A supplier grouped the capacity, \(v\) litres, of 60 fuel tanks in stock. The grouped frequency table shows the results. Capacity, \(c\) (litres) \(0 < c \l...
Question 1 Report
A supplier grouped the capacity, \(v\) litres, of 60 fuel tanks in stock. The grouped frequency table shows the results.
| Capacity, \(c\) (litres) | \(0 < c \leq 50\) | \(50 < c \leq 100\) | \(100 < c \leq 150\) | \(150 < c \leq 200\) | \(200 < c \leq 250\) |
|---|---|---|---|---|---|
| Frequency | 8 | 20 | 18 | 10 | 4 |
- Write down the class with the highest frequency. (1)
- Work out an estimate for the mean capacity. (3)
- Write down the class that contains the median. (1)
Answer Details
This question tests identifying the highest-frequency class from a grouped table, estimating the mean using midpoints, and locating the class containing the median from cumulative frequencies.
(a) The class \( 50 < c \leq 100 \) has the highest frequency, 20, so it has the highest frequency of the five classes. [1 mark]
(b) Using the midpoint of each class as a representative capacity:
\[ \sum fx = (8 \times 25) + (20 \times 75) + (18 \times 125) + (10 \times 175) + (4 \times 225) = 200 + 1500 + 2250 + 1750 + 900 = 6600 \]
Dividing by the 60 tanks gives the estimated mean:
\[ \frac{6600}{60} = 110 \text{ litres} \] [3 marks]
(c) The running (cumulative) totals are 8, 28, 46, 56, 60. With 60 tanks, the median lies between the 30th and 31st values; since the cumulative total reaches 28 by the end of \( 50 < c \leq 100 \) but 46 by the end of \( 100 < c \leq 150 \), both the 30th and 31st values fall in \( 100 < c \leq 150 \). [1 mark]
The class with the highest frequency and the class containing the median are not automatically the same class, since the median depends on where the running total crosses the halfway point of 60 tanks, not on which single class is individually largest.
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