2025 RVHS P2 Q10

As part of the sound calibration for the National Day Parade at the Padang, sound engineers test how sound intensity changes with distance from the main stage speakers during rehearsals. This is helpful for calibrating the speaker system to avoid sound distortion or echo across seating zones.

At a particular rehearsal, the engineers record the sound intensity, \(y\), measured in decibels, at various distances, \(x\), measured in metres from the stage speakers and recorded the data in the table below.

1. Sketch a scatter diagram for these values. Explain, using your diagram, which of the following equations provides the most accurate model of the relationship between \(x\) and \(y\).

   \((A) \quad y = a - bx^2,\)

   \((B) \quad y = a + be^{-x},\)

   \((C) \quad y = a + b \ln(10 - x),\)

   where \(a\) and \(b\) are constants such that \(a > 0\) and \(b > 0\).

   [3]
2. Using the model you chose in part (a), find the equation of the least squares regression line. State the corresponding value of the product moment correlation coefficient, correct to \(5\) decimal places.[2]
3. At a particular spectator seat in the Padang, the sound intensity was recorded to be \(20\) decibels. Estimate the distance of this spectator seat from the stage speakers. Explain whether your estimate is reliable.[2]

As another part of sound calibration for the National Day Parade, engineers also measured the time that it takes for sound to reach the spectator stands, \(t\), measured in milliseconds, at various distances, \(x\), measured in metres from the stage speakers.

4. The engineers would like to investigate if the model \(t = 2.916x\) is a good fit of the above data. For a model \(t = \mathrm{f}(x)\), the residual for a point \((p, q)\) is \(q - \mathrm{f}(p)\).
   1. Explain why they should use the sum of the squares of the residuals rather than the sum of the residuals when assessing the fit of the model.[1]
   2. Calculate the sum of the squares of the residuals for the line \(t = 2.916x\).[2]
   3. It is further found that another line \(t = 2.916x + D\) where \(D \neq 0\) is a better fit for the above data. Explain how the sum of the squares of the residuals for the line \(t = 2.916x + D\) differs from the sum of the squares of the residuals of \(t = 2.916x\).[1]