$X_1$
", "$X_2$
", "$X_3$
"], "displayColumns": 0, "distractors": ["", "", ""], "shuffleChoices": true, "scripts": {}, "minMarks": 0, "type": "1_n_2", "maxMarks": 0, "showCorrectAnswer": true, "matrix": [0, 0, 1], "marks": 0}], "type": "gapfill", "prompt": "Which of these variables is an indicator variable?
\n[[0]]
", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "ansa+tol", "minValue": "ansa-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "ansb+tol", "minValue": "ansb-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "ansc+tol", "minValue": "ansc-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "ansd+tol", "minValue": "ansd-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "ansa+tol", "minValue": "ansa-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "ansc+tol", "minValue": "ansc-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "ansd+tol", "minValue": "ansd-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "si+tol", "minValue": "si-tol", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "Based on a recent survey of some of the restaurants in the North-East, the following (edited) Minitab output was obtained:
\nRegression Analysis: $y$ versus $x_1,\\;x_2,\\;x_3$
\nThe regression equation is: y = ********
\n\n| \n Predictor \n | \n Coef \n | \n SE Coef \n | \n T \n | \n P \n |
|---|---|---|---|---|
| \n Constant \n | \n\n $A$ \n | \n\n {sea} \n | \n\n 3.887 \n | \n\n 0.002$ \n | \n
| \n $x_1$ \n | \n\n {cb} \n | \n\n $B$ \n | \n\n -2.64 \n | \n\n 0.021 \n | \n
| \n $x_2$ \n | \n\n $C$ \n | \n\n {se} \n | \n\n 3.82 \n | \n\n 0.002 \n | \n
| \n $x_3$ \n | \n\n $D$ \n | \n\n {sed} \n | \n\n 2.53 \n | \n\n 0.024 \n | \n
s={sval} R-sq= 92.1% R-Sq(adj)=93.9%
\n(i) Find the values of $A,\\;B,\\;C$ and $D$ to 3 decimal places.
\n$A=\\;$[[0]], $B=\\;$[[1]]
\n$C=\\;$[[2]], $D=\\;$[[3]]
\n(ii) Write down the full fitted regression equation:
\n\\[Y = \\beta_0+ \\beta_1 X_1 + \\beta_2 X_2 + \\beta_3X_3 + \\epsilon,\\;\\;\\epsilon \\sim N(0,\\sigma^2)\\]
\n$Y=\\;$[[4]]-$\\var{abs(cb)}X_1$+[[5]]$X_2$+[[6]]$X_3+\\epsilon$
\nNote that you are given the coefficient of $X_1$ from the Minitab table.
\nAlso find $\\sigma^2=\\;$[[7]] to 3 decimal places where $\\epsilon \\sim N(0,\\sigma^2)$
", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "pred+0.1", "minValue": "pred-0.1", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "Predict sales for a restaurant with {thatmany} competitors, a population of {thismany} within 1 kilometre and that {hascond}.
\nEnter the predicted value to one decimal place:
\n$Y=\\;$[[0]]
", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "ind-1", "minValue": "ind-1", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "It is thought that a fourth variable - customer satisfaction based on a recent survey - could also be a predictor of sales.
\nEach of the restaurants was given an overall satisfaction rating by choosing one of the following in the survey:
\n{sv[0]} {sv[1]} {sv[2]} {sv[3]} {sv[4]} {sv[5]}
\nHow many indicator variables would need to be used here?: [[0]]
", "showCorrectAnswer": true, "marks": 0}], "statement": "The management at {thisrest} propose the following model to predict sales Y at their {place} branch.
\n\\[Y = \\beta_0+ \\beta_1 X_1 + \\beta_2 X_2 + \\beta_3X_3 + \\epsilon\\]
\nwhere
\n$X_1=\\;$number of competitors within one kilometre.
\n$X_2=\\;$population within one kilometre (in 1000s).
\n$X_3=\\;$ 1 if {cond}, 0 otherwise.
", "tags": ["ACE2013", "checked2015", "indicator variables", "minitab output", "multiple regression", "regression", "regression equation", "statistics"], "rulesets": {}, "preamble": {"css": ".minitab {\nfont-family: 'Courier', monospace;\n}", "js": ""}, "type": "question", "metadata": {"notes": "09/02/2014:
\nFirst draft. Based on an i-assess question for ACE.
", "licence": "Creative Commons Attribution 4.0 International", "description": "Asking users to interpret a minitab output to give the coefficients of a multiple regression together with a prediction based on the subsequent equation.
"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "a) $X_3$ is the indicator variable.
\nb)
\n\n(i) The values of A, B, C and D are given by:
\nA = 3.887 x SE Coef (A) = 3.887 x {sea} = {ansa} to 3 decimal places.
\nB = Coef (B)/-2.64 = {cb}/-2.64 = {ansb} to 3 decimal places.
\nC = 3.82 x SE Coef (C) = 3.82 x {se} = {ansc} to 3 decimal places.
\nD = 2.53 x SE Coef (D) = 2.53 x {sed} = {ansd} to 3 decimal places.
\nii) The fitted regression equation is:
\n\\[Y=\\var{ansa}-\\var{abs(cb)}X_1+\\var{ansc}X_2+\\var{ansd}X_3+\\epsilon\\] with $\\sigma^2=\\var{sval}^2=\\var{si}$ all to 3 decimal places.
\nc)
\n\nUsing the above fitted model where $X_1=\\var{thatmany}$ and $X_2= \\frac{\\var{thismany}}{1000}=\\var{thismany/1000}$ and since $X_3=\\var{q}$ as the restaurant {hascond} we find :
\n\\[Y=\\var{ansa}-\\var{abs(cb)}\\times \\var{thatmany}+\\var{ansc}\\times \\var{thismany/1000}+\\var{ansd}\\times \\var{q}=\\var{pred}\\] to one decimal place.
\nd)
\nThere are {ind} categories in the survey and so the number of indicator variables is {ind}-1={ind-1}.
", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}