| Date | None Specimen | Marks available | 3 | Reference code | SPNone.1.hl.TZ0.10 |
| Level | HL only | Paper | 1 | Time zone | TZ0 |
| Command term | Calculate and Estimate | Question number | 10 | Adapted from | N/A |
Question
Bill is investigating whether or not there is a positive association between the heights and weights of boys of a certain age. He defines the hypotheses\[{{\rm{H}}_0}:\rho = 0;{{\rm{H}}_1}:\rho > 0 ,\]where \(\rho \) denotes the population correlation coefficient between heights and weights of boys of this age. He measures the height, \(h\) cm, and weight, \(w\) kg, of each of a random sample of \(20\) boys of this age and he calculates the following statistics.\[\sum {w = 340,\sum {h = 2002,\sum {{w^2} = 5830} } } ,\sum {{h^2} = 201124} ,\sum {hw = 34150} \]
(i) Calculate the correlation coefficient for this sample.
(ii) Calculate the \(p\)-value of your result and interpret it at the \(1\% \) level of significance.
(i) Calculate the equation of the least squares regression line of \(w\) on \(h\) .
(ii) The height of a randomly selected boy of this age of \(90\) cm. Estimate his weight.
Markscheme
(i) \(r = \frac{{34150 - 340 \times \frac{{2002}}{{20}}}}{{\sqrt {\left( {5830 - \frac{{{{340}^2}}}{{20}}} \right)} \left( {201124 - \frac{{{{2002}^2}}}{{20}}} \right)}}\) (M1)(A1)
Note: Accept equivalent formula.
\( = 0.610\) A1
(ii) (\(T = R \times \sqrt {\frac{{n - 2}}{{1 - {R^2}}}} \) has the t-distribution with \(n - 2\) degrees of freedom)
\(t = 0.6097666 \ldots \sqrt {\frac{{18}}{{1 - 0.6097666{ \ldots ^2}}}} \) M1
\( = 3.2640 \ldots \) A1
\({\rm{DF}} = 18\) A1
\(p{\rm{ - value}} = 0.00215 \ldots \) A1
this is less than \(0.01\), so we conclude that there is a positive association between heights and weights of boys of this age R1
[8 marks]
(i) the equation of the regression line of \(w\) on \(h\) is
\(w - \frac{{340}}{{20}} = \left( {\frac{{20 \times 34150 - 340 \times 2002}}{{20 \times 201124 - {{2002}^2}}}} \right)\left( {h - \frac{{2002}}{{20}}} \right)\) M1
\(w = 0.160h + 0.957\) A1
(ii) putting \(h = 90\) , \(w = 15.4\) (kg) A1
Note: Award M0A0A0 for calculation of \(h\) on \(w\).
[3 marks]
