00fcr einen Datensatz bestimmen", "tags": [], "metadata": {"description": "<p>This question provides a list of data to the student. They are asked to find the&nbsp;mean, median, mode and range.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "rulesets": {}, "preamble": {"css": "", "js": ""}, "functions": {}, "variable_groups": [{"name": "final list", "variables": ["a", "a_s", "mean", "median", "mode", "mode1", "range", "modetimes"]}], "extensions": ["stats"], "statement": "<p>Eine Zufallsstichprobe von 20 Einwohnern aus Newcastle wurde nach der Anzahl der Besuche im Theater im vergangenen Jahr gefragt.</p>\n<p>Hier ist die Liste ihrer Antworten:</p>\n<table border=\"1\">\n<tbody>\n<tr>\n<td>$\\var{a[0]}$</td>\n<td>$\\var{a[1]}$</td>\n<td>$\\var{a[2]}$</td>\n<td>$\\var{a[3]}$</td>\n<td>$\\var{a[4]}$</td>\n<td>$\\var{a[5]}$</td>\n<td>$\\var{a[6]}$</td>\n<td>$\\var{a[7]}$</td>\n<td>$\\var{a[8]}$</td>\n<td>$\\var{a[9]}$</td>\n</tr>\n<tr>\n<td>$\\var{a[10]}$</td>\n<td>$\\var{a[11]}$</td>\n<td>$\\var{a[12]}$</td>\n<td>$\\var{a[13]}$</td>\n<td>$\\var{a[14]}$</td>\n<td>$\\var{a[15]}$</td>\n<td>$\\var{a[16]}$</td>\n<td>$\\var{a[17]}$</td>\n<td>$\\var{a[18]}$</td>\n<td>$\\var{a[19]}$</td>\n</tr>\n</tbody>\n</table>\n<p style=\"text-align: center;\"></p>", "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<h4>a)</h4>\n<p>The mean is the sum of all the&nbsp;responses ($\\sum x$) divided by the number of responses ($n$).</p>\n<p>Here, $n = 20$.</p>\n<p>\\begin{align} <br/>\\sum x &amp;= \\var{a[0]} + \\var{a[1]} +\\var{a[2]} +\\var{a[3]} +\\var{a[4]} +\\var{a[5]} +\\var{a[6]} +\\var{a[7]} +\\var{a[8]} +\\var{a[9]} + \\var{a[10]} + \\var{a[11]} +\\var{a[12]} +\\var{a[13]} +\\var{a[14]} +\\var{a[15]} +\\var{a[16]} +\\var{a[17]} +\\var{a[18]} +\\var{a[19]}&nbsp;\\\\<br/>&amp;= \\var{sum(a)} \\text{.}<br/>\\end{align}</p>\n<p><span face=\"Courier New\">Therefore we calculate the mean</span></p>\n<p><span face=\"Courier New\">\\begin{align} <br/>\\overline{x} &amp;= \\frac{\\sum x}{n}&nbsp;\\\\[0.5em]<br/>&amp;= \\frac{\\var{sum(a)}}{20}&nbsp;\\\\[0.5em]<br/>&amp;= \\var{mean} \\text{.}<br/>\\end{align}</span></p>\n<p><span face=\"Courier New\">&nbsp;</span></p>\n<h4>b)</h4>\n<p>The median is&nbsp;the middle value. We need to sort the list in order:</p>\n<p style=\"text-align: center;\">\\[ \\var{a_s[0]}, \\quad \\var{a_s[1]}, \\quad&nbsp;\\var{a_s[2]}, \\quad \\var{a_s[3]}, \\quad \\var{a_s[4]}, \\quad \\var{a_s[5]}, \\quad \\var{a_s[6]}, \\quad \\var{a_s[7]}, \\quad \\var{a_s[8]}, \\quad \\var{a_s[9]}, \\quad \\var{a_s[10]}, \\quad \\var{a_s[11]}, \\quad&nbsp;\\var{a_s[12]}, \\quad \\var{a_s[13]}, \\quad \\var{a_s[14]}, \\quad \\var{a_s[15]}, \\quad \\var{a_s[16]}, \\quad \\var{a_s[17]}, \\quad \\var{a_s[18]}, \\quad \\var{a_s[19]} \\]</p>\n<p>There is an even number of responses, so there are&nbsp;two numbers in the middle (10<sup>th</sup> and 11<sup>th</sup> place). To&nbsp;find the median, we need to&nbsp;find the mean of these two numbers $\\var{a_s[9]}$ and $\\var{a_s[10]}$:</p>\n<p>\\begin{align} <br/>\\frac{\\var{a_s[9]} + \\var{a_s[10]}}{2} &amp;= &nbsp;\\frac{\\var{a_s[9] + a_s[10]}}{2} \\\\<br/>&amp;= \\var{median} \\text{.