2023 SAJC P2 Q10

Tom records the maximum speeds of an athlete running in a temperature controlled environment. The table below shows the maximum speed, \(y\) metres per second, recorded for various specific temperatures, \(x\) degree Celsius.

1. Tom decides to model the data using the line \(y=-0.02x+3.1\).
   1. On the grid opposite
      • draw a scatter diagram of the data, and
      • draw the line \(y=-0.02x+3.1\).[3]
   2. For a line of best fit \(y=\mathrm{f}\left( x \right)\), the residual for a point \(\left( a,b \right)\) plotted on the scatter diagram is the vertical distance between \(\left( a,\mathrm{f}\left( a \right) \right)\) and \(\left( a,b \right)\). Mark the residual for each point on your diagram.[1]
   3. Calculate the sum of the squares of the residuals for Tom’s line.[1]
   4. Explain why, in general, the sum of the squares of the residuals rather than the sum of the residuals is used.[1]
2. State the value of the product moment correlation coefficient.[1]
3. Use your calculator to find the equation of a suitable regression line to estimate the temperature for a given maximum speed. Hence estimate the temperature when the maximum speed is \(2.48\) metres/second. Explain whether you would expect this value to be reliable.[4]
4. Explain whether the product moment correlation coefficient would differ if Tom converted the speed to kilometres per hour.[1]