Exam-Style Questions on Correlation
Problems on Correlation adapted from questions set in previous exams.
The number of visitors to a cycle track and the number of drinks sold by a café at the location are recorded in the table below.
|Numver of visitors||32||45||39||43||58||84||65|
The data is shown in the following scatter diagram:
(a) Add Sunday's data to the scatter diagram.
(b) Draw, by eye, a line of best fit on the scatter diagram.
(c) Describe the relationship between the number of visitors and the number of drinks sold.
(d) Which particular day does not fit the relationship?
(e) If one day there were 50 visitors, estimate how many drinks would be sold.
The table below shows the scores for 12 students on two Mathematic exam papers. For the first paper calculators were allowed and for the second paper they were not.
|Paper 1 (\(x\))||74||73||65||75||68||72||69||71||83||68||68||73|
|Paper 2 (\(y\))||75||83||69||77||71||77||68||76||84||69||71||75|
(a) Write down the mean score on Paper 1.
(b) Write down the standard deviation of the scores for Paper 1.
(c) Find the number of students that had a score of more than one standard deviation below the mean on Paper 1.
(d) Write down the correlation coefficient, \(r\).
(e) Write down the equation of the regression line of \(y\) on \(x\).
Another student scored 75 on Paper 1.
(f) Calculate an estimate of his score on Paper 2
Another student scored 88 on Paper 1.
(g) Determine whether you can use the equation of the regression line to estimate his score on Paper 2. Give a reason for your answer.
The following table shows the average weights for given heights in a population of men.
|Heights (x cm)||160||165||170||175||180||185|
|Weights ( y kg)||65.1||67.9||70.1||72.8||75.4||77.2|
(a) The relationship between the variables is modelled by the regression equation \(y = ax + b\). Write down the value of \(a\) and of \(b\).
(b) Use this relationship to estimate the weight of a man whose height is 177 cm.
(c) Find the correlation coefficient.
(d) State which two of the following describe the correlation between the variables.
The following table shows the relationship between the number of workers and the amount of time in minutes it takes them to harvest the sugar cane in a particular field.
|Workers (\(n\))||Time (\(t\))|
(a) Find the equation of the regression line of \(t\) on \(n\).
(b) Find the value of the Pearson’s product–moment correlation coefficient, r.
(c) Use the regression equation to find how long it would take seven workers to harvest the sugar cane.
As part of a conservation project, Darren was asked to measure the circumference of trees that were growing at different distances from a beach.
His results are shown in the following table.
|Distance, \(x\) (metres)||6||14||20||25||35||48||46||48||52|
|Circumference, \(y\) (centimetres)||52||57||57||68||65||70||75||80||82|
(a) State whether distance from the beach is a continuous or discrete variable.
(b) On graph paper, draw a scatter diagram to show Darren’s results. Use a scale of 1 cm to represent 5 m on the x-axis and 1 cm to represent 10 cm on the y-axis.
(c) Calculate the mean distance, \(\bar x\) , of the trees from the beach.
(d) Work out the mean circumference, \(\bar y\) , of the trees.
(e) Plot and label the point M(\(\bar x,\bar y\)) on your graph.
(f) Write down the Pearson’s product–moment correlation coefficient, \(r\) , for Darren's results.
(g) Find the equation of the regression line \(y\) on \(x\), for Darren’s results.
(h) Draw the regression line \(y\) on \(x\) on your graph.
(i) Use the equation of the regression line \(y\) on \(x\) to estimate the circumference of a tree that is 42 m from the beach.
In a survey of insect life near a stream, a student collected data about the number of different insect species \((y)\) that were found at different distances \((x)\) in metres from the stream.
|Insect species \((y)\)||26||25||19||19||14||9||5||3||2|
(a) Draw a scatter diagram to show the data.
(b) Using your scatter diagram, describe the correlation between the number of different insect species and the distance from the stream.
(c) Find \(\bar x\), the mean of the distances from the stream;
(d) Find \(\bar y\), the mean number of insect species.
(e) Plot the point \((\bar x,\bar y)\) on your scatter diagram. Label this point M.
(f) Write down the equation of the regression line \(y\) on \(x\) for the above data.
(g) Draw the regression line \(y\) on \(x\) on your scatter diagram.
(h) Estimate the number of insect species to be found 30 metres from the stream.
The following table shows the amount of diesel required by a train to travel certain distances.
|Distance (\(x\) km)||90||150||230||310||390|
|Diesel used (\(y\) litres)||19.2||33.9||49.0||79.5||89.9|
This data can be modelled by the regression line with equation \(y=ax+b\).
(a) Find the values of \(a\) and of \(b\).
(b) Explain what the gradient a represents.
(c) Use the model to estimate the amount of diesel the train would use if it is driven 270 km.