2023 NYJC P2 Q10

2023 NYJC P2 Q10

12 marks

An anemometer is used to estimate wind speed by observing the rotational speed of its vanes. This speed is converted to wind speed by means of an equation obtained from calibrating the instrument in a wind tunnel. In this calibration process the wind speed is fixed precisely and the resulting anemometer speed noted. For a particular anemometer, this process produced the following set of data.

Actual wind speed, \(x\)
(\(\text{m/s}\))		\(1.0\)\(1.2\)\(1.4\)\(1.6\)\(1.8\)\(2.0\)\(2.2\)
Anemometer, \(y\)
(\(\text{revs/min}\))		\(15\)\(31\)\(71\)\(112\)\(168\)\(230\)\(317\)
  1. Draw a scatter diagram of these data. Use your diagram to explain whether the relationship between \(y\) and \(x\) is likely to be well modelled by an equation of the form \(y = ax + b\), where \(a\) and \(b\) are constants.[2]
  2. By calculating the relevant product moment correlation coefficients, determine whether the relationship between \(y\) and \(x\) is modelled better by \(y = ax + b\) or \(y = ax^2 + b\). Explain how you decide which model is better and state the equation in this case.[5]
  3. Use your equation to estimate the wind speed when the rotational speed of the anemometer is \(100\text{ revs/min}\). Explain whether your estimate is reliable.[2]
  4. Use your equation to estimate the rotational speed of the anemometer when the actual wind speed is \(2.4\text{ m/s}\). Explain whether your estimate is reliable.[2]

  5. Suppose there was an error in recording the values of \(y\) and these values must be increased by \(5\text{ revs/min}\), re-write your equation from part (b) so that it can be used for the correct values of \(y\).

    [1]
Similar questions are unavailable for this question.
Answer:(a) It is unlikely to be well modelled by the equation \({y = ax + b}\) as the scatter diagram shows a curvilinear relationship, that is as \({x}\) increases, \({y}\) increases at an increasing rate. (b) \({y = 79.0x^2 - 80.0}\) (\({3}\) s.f.) (c) \({x = 1.51\text{ m/s}}\). Since \({y = 100}\) is within the data range and \({r}\) is close to \({1}\), the estimate is reliable. (d) \({y = 375\text{ revs/min}}\). Since \({x = 2.4}\) is out of the data range of \({x}\), the estimate is not reliable. (e) \({y = 78.982x^2 - 74.973}\)

Need help? Join our JC Math tuition classes.

Learn more