2024 TJC H1 Q10
2024 TJC H1 Q10
9 marks
The girth of a tree is the circumference of the tree trunk measured \(1\) metre above the ground. A random sample of \(9\) oak trees in a forest was taken and the age, \(x\) years, and the girth, \(y\) cm, of each tree was recorded. The data collected is shown in the following table.
| \(x\) | \(20\) | \(23\) | \(30\) | \(38\) | \(39\) | \(45\) | \(48\) | \(55\) | \(71\) |
|---|---|---|---|---|---|---|---|---|---|
| \(y\) | \(35\) | \(41\) | \(59\) | \(70\) | \(67\) | \(87\) | \(91\) | \(109\) | \(124\) |
- Give a sketch of the scatter diagram for the data, as shown on your calculator.[2]
- Find the value of 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 \(y\) on \(x\), in the form \(y = a + bx\), giving 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 girth of a \(50\)-year-old oak tree, and comment on the reliability of your estimate.[2]
- Rewrite your equation in (c) so that it can be used to estimate the girth when the age is measured in months.[1]
Solution:
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Answer:(a), (c) See scatter diagram and fitted line. (b) \(r\approx0.988\), strong positive linear correlation. (c) \(y=0.954+1.83x\). (d) \(92.3\) cm, reasonably reliable interpolation. (e) \(y\approx0.954+0.152u\), where \(u\) is age in months.
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