}&nbsp;<br/>\\end{align}</p>\n<p>&nbsp;</p>\n<h4>c)</h4>\n<p>The mode is the value that occurs the most often in the data.</p>\n<p>To find a mode, we can look at our sorted list:</p>\n<p style=\"text-align: center;\">$\\var{a_s[0]}, \\var{a_s[1]}, \\var{a_s[2]}, \\var{a_s[3]}, \\var{a_s[4]}, \\var{a_s[5]}, \\var{a_s[6]}, \\var{a_s[7]}, \\var{a_s[8]}, \\var{a_s[9]}, \\var{a_s[10]}, \\var{a_s[11]}, \\var{a_s[12]}, \\var{a_s[13]}, \\var{a_s[14]}, \\var{a_s[15]}, \\var{a_s[16]}, \\var{a_s[17]}, \\var{a_s[18]}, \\var{a_s[19]}$.</p>\n<p>We&nbsp;notice that $\\var{mode1}$ occurs the most ($\\var{modetimes[mode1]}$ times) so $\\var{mode1}$ is the mode.</p>\n<p>&nbsp;</p>\n<h4>d)</h4>\n<p>Range is the difference between the highest and the lowest value in the data.</p>\n<p>To find this, we subtract the lowest value from the highest value:</p>\n<p>\\[ \\var{max(a)} - \\var{min(a)} = \\var{range} \\text{.}\\]</p>", "parts": [{"maxValue": "mean", "variableReplacementStrategy": "originalfirst", "prompt": "<p>Bestimmen Sie das arithmetische Mittel.</p>", "type": "numberentry", "unitTests": [], "showCorrectAnswer": true, "allowFractions": false, "minValue": "mean", "notationStyles": ["eu", "plain-eu"], "marks": 1, "showFractionHint": true, "correctAnswerFraction": false, "variableReplacements": [], "mustBeReduced": false, "customName": "", "mustBeReducedPC": 0, "customMarkingAlgorithm": "", "correctAnswerStyle": "eu", "useCustomName": false, "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true}, {"maxValue": "median", "variableReplacementStrategy": "originalfirst", "prompt": "<p>Bestimmen Sie den Median.</p>", "type": "numberentry", "unitTests": [], "showCorrectAnswer": true, "allowFractions": false, "minValue": "median", "notationStyles": ["eu", "plain-eu"], "marks": 1, "showFractionHint": true, "correctAnswerFraction": false, "variableReplacements": [], "mustBeReduced": false, "customName": "", "mustBeReducedPC": 0, "customMarkingAlgorithm": "", "correctAnswerStyle": "eu", "useCustomName": false, "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true}, {"maxValue": "mode1", "variableReplacementStrategy": "originalfirst", "prompt": "<p>Finden Sie den Modus (Modalwert).</p>", "type": "numberentry", "unitTests": [], "showCorrectAnswer": true, "allowFractions": false, "minValue": "mode1", "notationStyles": ["eu", "plain-eu"], "marks": 1, "showFractionHint": true, "correctAnswerFraction": false, "variableReplacements": [], "mustBeReduced": false, "customName": "", "mustBeReducedPC": 0, "customMarkingAlgorithm": "", "correctAnswerStyle": "eu", "useCustomName": false, "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true}, {"maxValue": "range", "variableReplacementStrategy": "originalfirst", "prompt": "<p>Bestimmen Sie die Spannweite.</p>", "type": "numberentry", "unitTests": [], "showCorrectAnswer": true, "allowFractions": false, "minValue": "range", "notationStyles": ["plain", "en", "si-en"], "marks": 1, "showFractionHint": true, "correctAnswerFraction": false, "variableReplacements": [], "mustBeReduced": false, "customName": "", "mustBeReducedPC": 0, "customMarkingAlgorithm": "", "correctAnswerStyle": "plain", "useCustomName": false, "showFeedbackIcon": true, "scripts": {}, "extendBaseMarkingAlgorithm": true}], "variables": {"a2": {"description": "<p>Option&nbsp;2 for the list. Only used if there is only one mode and option 1 was not used.