Foundation November 2024 Paper 3 Q23
23 A farmer has 60 pear trees.
The table shows the heights, \(h\) metres, of the pear trees.
| Height (\(h\) metres) | Frequency | ||
|---|---|---|---|
| \(1 < h \leqslant 2\) | 5 | ||
| \(2 < h \leqslant 3\) | 8 | ||
| \(3 < h \leqslant 4\) | 32 | ||
| \(4 < h \leqslant 5\) | 15 |
(a) Calculate an estimate of the mean height of the 60 pear trees. [4]
(b) Explain why it is not possible to use the information from this table to calculate the exact value of the mean height. [1]
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| 3.45 | 4 | B1 for at least three of 1.5, 2.5, 3.5, 4.5 | May be implied by three correct products (7.5, 20, 112, 67.5) or [total =] 207 |
| M1 FT for \(\Sigma mf\) where \(m\) is a consistent value within each group \(1.5 \times 5 + 2.5 \times 8 + 3.5 \times 32 + 4.5 \times 15\) soi by 7.5 + 20 + 112 + 67.5 or 207 | FT their “midpoints” seen M1 may be implied by Lower: 5+16+96+60 (177) Upper: 10+24+128+75 (237) Allow one error in calculation | ||
| M1 FT dep on M1 for their \(207 \div 60\) | FT from lower 2.95, upper 3.95 | ||
| Answer | Marks | Part marks and guidance | |
|---|---|---|---|
| Exact heights are not known oe | 1 | See appendix Do not accept comments on the method used Do not accept “estimate” unless clarified in comment | |
Appendix
Question 23b
| Response | Mark | |
|---|---|---|
| Because it does not give us an exact height of a pear tree | Correct as mentions “exact height” | 1 |
| We don’t know the exact measurements | Accept measurements for heights | 1 |
| The heights are not exactly accurate | BOD | 1 |
| The data are not accurate | No as could refer to frequencies | 0 |
| We don’t know the value of H | Exact value not mentioned | 0 |
| We can only estimate the height, we won’t know the exact height | The first part suggests a comment on part (a) | 0 |
| Because the metres are in ranges and not precise | Comment on method | 0 |
| Because it’s an estimate between two heights | This is a comment on method | 0 |
| We have to use the middle value of height each time | This is a comment on method | 0 |
| Because the heights are not given | They are but not exact heights | 0 |
| It is not accurate | No mention of exact heights not known | 0 |
| Due to it being an estimate so the exact can be far off | This is a comment on part (a) and the method | 0 |
| Because the heights are given as \(1 < h \leqslant 2\) | This is a comment on method | 0 |
| Because it is an estimate and isn’t an exact number | This is a comment on the answer in (a) | 0 |
| Because the frequency isn’t exact, it’s an estimate of the height | Incorrect term. “Because the height isn’t exact” would get the mark | 0 |
| There is not enough data to tell the exact value of mean height | This is some comment about small samples | 0 |
| Because the mid-point is used | Comment on method | 0 |
| Because it is grouped data | Comment on method | 0 |