2024 MI H1 Q11
2024 MI H1 Q11
9 marks
Scientists studying the climate of a national park placed recording instruments at \(8\) locations of differing altitudes. The altitudes above sea level, \(x\) m, and the average daily temperatures, \(t\) \(^\circ\)C, and are shown in the following table.
| \(x\) | 110 | 237 | 502 | 798 | 1305 | 1974 | 2358 | 3521 |
|---|---|---|---|---|---|---|---|---|
| \(t\) | 22.4 | 20.1 | 16.9 | 15.6 | 12.8 | 8.7 | 4.5 | -5.2 |
- Give a sketch of the scatter diagram for the data, as shown on your calculator.[2]
- Find the product moment correlation coefficient and comment on its value in the context of the data.[2]
- Find the equation of the regression line of \(t\) on \(x\), giving your answer in the form \(t = ax + b\), with the values of \(a\) and \(b\) correct to 3 significant figures. Sketch this line on your scatter diagram.[2]
- Use the equation of your regression line to calculate an estimate of the average daily temperature of a location at \(3000\) m above sea level, leaving your answer to 1 decimal place. Comment on the reliability of your estimate.[2]
- A hiker came across this set of data and would like to make estimations for his upcoming hike to another national park of similar altitude. State, with a reason, if he would obtain reliable estimates.[1]
Solution:
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Answer:(i) Scatter and regression line shown. (ii) \(r\approx-0.994\), strong negative linear association. (iii) \(t=-0.00756x+22.2\) (3 s.f. coefficients). (iv) \(-0.5^\circ\mathrm C\), using unrounded line; reasonably reliable interpolation. (v) Not established as reliable for a different national park.
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