2009 H2 Mathematics Paper 2 Question 6

Linear Correlation and Regression

Answers

(i)

(ii)

The scatter diagram suggests that as x{x} increases, t{t} is decreasing at a decreasing rate, which is not consistent with a linear model. Hence a linear model may not be appropriate.
Moreover, a linear model will predict that the world record time will continually decrease at the same rate over the years. This is inappropriate as there are physical limits to the time in running 1 mile and the world record time cannot decrease indefinitely.

(iii)

A quadratic model will predict a minimum world record time, after which the world record time will increase. This is not appropriate for long-term predictions as the world record time cannot increase.

(iv)

ln⁡t=34.9−0.0161x{\ln t = 34.9-0.0161x}
3{3} minutes 41.4{41.4} seconds
The prediction is not reliably as x=2010{x=2010} is outside the given data range 1930≤x≤2000{1930 \leq x \leq 2000} and the observed data trend may no longer hold.

Full solutions

(i)

(ii)

The scatter diagram suggests that as x{x} increases, t{t} is decreasing at a decreasing rate, which is not consistent with a linear model. Hence a linear model may not be appropriate. ■{\blacksquare}
Moreover, a linear model will predict that the world record time will continually decrease at the same rate over the years. This is inappropriate as there are physical limits to the time in running 1 mile and the world record time cannot decrease at the same rate indefinitely. ■{\blacksquare}

(iii)

A quadratic model will predict a minimum world record time, after which the world record time will increase. This is not appropriate for long-term predictions as the world record time cannot increase. ■{\blacksquare}

(iv)

Using a GC, least squares regression line of ln⁡t{\ln t } on x:{x:}
ln⁡t=34.853−0.016128xln⁡t=34.9−0.0161x (3 sf)  ■\begin{align*} & \ln t = 34.853-0.016128x \\ & \ln t = 34.9-0.0161x \textrm{ (3 sf)} \; \blacksquare \end{align*}
When x=2010,{x=2010,}
ln⁡t=34.853−0.016128(2010)t=11.4 (3 sf)\begin{align*} \ln t &= 34.853 -0.016128 (2010) \\ t &= 11.4 \textrm{ (3 sf)} \end{align*}
Hence the predicted world record time as at 1st January 2010:
3 minutes 41.4 seconds  ■3 \textrm{ minutes } 41.4 \textrm{ seconds} \; \blacksquare
The prediction is not reliably as x=2010{x=2010} is outside the given data range 1930≤x≤2000{1930 \leq x \leq 2000} and the observed data trend may no longer hold. ■{\blacksquare}