</p>", "name": "a2", "definition": "repeat(random(0..8), 20)", "group": "Ungrouped variables", "templateType": "anything"}, "modea2": {"description": "", "name": "modea2", "definition": "mode(a2)", "group": "Ungrouped variables", "templateType": "anything"}, "mean": {"description": "", "name": "mean", "definition": "mean(a)", "group": "final list", "templateType": "anything"}, "a1": {"description": "<p>Option 1 for the list. Only used if there is only one mode.</p>", "name": "a1", "definition": "repeat(random(0..8), 20)", "group": "Ungrouped variables", "templateType": "anything"}, "a": {"description": "<p>The final list.</p>", "name": "a", "definition": "if(len(modea1) = 1, a1, if(len(modea2) = 1, a2, a3))", "group": "final list", "templateType": "anything"}, "a_s": {"description": "<p>Sorted list.</p>", "name": "a_s", "definition": "sort(a)", "group": "final list", "templateType": "anything"}, "a3": {"description": "<p>Option 3 for the list. Ensures there is only one mode (2) while still randomising the data.</p>", "name": "a3", "definition": "shuffle([ random(0..1),\n  2, \n  random(4..6),\n  random(0..3 except 2), \n  random(0..3 except 2),\n  random(4..6),\n  2,\n  2,\n  random(4..6),\n  random(7..8),\n  random(0..3 except 2 except 1), \n  random(4..6),\n  2,\n  random(1..3 except 2), \n  random(7..8),\n  2,\n  random(7..8),\n  random(4..6), \n  random(0..3 except 2), \n  2\n])", "group": "Ungrouped variables", "templateType": "anything"}, "range": {"description": "", "name": "range", "definition": "max(a) - min(a)", "group": "final list", "templateType": "anything"}, "median": {"description": "", "name": "median", "definition": "median(a)", "group": "final list", "templateType": "anything"}, "modea1": {"description": "", "name": "modea1", "definition": "mode(a1)", "group": "Ungrouped variables", "templateType": "anything"}, "modetimes": {"description": "<p>The vector of&nbsp;number of times of each value in the data.</p>", "name": "modetimes", "definition": "map(\nlen(filter(x=j,x,a)),\nj, 0..8)", "group": "final list", "templateType": "anything"}, "mode1": {"description": "<p>Mode as a value.</p>", "name": "mode1", "definition": "mode[0]", "group": "final list", "templateType": "anything"}, "mode": {"description": "<p>Mode as a vector.</p>", "name": "mode", "definition": "mode(a)", "group": "final list", "templateType": "anything"}}, "ungrouped_variables": ["modea1", "modea2", "a1", "a2", "a3"], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}, {"name": "Stanislav Duris", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1590/"}, {"name": "Andreas Vohns", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3622/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Chris Graham", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/369/"}, {"name": "Stanislav Duris", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1590/"}, {"name": "Andreas Vohns", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3622/"}]}