// Numbas version: exam_results_page_options
{"name": "Limits of accuracy in measuring weight in a gym scenario", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"metadata": {"description": "<p>Work with lower and upper bounds for estimates of weight.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "ungrouped_variables": ["weight1", "weight2", "total", "limit"], "extensions": [], "statement": "<p>We use lower and upper bounds&nbsp;to describe intervals. We can use them to identify all the possible values that could round to a given number.</p>\n<ul>\n<li>The <strong>lower</strong> bound is the lowest value that would round up to&nbsp;our rounded&nbsp;number.</li>\n<li>The <strong>upper</strong> bound is the&nbsp;lowest&nbsp;value after the lower bound that would <em>not</em>&nbsp;round down&nbsp;to our rounded number.</li>\n</ul>\n<p>For example, for&nbsp;<strong>50</strong> rounded&nbsp;to the nearest&nbsp;ten, the <strong>lowest bound&nbsp;is 45</strong> since 45 is the lowest value that rounds to 50. The <strong>upper bound is 55</strong> since 55 is the lowest value that rounds to 60 (the&nbsp;following ten).</p>", "variable_groups": [], "rulesets": {}, "type": "question", "name": "Limits of accuracy in measuring weight in a gym scenario", "advice": "<h4>a)</h4>\n<p>When using inequality symbols to show an error interval for $x$, we use</p>\n<p>\\[\\text{lower bound} \\leq x \\lt \\text{upper bound}\\]</p>\n<p>since the lower bound is included within the interval, while the upper bound is not.</p>\n<p></p>\n<h4>b)</h4>\n<p>To find bounds, we divide the degree of accuracy by 2 and subtract/add to our estimate to get lower and upper bounds respectively.</p>\n<p>The degree of accuracy is to the nearest 10 kg.</p>\n<p>\\[10 \\text{ kg} \\div 2 = 5 \\text{ kg}\\]</p>\n<p>\\[\\begin{align} \\text{Lower bound} = \\var{weight1}&nbsp;- 5 &amp;= \\var{weight1&nbsp;- 5} \\text{ kg}<br/>\\\\\\text{Upper bound} = \\var{weight1} + 5 &amp;= \\var{weight1 + 5} \\text{ kg}\\end{align}\\]</p>\n<h4></h4>\n<h4>c)</h4>\n<p>To calculate the lower bound of Alex and Cathy's&nbsp;total weight, we add the lower bounds of their individual weights:</p>\n<p>\\[ \\var{weight1&nbsp;- 5} \\text{ kg} + \\var{weight2&nbsp;- 5} \\text{ kg} = \\var{weight1 + weight2&nbsp;- 10} \\text{ kg.}\\]</p>\n<p>Similarly for the upper bound,</p>\n<p>\\[ \\var{weight1 + 5} \\text{ kg} + \\var{weight2 + 5} \\text{ kg} = \\var{weight1 + weight2&nbsp;+ 10} \\text{ kg.}\\]</p>\n<p></p>\n<p><strong>d)</strong></p>\n<p>The upper bound for their total weight is $\\var{weight1 + weight2&nbsp;+ 10}$ kg.</p>\n<p>The weight limit is $\\var{limit}$ kg.</p>\n<p>Hence, the upper bound for their total weight is greater than the weight limit.</p>\n<p></p>\n<p><strong>e)</strong></p>\n<p>We have found that the upper bound is above the weight limit, so we won't consider the&nbsp;first option.</p>\n<p>To decide between the second and&nbsp;the third option, we compare the&nbsp;lower bound for&nbsp;Alex's and Cathy's total weight to the weight limit.</p>\n<p>Here, the lower bound ($\\var{weight1 + weight2&nbsp;- 10}$ kg) is in fact less than the weight limit $\\var{limit}$ kg.</p>\n<p>Therefore, we cannot&nbsp;confirm nor deny whether it is safe for them to use the lift given the information we know.