Numeracy Intermediate November 2022 Unit 2 Q6
6. Delyth and Ronnie are both students at the local college.
Their houses and the college are all joined by straight roads, as shown in the diagram.
Diagram not drawn to scale
Delyth usually walks directly to college.
Calculate how much further Delyth has to walk if she passes Ronnie’s house on her way to college. [5]
35 students were asked how far they travelled to the college.
The results are recorded in the table below.
| Distance, \(d\) (metres) | Frequency |
|---|---|
| \(100 \lt d \leqslant 200\) | 9 |
| \(200 \lt d \leqslant 1000\) | 10 |
| \(1000 \lt d \leqslant 3000\) | 15 |
| \(3000 \lt d \leqslant 7000\) | 1 |
He walks 200 m directly to college.
Does Ronnie travel further than the median distance travelled by these 35 students?
Yes ☐ No ☐ Can’t tell ☐
You must give a reason for your answer. [1]
Estimate of the mean distance travelled by these 35 students is …………… m
There are 140 students who travel by bus to and from college.
Delyth wants to find out why these students do not walk to college.
She has decided to use a systematic sampling method to select 7 of these students to form a discussion group.
The names of all the 140 students are in a list.
Delyth has randomly selected the 2nd student in the list to join the discussion group.
Complete the table below to give the positions in the list of the 7 students selected to join the discussion group. [2]
| Student | 1 | 2 | 3 | 4 | 5 | 6 | 7 |
|---|---|---|---|---|---|---|---|
| Position in the list | 2nd | …………… | …………… | …………… | …………… | …………… | …………… |
| Scheme | Marks | Comments |
|---|---|---|
| (Direct2 =) 2002 + 3502 | M1 | |
| Direct2 = 162500 or (Direct =) √162500 | A1 | |
| (Direct =) 403(.11... m) or 50√65 (m) or √162500 (m) | A1 | FT from M1 for the correctly evaluated square root of ‘their 162500’ provided ‘their answer’ > 350 (m) May be implied in further working Mark final answer or the answer they go on to use, but then FT |
| (Extra distance =) 200 + 350 – 403(.1...) or 200 + 350 – 50√65 or 200 + 350 – √162500 | M1 | FT ‘their derived 403(.11…)’ > 350 and from an attempt to use Pythagoras’ Theorem |
| 146.8(87....m) or 146.9(m) or 147(m) | A1 |
| Scheme | Marks | Comments |
|---|---|---|
| (i) Selects or unambiguously implies ‘No’ with a reason, e.g. ‘the median is in group >200m to 1000m (and he lives 200m away)’, ‘median is more than 200m away (but Ronnie is 200m away)’ | E1 | Needs to compare 200(m) with median >200(m) The 200(m) can be implied from selecting ‘No’ Ignore additional spurious statements Allow ‘No’ with a reason, e.g. ‘Ronnie’s distance is in the first group, the median is in the second group’ ‘Ronnie only travels 200m which is less than the median (distance)’ ‘because the median distance travelled is between 200m and 1000m’ ‘Ronnie doesn’t travel the distance of the 17.5(th) person’ ‘Ronnie doesn’t travel the distance of the 17(th) (or 18th) person’ ‘the median 17.5(th)’ ‘the median 17(th) (or 18(th))’ ‘he only walks 200m when the (median) distance is higher’ ‘he only walks 200m which is less than the median’ ‘can’t estimate exact number from the group 200< d ≤ 1000’ ‘the median could be 880’ ‘9 less than half of 35’ ’26 students walk further than him’ Do not accept ‘No’ with a reason e.g. ‘Ronnie’s distance is in the first group’ ‘the median is 250m’ |
| (ii) Midpoints 150, 600, 2000, 5000 | B1 | Check the table Sight of 7750 implies correct midpoints |
| 150×9 + 600×10 + 2000×15 + 5000×1 (= 1350 + 6000 + 30000 + 5000 = 42350 m) | M1 | FT ‘their midpoints’ provided at least 3 are within or at the bounds of the appropriate groups |
| ÷ 35 | m1 | |
| 1210 (m) | A1 | Answer space takes precedence |
| Scheme | Marks | Comments | ||||||||||||||
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| (140 ÷ 7 =) 20 or 140 ÷ 20 = 7 or 7 × 20 = 140 | B1 | May be implied by any of the following:
| ||||||||||||||
| B1 | CAO |