S1 June 2011 Q7
7. A teacher took a random sample of 8 children from a class. For each child the teacher recorded the length of their left foot, \(f\) cm, and their height, \(h\) cm. The results are given in the table below.
| \(f\) | 23 | 26 | 23 | 22 | 27 | 24 | 20 | 21 |
|---|---|---|---|---|---|---|---|---|
| \(h\) | 135 | 144 | 134 | 136 | 140 | 134 | 130 | 132 |
(You may use \(\sum f = 186 \quad \sum h = 1085 \quad \mathrm{S}_{ff} = 39.5 \quad \mathrm{S}_{hh} = 139.875 \quad \sum fh = 25291\))
Give the value of \(a\) and the value of \(b\) correct to 3 significant figures. (5)
The left foot length of the teacher is 25 cm.
| Scheme | Marks |
|---|---|
| \((\mathrm{S}_{fh} =)\ 25291 - \dfrac{186 \times 1085}{8}\) | M1 |
| = 64.75 (accept 64.8) | A1 |
| (2) |
Notes
[NB \(r\) = 0.871 so do not confuse this with question 1]
M1 for attempting a correct expression [allow a copying slip e.g. 25921]
| Scheme | Marks |
|---|---|
| \(b = \dfrac{\text{"}64.75\text{"}}{39.5}\), = 1.6392…. (awrt 1.6) | M1, A1 |
| \(a = \dfrac{1085}{8} - b \times \dfrac{186}{8}\), = 97.512… (awrt 97.5) | M1, A1 |
| \(h = 97.5 + 1.64f\) | A1ft (dep on M1M1) |
| (5) |
Notes
1st M1 for a correct expression for \(b\), ft their part (a) but not \(\mathrm{S}_{fh}\) = 25291
1st A1 for awrt 1.6
2nd M1 for use of \(a = \bar{h} - b \times \bar{f}\), ft their value for \(b\). Must use \(\bar{h}\) and \(\bar{f}\) not values from table.
2nd A1 for awrt 97.5 [NB \(a\) = 135 – 1.63\(\times\)23 = 97.51 but M0A0 since not using \(\bar{h}\) and \(\bar{f}\) ]
3rd A1ft for an equation for \(h\) and \(f\) with their coefficients to 3sf. Dependent on both Ms
Must be 3sf not awrt. Give this mark if seen in (c). Equation must be in \(h\) and \(f\) not \(y\) and \(x\).
| Scheme | Marks |
|---|---|
| \(h = 97.5 + 1.64 \times 25\), = 138~139 (final answer in [138, 139]) | M1, A1 |
| (2) |
Notes
M1 for using their equation and \(f\) = 25 to find \(h\)
A1 for their final answer in [138, 139]. Can give if they have 137.7... but round to 138
| Scheme | Marks |
|---|---|
| Should be reliable, since 25 cm(or \(f\) or footlength) is within the range of the data | B1, B1 |
| (2) |
Notes
1st B1 for suggesting it is reliable
2nd B1 for mentioning that 25 cm is within range of data. “interpolation”or“not extrapol’B1
Use of “it” or a comment that height is in range is B0 but apply ISW
| Scheme | Marks |
|---|---|
| Line is for children – a different equation would apply to adults or Children are still growing, height will increase more than foot length | B1 |
| (1) | |
| (12 marks) |
Notes
B1 for some comment that states a difference between children and teachers(adults)
Must mention teacher/adults and children
e.g. ".teacher is not in same age group as the children", "equation is for children not adults"
“children and adults are different populations”
"teacher will be taller" is B0 since no mention of children.
“equation is only valid for children” is OK since “only” implies not suitable for adults
Or Reference to different growth rates