(i) Finding x and the mean mass:
Total apples = 250
$$20 + 60 + 70 + x + 60 = 250$$
$$210 + x = 250 \implies x = 40$$
Mean by Direct Method:
| Mass (g) | $f_i$ | $x_i$ (mid-point) | $f_i x_i$ |
|---|---|---|---|
| 80–100 | 20 | 90 | 1800 |
| 100–120 | 60 | 110 | 6600 |
| 120–140 | 70 | 130 | 9100 |
| 140–160 | 40 | 150 | 6000 |
| 160–180 | 60 | 170 | 10200 |
| Total | 250 | | 33700 |
$$\bar{x} = \frac{\Sigma f_i x_i}{\Sigma f_i} = \frac{33700}{250} = \textbf{134.8 g}$$
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(ii) Finding the modal mass:
The class 120–140 has the highest frequency (70), so it is the modal class.
Here: $l = 120$, $f_1 = 70$, $f_0 = 60$, $f_2 = 40$, $h = 20$
$$\text{Mode} = l + \frac{f_1 - f_0}{2f_1 - f_0 - f_2} \times h = 120 + \frac{70-60}{2(70)-60-40} \times 20$$
$$= 120 + \frac{10}{40} \times 20 = 120 + 5 = \textbf{125 g}$$
Source: Statistics, Chapter 13, Sections 13.2 and 13.4
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