To find $a$ and $b$ you first find $\\displaystyle b = \\frac{n\\Sigma xy-\\Sigma x \\Sigma y}{n\\Sigma x^2 -(\\Sigma x)^2}$
\nThen $\\displaystyle a = \\frac{\\Sigma y - b \\Sigma x}{n}$
\nNote that $n$ is the number of data points. In this case $\\var{n}$
\nNow go back and fill in the values for $a$ and $b$.
\n", "variableReplacementStrategy": "originalfirst", "variableReplacements": []}], "prompt": "
\\[Y = a + b x.\\] Using the least squares method to find $a$ and $b$ to 5 decimal places, then input them below to 3 decimal places. You will use these approximate values in the rest of the question.
\n$a=\\;$[[0]] $b=\\;$[[1]], (enter both to 3 decimal places).
", "gaps": [{"showPrecisionHint": false, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "maxValue": "ja+tol", "allowFractions": false, "variableReplacements": [], "correctAnswerFraction": false, "minValue": "ja-tol", "variableReplacementStrategy": "originalfirst"}, {"showPrecisionHint": false, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "maxValue": "jb+tol1", "allowFractions": false, "variableReplacements": [], "correctAnswerFraction": false, "minValue": "jb-tol1", "variableReplacementStrategy": "originalfirst"}], "variableReplacements": [], "stepsPenalty": 0, "variableReplacementStrategy": "originalfirst", "type": "gapfill"}, {"marks": 0, "showCorrectAnswer": true, "scripts": {}, "type": "gapfill", "prompt": "A scatter diagram for the data is plotted below.
\nUsing the equation of the least squares regression line that you found in part (a) plot the line on the diagram below. Find two points on line from the equation and then drag the points A and B to these points.
\n\\[Y = a + b x.\\]
\nThe equation of the line you ploted is
\n$Y=$[[1]]$+$[[0]]$x$
\n\n{regressline(r1,r2,min(r1)-10,max(r1)+10,min(r2)-10,max(r2)+10)}
\nSSE=
\nSSE for fitted regression line: {sumr}
\nThe SSE is displayed which gives an indication of the fit. The nearer the SSE for fitted regression line is the SSE the better.
\n", "gaps": [{"showPrecisionHint": false, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "maxValue": "b+tol2", "allowFractions": false, "variableReplacements": [], "correctAnswerFraction": false, "minValue": "b-tol2", "variableReplacementStrategy": "originalfirst"}, {"showPrecisionHint": false, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "maxValue": "a+tol", "allowFractions": false, "variableReplacements": [], "correctAnswerFraction": false, "minValue": "a-tol", "variableReplacementStrategy": "originalfirst"}], "variableReplacementStrategy": "originalfirst", "variableReplacements": []}, {"marks": 0, "showCorrectAnswer": true, "scripts": {}, "steps": [{"marks": 0, "showCorrectAnswer": true, "scripts": {}, "type": "information", "prompt": "\\[r=\\frac{n\\Sigma xy -\\Sigma x \\Sigma y}{\\sqrt{n\\Sigma x^2-(\\Sigma x)^2}\\sqrt{n\\Sigma y^2-(\\Sigma y)^2}}\\]
", "variableReplacementStrategy": "originalfirst", "variableReplacements": []}], "prompt": "Calculate the coefficient of correlation $r$ for these data:
\n$r=\\;$[[0]] (enter to 2 decimal places).
", "gaps": [{"showPrecisionHint": false, "marks": 1, "showCorrectAnswer": true, "scripts": {}, "type": "numberentry", "maxValue": "corr+tol1", "allowFractions": false, "variableReplacements": [], "correctAnswerFraction": false, "minValue": "corr-tol1", "variableReplacementStrategy": "originalfirst"}], "variableReplacements": [], "stepsPenalty": 0, "variableReplacementStrategy": "originalfirst", "type": "gapfill"}, {"marks": 0, "showCorrectAnswer": true, "scripts": {}, "type": "gapfill", "prompt": "Next month, the average temperature in {owner}'s town is forecast to be {thisval} Celsius. Use the regression equation $Y = a + b x$, with the values of $a$ and $b$ that you calculated in part (a), to predict sales of the {beverage} in that month.
\nWhat is the predicted value of sales (in hundreds of euros) ?
\nUse the values of $a$ and $b$ you input above to 3 decinal places.
\nEnter the predicted sales here: [[0]] (hundreds of euros to the nearest whole number).
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\n| Month | $\\var{obj[0]}$ | $\\var{obj[1]}$ | $\\var{obj[2]}$ | $\\var{obj[3]}$ | $\\var{obj[4]}$ | $\\var{obj[5]}$ | $\\var{obj[6]}$ | $\\var{obj[7]}$ | $\\var{obj[8]}$ | $\\var{obj[9]}$ | $\\var{obj[10]}$ | $\\var{obj[11]}$ |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| $X$ (temperature) | \n$\\var{r1[0]}$ | \n$\\var{r1[1]}$ | \n$\\var{r1[2]}$ | \n$\\var{r1[3]}$ | \n$\\var{r1[4]}$ | \n$\\var{r1[5]}$ | \n$\\var{r1[6]}$ | \n$\\var{r1[7]}$ | \n$\\var{r1[8]}$ | \n$\\var{r1[9]}$ | \n$\\var{r1[10]}$ | \n$\\var{r1[11]}$ | \n
| $Y$ (sales, €100s) | \n$\\var{r2[0]}$ | \n$\\var{r2[1]}$ | \n$\\var{r2[2]}$ | \n$\\var{r2[3]}$ | \n$\\var{r2[4]}$ | \n$\\var{r2[5]}$ | \n$\\var{r2[6]}$ | \n$\\var{r2[7]}$ | \n$\\var{r2[8]}$ | \n$\\var{r2[9]}$ | \n$\\var{r2[10]}$ | \n$\\var{r2[11]}$ | \n
You are given the following information:
\n| $X$ | \n$\\sum x=\\;\\var{t[0]}$ | \n$\\sum x^2=\\;\\var{ssq[0]}$ | \n
|---|---|---|
| $Y$ | \n$\\sum y=\\;\\var{t[1]}$ | \n$\\sum y^2=\\;\\var{ssq[1]}$ | \n
Also you are given $\\sum xy = \\var{sxy}$.
", "extensions": ["stats", "jsxgraph"], "ungrouped_variables": ["ch", "prediction", "b1", "owner", "sxy", "res", "spxy", "ls", "tol", "tcorr", "tsqovern", "ssq", "sumr", "thisval", "a1", "pub", "corr", "a", "b", "obj", "r1", "r2", "ss", "tol1", "n", "beverage", "t", "sc", "rsquared", "ja", "jb", "tol2"], "advice": "{regfun(r1,r2,max(r1)+10,max(r2)+10,rsquared,sumr)}
", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "Find a regression equation given 12 months data on temperature and sales of a drink. Includes an interactive diagram for experimenting with fitting a regression line.
\nrebelmaths
"}, "variablesTest": {"maxRuns": 100, "condition": ""}, "showQuestionGroupNames": false, "type": "question", "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}]}], "contributors": [{"name": "Julie Crowley", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/113/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}