Numeracy Higher June 2023 Unit 2 Q2
2. The table gives the heights of the 27 musicians in the Camwen Youth Orchestra.
| Height, \(h\) cm | Frequency |
|---|---|
| \(150 \leqslant h \lt 158\) | 3 |
| \(158 \leqslant h \lt 166\) | 10 |
| \(166 \leqslant h \lt 174\) | 9 |
| \(174 \leqslant h \lt 182\) | 4 |
| \(182 \leqslant h \lt 190\) | 0 |
| \(190 \leqslant h \lt 198\) | 1 |
(a)
(i)
In which group does the median height of a musician lie?
You must give a reason for your answer. [2]
Group: ……………………………………
Reason: ……………………………………
(ii)
Is it certain that there is at least one musician in the orchestra who is less than 154 cm tall?
Give a reason for your answer. [1]
Yes ☐ No ☐
(b)
Calculate an estimate of the mean height of a musician in the Camwen Youth Orchestra. [4]
| Scheme | Marks | Comments |
|---|---|---|
| (i) (Median group) 166 ≤ \(h\) < 174 | B1 | Accept ‘166 to 174’ or ‘166 – 174’ or ‘third group’ or ‘group 3’ or similar Do not accept 9 or 14 or 170 |
| Reason, e.g. ‘14th height’ | E1 | Depends on B1 previously awarded or previous B0 was due to giving the answer ‘9’, ‘14’ or ‘170’ E1 for clear indication that median height is the 14th Allow, e.g. sight of 14, ‘middle person’, ‘middle height’, ‘by counting the frequencies, ½ the people are taller’, ‘half the people are taller’, ‘13.5(th) (musician)’, ‘total of 27 (people), the middle of that is in the group’ Do not accept, e.g. ‘middle group’, ‘in the middle’, ‘middle’, ‘middle number’, ‘groups are not specific’, ‘because the median (height) is 174’ |
| (ii) Indicates unambiguously ‘No’ with a valid reason, such as ‘only know the group’ ‘it doesn’t show raw data’ ‘the actual heights are not given’, ‘the 3 people could be anywhere in the group 150cm to (less than) 158cm’, ‘no way of knowing individual heights’ | E1 | Ignore spurious additional comments Allow ‘No’ with, e.g. ‘don’t know the height of these 3 people ‘, ‘all 3 people could all be 155cm tall’, ‘everyone in group 150cm to 158cm could be 157cm’, ‘could all be taller than 154(cm)’, ‘3 of them from 150 to 158 but not certain of height’, ‘we only know they are between 150 and 158’, ‘of the 3 people there may be, (but it is not certain)’, ‘(data is) not specific’, ‘little chance as there are only 3 people in the group’, ‘the groups are not that specific’, ‘it’s not specific enough’, ‘there is a possibility that there is one person shorter than 154cm as the midpoint is 154cm’ Do not accept, e.g. ‘everyone in the group 150cm to 158cm could be 158cm tall’ |
| Scheme | Marks | Comments |
|---|---|---|
| Midpoints 154, 162, 170, 178, (186,) 194 | B1 | |
| 154×3 + 162×10 + 170×9 + 178×4 (+186×0) + 194×1 (= 462 + 1620 + 1530 + 712 + 0 + 194 = 4518) | M1 | 186×0 may not be seen FT ‘their midpoints’ or at the bounds of the appropriate groups, provided no more than one of ‘their midpoints’ lies outside the group |
| ÷ 27 | m1 | |
| 167(.333.... cm) or equivalent | A1 | ISW Treat an error of e.g. 186 × 0 written as 186, leading to total 4704, 4704 ÷ 27 = 174(.222….) as follows: B1 M1 m1 possible but A0 or equivalents on FT |