</p>", "parts": [{"scripts": {}, "gaps": [{"shuffleChoices": true, "type": "1_n_2", "minMarks": 0, "showCorrectAnswer": true, "displayColumns": 0, "showFeedbackIcon": true, "choices": ["<p>&lt;</p>", "<p>&gt;</p>", "<p><span class=\"huge\">&le;</span></p>", "<p>&ge;</p>"], "distractors": ["Incorrect, as the lower bound is included in the interval", "Incorrect, as the lower bound is the lowest possible value within the interval", "", "Incorrect, as the lower bound is the lowest possible value within the interval"], "scripts": {}, "variableReplacementStrategy": "originalfirst", "maxMarks": 0, "marks": 0, "variableReplacements": [], "matrix": [0, 0, "1", 0], "displayType": "dropdownlist"}, {"shuffleChoices": true, "type": "1_n_2", "minMarks": 0, "showCorrectAnswer": true, "displayColumns": 0, "showFeedbackIcon": true, "choices": ["<p>&lt;</p>", "<p>&gt;</p>", "<p><span class=\"huge\">&le;</span></p>", "<p><span class=\"huge\">&ge;</span></p>"], "distractors": ["", "Incorrect, as the upper bound is higher than any value within the interval", "Incorrect, as the upper bound is NOT included in the interval", "Incorrect, as the upper bound is higher than any value within the interval"], "scripts": {}, "variableReplacementStrategy": "originalfirst", "maxMarks": 0, "marks": 0, "variableReplacements": [], "matrix": ["1", 0, 0, 0], "displayType": "dropdownlist"}], "type": "gapfill", "showCorrectAnswer": true, "marks": 0, "variableReplacements": [], "showFeedbackIcon": true, "prompt": "<p>Suppose we are given a rounded variable such as weight. Select the inequality signs that denote the interval that contains the true value before rounding:</p>\n<p style=\"padding-left: 30px;\">Lower bound &nbsp;[[0]]weight &nbsp;[[1]]upper bound</p>", "variableReplacementStrategy": "originalfirst"}, {"scripts": {}, "gaps": [{"correctAnswerFraction": false, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "marks": 1, "minValue": "{weight1} - 5", "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "{weight1} - 5", "mustBeReduced": false, "allowFractions": false, "variableReplacements": [], "correctAnswerStyle": "plain", "showCorrectAnswer": true}, {"correctAnswerFraction": false, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "marks": 1, "minValue": "{weight1} + 5", "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "{weight1} + 5", "mustBeReduced": false, "allowFractions": false, "variableReplacements": [], "correctAnswerStyle": "plain", "showCorrectAnswer": true}], "type": "gapfill", "showCorrectAnswer": true, "marks": 0, "variableReplacements": [], "showFeedbackIcon": true, "prompt": "<p>Alex is filling&nbsp;out an online&nbsp;form to join a new gym in Newcastle.&nbsp;One of the fields&nbsp;is asking&nbsp;for his weight in kg rounded to the nearest 10kg.&nbsp;Alex&nbsp;rounds&nbsp;his weight and writes down <strong>{weight1} </strong>kg. What are the upper and lower bounds for his weight?</p>\n<p style=\"padding-left: 30px;\">Lower&nbsp;bound =&nbsp; [[0]]kg</p>\n<p style=\"padding-left: 30px;\">Upper&nbsp;bound =&nbsp; [[1]]kg</p>", "variableReplacementStrategy": "originalfirst"}, {"scripts": {}, "gaps": [{"correctAnswerFraction": false, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "marks": 1, "minValue": "{weight1} + {weight2} - 10", "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "{weight1} + {weight2} - 10", "mustBeReduced": false, "allowFractions": false, "variableReplacements": [], "correctAnswerStyle": "plain", "showCorrectAnswer": true}, {"correctAnswerFraction": false, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "showFeedbackIcon": true, "marks": 1, "minValue": "{weight1} + {weight2} + 10", "scripts": {}, "variableReplacementStrategy": "originalfirst", "type": "numberentry", "maxValue": "{weight1} + {weight2} + 10", "mustBeReduced": false, "allowFractions": false, "variableReplacements": [], "correctAnswerStyle": "plain", "showCorrectAnswer": true}], "type": "gapfill", "showCorrectAnswer": true, "marks": 0, "variableReplacements": [], "showFeedbackIcon": true, "prompt": "<p>Alex's friend Cathy is also joining the gym. She writes <strong>{weight2} </strong>kg in the weight field.