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Nov 2007 p6 q5
2480
The arrival times of 204 trains were noted and the number of minutes, t, that each train was late was recorded. The results are summarised in the table.
Number of minutes late (t)
-2 ≤ t < 0
0 ≤ t < 2
2 ≤ t < 4
4 ≤ t < 6
6 ≤ t < 10
Number of trains
43
51
69
22
19
Explain what -2 ≤ t < 0 means about the arrival times of trains.
Draw a cumulative frequency graph, and from it estimate the median and the interquartile range of the number of minutes late of these trains.
Solution
(i) The interval
\(-2 \leq t < 0\) indicates that some trains arrived up to 2 minutes early.
(ii) To draw the cumulative frequency graph, first construct the cumulative frequency table:
Min late, less than
Cumulative frequency
0
43
2
94
4
163
6
185
10
204
Plot these points on a graph with the x-axis representing the number of minutes late and the y-axis representing the cumulative frequency. Connect the points with a smooth curve or straight lines.
To estimate the median, find the value corresponding to the cumulative frequency of 102 (half of 204). This is approximately between 2.1 and 2.4 minutes.
To estimate the interquartile range, find the values corresponding to the cumulative frequencies of 51 (quarter of 204) and 153 (three-quarters of 204). The interquartile range is approximately between 3.2 and 3.6 minutes.