2017 ACJC P2 Q8
A swim school takes in both male and female primary school students for competitive swimming lessons. The school assesses its students’ progress each year by recording the time, $t$ seconds, each student takes to swim a $50$-metre lap in breaststroke, and the number of months, $m$, that he or she has been at the school. The records for $8$ randomly chosen students are shown in the following table.
(i)
Labelling the axes clearly, draw a scatter diagram for the data and explain, in context, why a linear model would not be suitable to predict the time taken by a student to swim a lap of breaststroke given the number of months that he or she has been at the school.
[2]
It is desired to fit a model of the form $\ln \left( t-C \right)=a+bm$, where $C$ is a suitable constant. The product moment correlation coefficient $r$ between $m$ and $\ln \left( t-C \right)$ for some possible values of $C$ are shown in the table below.
(ii)
Calculate the value of $r$ for $C=37$, giving your answer correct to 6 decimal places.
[1]
(iii)
Use the table and your answer to (ii) to choose the most appropriate value for $C$. Explain your choice.
[2]
For the remainder of this question, use the value of $C$ that you have chosen in (iii).
(iv)
Find the equation of the least squares regression line of $\ln \left( t-C \right)$ on $m$. Give an interpretation of $C$ in the context of the question.
[2]
(v)
Another student who has been swimming at the school for $9$ months clocked a time of $60.33$ seconds for a lap of breaststroke. Using your regression line, comment on the student’s swimming ability.
[2]
(vi)
Suggest an improvement to the data collection process so that the results could provide a fairer gauge of the expected outcome for the students in the first $2$ years of lessons.
[1]
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