15 80 cyclists take part in a race. A summary of their times is shown in the table.
Time (\(t\) minutes)
Frequency
\(20 < t \leqslant 25\)
5
\(25 < t \leqslant 30\)
15
\(30 < t \leqslant 35\)
24
\(35 < t \leqslant 40\)
25
\(40 < t \leqslant 45\)
7
\(45 < t \leqslant 50\)
4
(a) Complete the cumulative frequency table.
Time (\(t\) minutes)
Cumulative frequency
\(t \leqslant 25\)
5
\(t \leqslant 30\)
\(t \leqslant 35\)
\(t \leqslant 40\)
\(t \leqslant 45\)
\(t \leqslant 50\)
[2]
(b) Draw the cumulative frequency graph to show the information.
[3]
(c) Reece makes two comments about the times taken to complete the race. For each comment, decide if Reece is right or wrong and give a reason for your answer.
(i) \(\frac{3}{4}\) of the 80 cyclists took more than 30 minutes to complete the race. [2]
(ii) The longest time that any of the 80 cyclists took to complete the race must have been 50 minutes. [1]
Mark scheme (a)
Answer
Marks
Part marks and guidance
20, 44, 69, 76, 80
2
M1 for cf calculated with one arithmetic error
Allow 80 or their 80 FT
Mark scheme (b)
Answer
Marks
Part marks and guidance
Plots at right-hand end of intervals
1
Plots of frequencies scores 0 across the whole of (b)
Plots at correct heights
1
Tol ½ square FT if M1 in (a)
0 for bars at correct heights since must miss off one end
Join with smooth curve or straight line segments
1
Tol ½ square FT their ascending plots only Condone curve or absence of curve below \(t = 25\)
if cf graph as well as bars, ignore bars
Mark scheme (c)
Answer
Marks
Part marks and guidance
(i)
Sight of 20 or 60
1
For 2 marks must say right/correct/true not yes
Must be separate from the fraction comment Do not accept more than/less than 60/20
See Appendix
\(\frac{20}{80} = \frac{1}{4}\) or \(\frac{1}{4}\) of 80 = 20 \(\frac{60}{80} = \frac{3}{4}\) or \(\frac{3}{4}\) of 80 = 60
1
Accept equivalent in words
(ii)
Wrong/incorrect/false and correct reason
1
e.g. cannot be sure as exact data not given; it could be anywhere between 45 and 50 etc.
Appendix: exemplar responses for Q15(c)(i)
Reason
Judgement
Mark
Correct, ¾ of 80 is 60 and 20 cyclists completed within 30 minutes, leaving the remaining ¾ of cyclists taking over 30 minutes
1 1
True, ¾ of 80 = 60 and 60 cyclists took more than 30 minutes
1 1
Right, 20 cyclists completed the race in under 30 minutes and 20/80 = ¼.
1 1
Right, 60 cyclists took more than 30 minutes and ¾ of 80 = 60.
1 1
Wrong, 60 cyclists took more than 30 minutes and ¾ of 80 = 60.
Despite “wrong”, mark the comments but do not award full marks
1 0
Yes, 60 cyclists took more than 30 minutes and ¾ of 80 = 60.
Do not award 2 marks with “yes”
1 0
Correct, if you add the amount of times after 30 you get 60 and there’s 80 cyclists.
1 0
Right, 60 cyclists took more than 30 minutes.
1 0
Wrong, 60 cyclists took more than 30 minutes.
Despite “wrong”, mark the comment
1 0
True, the lower quartile is at 30mins
Implies 20
1 0
¾ of 80 = 60
0 1
Appendix: exemplar responses for Q15(c)(ii)
Reason
Judgement
Mark
Wrong, all we know is that it is between 45 and 50
Condone 46 and 50
1
Wrong, there is a range of values for the time so not definitely 50
1
Wrong, as the time is given as a range
1
Wrong, we are not given the accurate values
1
Wrong, it might be but we need to know the exact values to find out
