Question

At a courier company, a daily report of parcels received for dispatch is prepared every evening, which classifies the parcels on the basis of their weights. The report of a certain day is as follows: Weight of parcels (in grams) (x) Number of parcels (f) Below 600 60 Below 500 58 Below 400 54 Below 300 35 Below 200 22 Below 100 10

Based on the above information, answer the following questions.

1. How many parcels have weights in the range of 300–400 grams?
2. How many parcels have weights in the range of 200–300 grams?
3. In which weight range the number of parcels is the lowest?
4. Find the mean weight of parcels received on that particular day.

Answer 1

Conversion to continuous frequency distribution

Converting the 'less than' cumulative distribution into normal class intervals:

Weight range (grams)	Cumulative Frequency	Frequency (Number of parcels, \(f_i)\)
0 – 100	10	10
100 – 200	22	22 – 10 = 12
200 – 300	35	35 – 22 = 13
300 – 400	54	54 – 35 = 19
400 – 500	58	58 – 54 = 4
500 – 600	60	60 – 58 = 2

i. Number of parcels in the range 300 – 400 grams

\[\text{Parcels in 300 – 400 g} = 54 - 35 = 19\]

ii. Number of parcels in the range 200 – 300 grams

\[\text{Parcels in 200 – 300 g} = 35 - 22 = 13\]

iii. Weight range with the lowest number of parcels

Comparing the frequencies:

0 – 100 g: 10

100 – 200 g: 12

200 – 300 g: 13

300 – 400 g: 19

400 – 500 g: 4

500 – 600 g: 2

The frequency is lowest (2 parcels) in the weight range 500 – 600 grams.

iv. Mean weight of parcels

Using the assumed mean method with assumed mean A = 350:

Class	Frequency \((f_i)\)	Mid-value \((x_i)\)	\(d_i = x_i - A\)	\(f_i d_i\)
0 – 100	10	50	–300	–3000
100 – 200	12	150	–200	–2400
200 – 300	13	250	–100	–1300
300 – 400	19	350	0	0
400 – 500	4	450	100	400
500 – 600	2	550	200	400
Total	\[\sum f_i = 60\]			\[\sum f_i d_i = -5900\]

Formula: \[\text{Mean } (\bar{x}) = A + \frac{\sum f_i d_i}{\sum f_i}\]

Calculation: \[\bar{x} = 350 + \left(\frac{-5900}{60}\right)\]

\[\bar{x} = 350 - 98.33\]

\[\bar{x} = 251.67\text{ g}\]