Problems adapted from questions set for previous Mathematics exams.
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.
The following table shows the 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 \(y = ax + b\)
(a) Find the value of \(a\) and of \(b\)
(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 \(y = mx + c\).
(a) Find the correlation coefficient.
(b) Write down the value of \(m\) and of \(c\).
(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 \(y = ax + b\). 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.
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.
|IB Applications and Interpretation|
The Scrumptious Sweet Company sell a variety pack of colourful, shaped sweets. The sweets are produced such that 60% are square and 40% are circular. It is known that 20% of the square shaped sweets and 40% of the circular sweets are coloured red.
(a) Show this information in a tree diagram.
A sweet is selected at random.
(b) Find the probability that the sweet is red.
(c) Given that the sweet is red, find the probability it is circular.
The Scrumptious Sweet Company also produce variety packs of Rainbow Gums. Their specifications state that the colours in each variety pack should be distributed as follows.
Inspector Lou Spowels opens a pack of 90 sweets and records the frequency of each colour:
To investigate if the sample is consistent with the company's specifications, Mr Spowels conducts a \(\chi^2\) goodness of fit test. The test is carried out at a 5% significance level.
(d) Write down the null hypothesis for this test.
(e) Copy and complete the following table giving the frequencies correct to one decimal place:
(f) Write down the number of degrees of freedom.
(g) Find the p-value for the test.
(h) State the conclusion of the test. Give a reason for your answer.
|IB Applications and Interpretation|
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.
|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, \(r\), of these scores.
(b) Using the value of \(r\), interpret the relationship between Anan’s scores and Jason’s score.
(c) Write down the equation of the regression line \(y\) on \(x\).
(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.
(g) Find the value of the Spearman’s rank correlation coefficient, \(r_s\).
(h) Comment on the result obtained for \(r_s\).
The adjudicator believes Jason’s score for competitor E is too high and so decreases the score from 6.9 to 6.5.
(i) Explain why the value of the Spearman’s rank correlation coefficient \(r_s\) does not change.
|IB Applications and Interpretation|
The 2nd Rutherford American Scouts joined the 37th Wolverhampton British Scouts for an International Camp. Skipper Jones is interested to see if the mean height of American Scouts, \( \mu_1\), is the same as the mean height of British Scouts, \( \mu_2\). The information is recorded in the following table.
|American Scout height (cm)||147||153||151||142||155||149||154||156||143||152||149||158|
|British Scout height (cm)||142||146||155||145||149||148||152||143||147||150||149||154||150||144||146|
At the 10% level of significance, a t-test was used to compare the means of the two groups. The data is assumed to be normally distributed and the standard deviations are equal between the two groups.
(a) State the null hypothesis.
(b) State the alternative hypothesis.
(c) Calculate the p-value for this test.
(d) State, giving a reason, whether Skipper Jones should accept the null hypothesis.
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.
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