// Numbas version: finer_feedback_settings
{"name": "Logistic regression: interpretation of software output.", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Logistic regression: interpretation of software output.", "tags": [], "metadata": {"description": "<p>Interpreting the minitab output from a logistic regression model of salary against obesity as measured by BMI.</p>\n<p>Adaptive marking is in place for Part b).</p>", "licence": "Creative Commons Attribution-ShareAlike 4.0 International"}, "statement": "<p>It is&nbsp;hypothesized that there is a relationship between salary and obesity, with employees in more highly-paid roles being more likely to be overweight than those in other roles.</p>\n<p>To test this hypothesis, the following data are collected for a random sample of 15 employees:</p>\n<ul>\n<li>their annual salary ($x$, in thousands of pounds);</li>\n<li>whether or not they are obese $\\Rightarrow$ body mass index of more than 30: $Y=1$ &nbsp;if yes, $Y=0$ if no.</li>\n</ul>\n<p>A logistic regression is performed using statistical software, obtaining the following (edited) output:</p>\n<table border=\"1\" class=\"minitab\" style=\"width: 425px;\">\n<tbody>\n<tr>\n<td style=\"width: 64.3646px;\">\n<p></p>\n</td>\n<td style=\"width: 33.0938px;\">\n<p></p>\n</td>\n<td style=\"width: 45.0417px;\">\n<p></p>\n</td>\n<td style=\"width: 39.8438px;\">\n<p></p>\n</td>\n<td style=\"width: 42.3333px;\">\n<p></p>\n</td>\n<td colspan=\"2\" style=\"width: 103.323px;\">\n<p>95% CI</p>\n</td>\n</tr>\n<tr>\n<td style=\"width: 64.3646px;\">\n<p>Predictor</p>\n</td>\n<td style=\"width: 33.0938px;\">\n<p>Coef</p>\n</td>\n<td style=\"width: 45.0417px;\">\n<p>SE Coef</p>\n</td>\n<td style=\"width: 39.8438px;\">\n<p>Z</p>\n</td>\n<td style=\"width: 42.3333px;\">\n<p>P</p>\n</td>\n<td style=\"width: 43.6667px;\">\n<p>Lower</p>\n</td>\n<td style=\"width: 43.6562px;\">\n<p>Upper</p>\n</td>\n</tr>\n<tr>\n<td align=\"left\" style=\"width: 64.3646px;\">\n<p>Constant</p>\n</td>\n<td style=\"width: 33.0938px;\">\n<p>$A$</p>\n</td>\n<td style=\"width: 45.0417px;\">\n<p>{sea}</p>\n</td>\n<td style=\"width: 39.8438px;\">\n<p>-2.24</p>\n</td>\n<td style=\"width: 42.3333px;\">\n<p>0.025</p>\n</td>\n<td style=\"width: 43.6667px;\">\n<p></p>\n</td>\n<td style=\"width: 43.6562px;\">\n<p></p>\n</td>\n</tr>\n<tr>\n<td align=\"left\" style=\"width: 64.3646px;\">\n<p>$x$</p>\n</td>\n<td style=\"width: 33.0938px;\">\n<p>$B$</p>\n</td>\n<td style=\"width: 45.0417px;\">\n<p>{seb}</p>\n</td>\n<td style=\"width: 39.8438px;\">\n<p>2.25</p>\n</td>\n<td style=\"width: 42.3333px;\">\n<p>0.025</p>\n</td>\n<td style=\"width: 43.6667px;\">\n<p>1.02</p>\n</td>\n<td style=\"width: 43.6562px;\">\n<p>1.37</p>\n</td>\n</tr>\n</tbody>\n</table>", "advice": "<p>Revise logistic regression, e.g., Section 9.5 of&nbsp;<a href=\"https://leanpub.com/openintro-statistics\">OpenIntro Statistics</a>.</p>\n<p>a)</p>\n<p>Use the Z-score definition ($Z=\\frac{x-\\mu_0}{SE}$, with $\\mu_0=0$) and solve for $x$:</p>\n<p>A is given by &nbsp;A = -2.24 x &nbsp;SE Coef (A) = -2.24 x {sea} = {ansa}.</p>\n<p>B is given by &nbsp;B = &nbsp;2.25 x &nbsp;SE Coef (B) = &nbsp;2.25 x {seb} = {ansb}.</p>\n<p>b)</p>\n<p>The probability is given by:</p>\n<p>\\[\\begin{align}P(Y=1 | X=\\var{thismuch})&amp;=\\frac{e^{A+B\\times\\var{thismuch}}}{1+e^{A+B\\times\\var{thismuch}}}\\\\&amp;=\\frac{e^{\\var{r1}}}{1+e^{\\var{r1}}}=\\var{prob1}\\end{align}\\]</p>\n<p>If using a spreadsheet program (e.g., Excel), note that $e^x$ corresponds to EXP($x$).</p>\n<p></p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"tol": {"name": "tol", "group": "Ungrouped variables", "definition": "0.2", "description": "", "templateType": "anything", "can_override": false}, 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"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>Find $A$ and $B$:</p>\n<p>$A=\\;$[[0]] &nbsp; &nbsp; $B=\\;$[[1]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2.5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ansa*(1-tol)", "maxValue": "ansa*(1+tol)", "correctAnswerFraction": false, "allowFractions": false, 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"", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [{"variable": "ansa", "part": "p0g0", "must_go_first": false}, {"variable": "ansb", "part": "p0g1", "must_go_first": false}], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Find the probability that an employee with an annual salary of {thismuch}&nbsp;thousand pounds is obese:</p>\n<p>Probability = [[0]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [{"variable": "ansa", "part": "p0g0", "must_go_first": false}, {"variable": "ansb", "part": "p0g1", "must_go_first": false}], "variableReplacementStrategy": "alwaysreplace", 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