</p>\n<p>Suppose Alex and Cathy meet up to go to the gym together for the first time. Their gym is located on the 8th floor of the building. They can either use the lift or the staircase to get there. Cathy wants to be active and go up the stairs but Alex doesn't want to do any exercise before he changes to his gym clothes and wants to use&nbsp;the lift. The weight limit for the&nbsp;lift is <strong>{limit} </strong>kg.</p>\n<p>Calculate the lower and upper bounds for their total weight.</p>\n<p style=\"padding-left: 30px;\">Lower&nbsp;bound for Alex's weight $+$ Cathy's weight $=$ &nbsp;[[0]]kg</p>\n<p style=\"padding-left: 30px;\">Upper bound for Alex's weight $+$ Cathy's weight $=$ &nbsp;[[1]]kg</p>", "variableReplacementStrategy": "originalfirst"}, {"scripts": {}, "gaps": [{"shuffleChoices": false, "type": "1_n_2", "minMarks": 0, "showCorrectAnswer": true, "displayColumns": 0, "showFeedbackIcon": true, "choices": ["<p>&gt;</p>", "<p>&lt;</p>", "<p>=</p>"], "distractors": ["", "", ""], "scripts": {}, "variableReplacementStrategy": "originalfirst", "maxMarks": 0, "marks": 0, "variableReplacements": [], "matrix": ["1", 0, 0], "displayType": "dropdownlist"}], "type": "gapfill", "showCorrectAnswer": true, "marks": 0, "variableReplacements": [], "showFeedbackIcon": true, "prompt": "<p style=\"text-align: left;\">Select&nbsp;the&nbsp;correct inequality sign between the following:</p>\n<p style=\"text-align: left; padding-left: 30px;\">Upper bound for&nbsp;Alex's and Cathy's&nbsp;total weight&nbsp;[[0]] Weight limit for the lift</p>\n<p></p>", "variableReplacementStrategy": "originalfirst"}, {"shuffleChoices": false, "variableReplacements": [], "showCorrectAnswer": true, "displayColumns": 0, "showFeedbackIcon": true, "prompt": "<p>Given what we know, what&nbsp;is our conclusion?&nbsp;Is it&nbsp;safe for them to take the lift together without going over the allowed weight limit?</p>\n<p></p>\n<p></p>\n<p></p>", "type": "1_n_2", "scripts": {}, "variableReplacementStrategy": "originalfirst", "maxMarks": 0, "minMarks": 0, "choices": ["<p>It is perfectly safe for them to use the lift because the upper bound for their total weight is below the allowed weight limit.</p>", "<p>We cannot be sure&nbsp;whether it is safe for them to use the lift because the upper bound for their total weight is above the allowed weight limit, while the lower bound is still below the weight limit.</p>", "<p>We can confidently&nbsp;say&nbsp;that it is unsafe for them to use the lift because their total weight exceeds the weight limit in all posibilities.</p>"], "marks": 0, "distractors": ["", "", ""], "matrix": [0, "2", 0], "displayType": "radiogroup"}], "tags": ["limits of accuracy", "lower bounds", "taxonomy", "upper bounds"], "preamble": {"css": "", "js": ""}, "functions": {}, "variables": {"weight2": {"description": "", "definition": "random(total-weight1)", "group": "Ungrouped variables", "name": "weight2", "templateType": "anything"}, "weight1": {"description": "", "definition": "random(50..100 #10)", "group": "Ungrouped variables", "name": "weight1", "templateType": "anything"}, "total": {"description": "", "definition": "random(140..180 #10)", "group": "Ungrouped variables", "name": "total", "templateType": "anything"}, "limit": {"description": "", "definition": "total + random(0..9.5 #0.5)", "group": "Ungrouped variables", "name": "limit", "templateType": "anything"}}, "variablesTest": {"maxRuns": 100, "condition": ""}, "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Stanislav Duris", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1590/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Stanislav Duris", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1590/"}]}