A swim school takes in both male and female primary school students for competitive swimming lessons

Question

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.

$m$	$6$	$7$	$10$	$12$	$15$	$19$	$21$	$24$
$t$	$92.32$	$87.11$	$66.12$	$59.41$	$53.94$	$43.82$	$42.07$	$41.45$

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.

$C$	$36$	$37$	$38$	$39$
$r$	$-0.992114$		$-0.992681$	$-0.992192$

Calculate the value of $r$ for $C=37$, giving your answer correct to 6 decimal places.[1]
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).

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]
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]
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]

Accepted Answer

(i) Scatter diagram plotted; a linear model would eventually predict impossible negative times (ii) $r=-0.992555$ (iii) $C=38$ (iv) $\ln(t-38)=5.01236-0.163486m$; $C$ is the limiting fastest lap time (v) Predicted time $72.5$ seconds; $60.33$ seconds is substantially faster than expected (vi) Use a larger sample stratified by age, sex, and time at the school

2017 ACJC P2 Q8

2017 ACJC P2 Q8

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