The following table shows the population (in thousands) of a city over several years.
| Year \(t\) |
2010 |
2012 |
2014 |
2016 |
2018 |
2020 |
| Population (\(P\), thousands) |
50.2 |
53.1 |
56.3 |
59.8 |
63.6 |
67.7 |
Let \(t\) be the number of years since 2010. It is believed the population can be modelled by \(P = a \times b^t\).
(a) Use linear regression on transformed data (\(\ln P\) vs \(t\)) to find the value of \(a\) and the value of \(b\). [4]
(b) Use the model to predict the population in 2025. [2]
(c) Comment on the reliability of your prediction in part (b). [1]
Question
Use linear regression on transformed data (\(\ln P\) vs \(t\)) to find the value of \(a\) and the value of \(b\). [4]
Hint
Hint
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Take the natural logarithm of the population values and perform linear regression against \(t\) to find \(\ln a\) and \(\ln b\). Then find \(a\) by exponentiating \(\ln a\) and compute \(b\) as \(b = e^{slope}\).
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a ≈ 50.2 and b ≈ 1.03
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Question
Use the model to predict the population in 2025. [2]
Hint
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For 2025, calculate \(t\) by subtracting 2010 from 2025, then substitute \(t\) into the model \(P = 50.2 \times (1.03)^t\).
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Population ≈ 78.2 thousand
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Comment on the reliability of your prediction in part (b). [1]
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Consider the effects of extrapolation when using models based on limited data.
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The prediction is less reliable because it is based on extrapolation beyond the available data.
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