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Question
The speed, \(s\) , in \({\text{km }}{{\text{h}}^{ – 1}}\), of \(120\) vehicles passing a point on the road was measured. The results are given below.
Write down the midpoint of the \(60 < s \leqslant 70\) interval.[1]
Use your graphic display calculator to find an estimate for
(i) the mean speed of the vehicles;
(ii) the standard deviation of the speeds of the vehicles.[3]
Write down the number of vehicles whose speed is less than or equal to \({\text{60 km }}{{\text{h}}^{ – 1}}\).[1]
Consider the cumulative frequency table below.
Write down the value of \(a\) , of \(b\) and of \(c\) .[2]
Consider the cumulative frequency table below.
Draw a cumulative frequency graph for the information from the table. Use \(1\) cm to represent \({\text{10 km }}{{\text{h}}^{ – 1}}\) on the horizontal axis and \(1\) cm to represent \(10\) vehicles on the vertical axis.[4]
Use your cumulative frequency graph to estimate
(i) the median speed of the vehicles;
(ii) the number of vehicles that are travelling at a speed less than or equal to \({\text{65 km }}{{\text{h}}^{ – 1}}\).[4]
All drivers whose vehicle’s speed is greater than one standard deviation above the speed limit of \({\text{50 km }}{{\text{h}}^{ – 1}}\) will be fined.
Use your graph to estimate the number of drivers who will be fined.[3]
Answer/Explanation
Markscheme
\(65\) (A1)[1 mark]
(i) \(54{\text{ (km }}{{\text{h}}^{ – 1}})\) (G2)
Note: If the answer to part (b)(i) is consistent with the answer to part (a) then award (G2)(ft) even if no working seen.
(ii) \(19.2\) (\(19.2093 \ldots \)) (G1)
Note: Accept \(19\), do not accept \(20\).[3 marks]
\(76\) (A1)[1 mark]
\(a = 76\), \(b = 98\) (A1)(ft)
Note: Follow through from their answer to part (c) for \(a\) and \(b = \) their \(a + 22\) .
\(c = 118\) (A1)[2 marks]
(A1)(A1)(ft)(A1)(ft)(A1)
Notes: Award (A1) for axes labelled and correct scales. If the axes are reversed do not award this mark but follow through. Award (A2)(ft) for their 6 points correct, (A1)(ft) for at least 3 of these points correct. Award (A1) for smooth curve drawn through all points including (\(0\), \(0\)). If either the \(x\) or the \(y\) axis has a break in it to zero, do not award this final mark.[4 marks]
(i) \(57\) \({\text{(km }}{{\text{h}}^{ – 1}})\) \(( \pm 2)\) (M1)(A1)(ft)(G2)
Note: Award (M1) for clear indication of median on their graph. Follow through from their graph. If their answer is consistent with their incorrect graph but there is no working present on graph then no marks are awarded.
(ii) \(90\) vehicles \(( \pm 2)\) (M1)(A1)(ft)(G2)
Note: Award (M1) for clear indication of method on their graph. Follow through from their graph. If their answer is consistent with their incorrect graph but there is no working present on graph then no marks are awarded.[4 marks]
\(50 + 19.2 = 69.2\) (A1)(ft)
\(24\) \(( \pm 2)\) drivers will be fined (M1)(A1)(ft)(G2)
Notes: Follow through from their graph and from their part (b)(ii). Award (M1) for indication of method on their graph. If their answer is consistent with their incorrect graph but there is no working present on graph then no marks are awarded.[3 marks]
Question
200 people were asked the amount of time T (minutes) they had spent in the supermarket. The results are represented in the table below.
State if the data is discrete or continuous.[1]
State the modal group.[1]
Write down the midpoint of the interval 10 < T ≤ 20 .[1]
Use your graphic display calculator to find an estimate for
(i) the mean;
(ii) the standard deviation.[3]
The results are represented in the cumulative frequency table below, with upper class boundaries of 10, 20, 30, 40, 50.
Write down the value of
(i) q;
(ii) r.[2]
The results are represented in the cumulative frequency table below, with upper class boundaries of 10, 20, 30, 40, 50.
On graph paper, draw a cumulative frequency graph, using a scale of 2 cm to represent 10 minutes (T) on the horizontal axis and 1 cm to represent 10 people on the vertical axis.[4]
Use your graph from part (f) to estimate
(i) the median;
(ii) the 90th percentile of the results;
(iii) the number of people who shopped at the supermarket for more than 15 minutes.[6]
Answer/Explanation
Markscheme
continuous (A1)[1 mark]
20 < T ≤ 30 (A1)[1 mark]
15 (A1)[1 mark]
(i) 21.5 (G2)
(ii) 9.21 (9.20597…) (G1)[3 marks]
(i) q = 194 (A1)
(ii) r = 200 (A1)[2 marks]