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.
| Monday | Tuesday | Wednesday | Thursday | Friday | Saturday | Sunday | |
| Numver of visitors | 32 | 45 | 39 | 43 | 58 | 84 | 65 |
| Drinks sold | 17 | 20 | 23 | 7 | 24 | 49 | 38 |
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 (nn) | Time (tt) |
|---|---|
| 3 | 799 |
| 4 | 703 |
| 5 | 645 |
| 6 | 570 |
| 8 | 422 |
| 9 | 322 |
| 10 | 241 |
(a) Find the equation of the regression line of tt on nn.
(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.
The following table shows the amount of diesel required by a train to travel certain distances.
| Distance (xx km) | 90 | 150 | 230 | 310 | 390 |
|---|---|---|---|---|---|
| Diesel used (yy litres) | 19.2 | 33.9 | 49.0 | 79.5 | 89.9 |
This data can be modeled by the regression line with equation y=ax+by=ax+b.
(a) Find the values of aa and of bb.
(b) Explain what the gradient aa represents.
(c) Use the model to estimate the amount of diesel the train would use if it is driven 270 km.
The following table shows the mean weight in kilograms of members of a group of young children of various ages.
| Age (x years) | 1.6 | 2.5 | 3.3 | 4.4 | 5.6 |
|---|---|---|---|---|---|
| Weight ( y kg) | 12 | 15 | 16 | 17 | 20 |
The relationship between the variables is modelled by the regression line with equation y=ax+by=ax+b
(a) Find the value of aa and of bb
(b) Write down the correlation coefficient.
(c) Use your equation to estimate the mean weight of a child that is four years old.
The following table shows the average number of hours per night sleeping by seven men and their youngest child.
| Hours sleep per night of father (x) | 6.7 | 7.1 | 7.2 | 7.9 | 8.1 | 8.2 | 8.2 |
| Hours sleep per night of youngest child (y) | 7.9 | 8.0 | 8.5 | 8.7 | 9.1 | 9.2 | 9.5 |
The relationship can be modelled by a regression line with equation y=mx+cy=mx+c.
(a) Find the correlation coefficient.
(b) Write down the value of mm and of cc.
(c) Young Ramin sleeps for an average of 8.6 hours per day. Use your regression line to predict the average number of hours his father sleeps. Give your answer in hours and minutes correct to the nearest minute.
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+by=ax+b. Write down the value of aa and of bb.
(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.
In an experiment 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, xx (metres) | 6 | 14 | 20 | 25 | 35 | 48 | 46 | 48 | 52 |
| Circumference, yy (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, x¯x¯ , of the trees from the beach.
(d) Work out the mean circumference, y¯y¯ , of the trees.
(e) Plot and label the point M(x¯,y¯x¯,y¯) on your graph.
(f) Write down the Pearson’s product–moment correlation coefficient, rr , for Darren’s results.
(g) Find the equation of the regression line yy on xx, for Darren’s results.
(h) Draw the regression line yy on xx on your graph.
(i) Use the equation of the regression line yy on xx to estimate the circumference of a tree that is 42 m from the beach.
The Farang Parkour Team hosted a Free Running event. The judges, Anan and Jason awarded 7 competitors a score out of 10. The scores are shown in the following table.
| Free Runners | A | B | C | D | E | F | G |
| Anan’s Score (x) | 7.8 | 9.1 | 8.3 | 6.9 | 7.0 | 8.5 | 9.3 |
| Jason’s Score (y) | 7.2 | 9.0 | 8.7 | 7.5 | 6.9 | 8.7 | 8.9 |
(a) Find the Pearson’s product–moment correlation coefficient, rr, of these scores.
(b) Using the value of rr, interpret the relationship between Anan’s scores and Jason’s score.
(c) Write down the equation of the regression line yy on xx.
(d) Use your regression equation from part (c) to estimate Jason’s score to one decimal place when Anan awards a score of 5.
(e) State whether this estimate is reliable. Justify your answer.
(f) The adjudicator for the event would like to find the Spearman’s rank correlation coefficient of the scores. Copy and complete the information in the following table.
| Free Runners | A | B | C | D | E | F | G |
| Anan’s Rank | 7 | 1 | |||||
| Jason’s Rank | 1 | 7 |
(g) Find the value of the Spearman’s rank correlation coefficient,
(h) Comment on the result obtained.
In a survey of insect life near a stream, a student collected data about the number of different insect species (y)(y) that were found at different distances (x)(x) in metres from the stream.
| Distance (x)(x) | 2 | 5 | 8 | 11 | 14 | 17 | 22 | 33 | 39 |
|---|---|---|---|---|---|---|---|---|---|
| Insect species (y)(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 x¯x¯, the mean of the distances from the stream;
(d) Find y¯y¯, the mean number of insect species.
(e) Plot the point (x¯,y¯)(x¯,y¯) on your scatter diagram. Label this point M.
(f) Write down the equation of the regression line yy on xx for the above data.
(g) Draw the regression line yy on xx on your scatter diagram.
(h) Estimate the number of insect species to be found 30 metres from the stream.
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 (xx) | 74 | 73 | 65 | 75 | 68 | 72 | 69 | 71 | 83 | 68 | 68 | 73 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Paper 2 (yy) | 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, rr.
(e) Write down the equation of the regression line of yy on xx.
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.
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