N2023 P2 Q7
N2023 P2 Q7
Junior College 2
7 marks
Free
The numbers of mobile phone subscriptions in Singapore, \(y\) million, for certain years from 2004 are given in the following table. The variable \(x\) is the number of years after a base year of 2000.
| Year | \(2004\) | \(2006\) | \(2008\) | \(2010\) | \(2012\) | \(2014\) | \(2016\) | \(2018\) |
| Number of years after base year, \(x\) | \(4\) | \(6\) | \(8\) | \(10\) | \(12\) | \(14\) | \(16\) | \(18\) |
| Number of subscriptions, \(y\) million | \(3.99\) | \(4.79\) | \(6.41\) | \(7.38\) | \(8.07\) | \(8.10\) | \(8.46\) | \(8.39\) |
Ling thinks that the number of mobile phone subscriptions in Singapore can be modelled by one of the formulae \(y=ax+b\), \(e^y=cx+d\), where \(a\), \(b\), \(c\) and \(d\) are constants.
- Find, correct to 4 decimal places, the value of the product moment correlation coefficient
- between \(x\) and \(y\),[1]
- between \(x\) and \(e^y\).[1]
- Explain which of Ling’s models, \(y=ax+b\) or \(e^y=cx+d\), gives a better fit to the data and find the equation of the regression line for this model.[3]
- Use the equation of the regression line to estimate the number of mobile phone subscriptions in 2024. Explain whether your estimate is reliable.[2]
Default solution
- Using the given \((x,y)\) data in the calculator's correlation calculation gives\(r_{x,y}=0.9281271487\ldots=\boxed{0.9281}\)
- Create the transformed data column \(\mathrm e^y\) and correlate it with \(x\).\(r_{x,\mathrm e^y}=0.9697235231\ldots=\boxed{0.9697}\)
- The second correlation is closer to 1. Under the question's correlation comparison, \(\mathrm e^y=cx+d\) gives the better fit. Regress \(\mathrm e^y\) on \(x\).\(\mathrm e^y=375.6152263\ldots x-1881.4818267\ldots\)\(\boxed{\mathrm e^y=376x-1880}\quad(3\text{ s.f.})\)The equivalent logarithmic prediction is used only where \(375.6152263\ldots x-1881.4818267\ldots>0\), or \(x>5.009066978\ldots\). The fitted model should not be treated as valid for every year.
- The year 2024 corresponds to \(x=24\). Retain the unrounded regression coefficients.\(y\)\({}=\ln\left(375.6152263\ldots(24)-1881.4818267\ldots\right)\)\({}=8.872526941\ldots\)\({}\approx\boxed{8.87\text{ million subscriptions}}\)This estimate is not reliable: \(x=24\) lies outside the observed range \(4\le x\le18\), so it is an extrapolation and the relationship may not continue beyond those years.
Answer:(a)(i) \(0.9281\). (ii) \(0.9697\). (b) \(\mathrm e^y=cx+d\), \(c=375.6152263\ldots,d=-1881.4818267\ldots\) (or \(\mathrm e^y=376x-1880\), 3 s.f.). (c) \(8.87\) million; unreliable extrapolation.
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