EXERCISE 15.1
Page No 270:
Question 1:
A survey was conducted by a group of students as a part of their environment awareness programme, in which they collected the following data regarding the number of plants in 20 houses in a locality. Find the mean number of plants per house.
Number of plants | 0 − 2 | 2 − 4 | 4 − 6 | 6 − 8 | 8 − 10 | 10 − 12 | 12 − 14 |
Number of houses | 1 | 2 | 1 | 5 | 6 | 2 | 3 |
Which method did you use for finding the mean, and why?
Answer:
To find the class mark (x_{i}) for each interval, the following relation is used.
Class mark (x_{i}) =
x_{i }and f_{i}x_{i} can be calculated as follows.
Number of plants | Number of houses (f_{i}) | x_{i} | f_{i}x_{i} |
0 − 2 | 1 | 1 | 1 × 1 = 1 |
2 − 4 | 2 | 3 | 2 × 3 = 6 |
4 − 6 | 1 | 5 | 1 × 5 = 5 |
6 − 8 | 5 | 7 | 5 × 7 = 35 |
8 − 10 | 6 | 9 | 6 × 9 = 54 |
10 − 12 | 2 | 11 | 2 ×11 = 22 |
12 − 14 | 3 | 13 | 3 × 13 = 39 |
Total | 20 | 162 |
From the table, it can be observed that
Mean,
Therefore, mean number of plants per house is 8.1.
Here, direct method has been used as the values of class marks (x_{i}) and f_{i} are small.
Video Solution
Question 2:
Consider the following distribution of daily wages of 50 worker of a factory.
Daily wages (in Rs) | 100 − 120 | 120 − 140 | 140 −1 60 | 160 − 180 | 180 − 200 |
Number of workers | 12 | 14 | 8 | 6 | 10 |
Find the mean daily wages of the workers of the factory by using an appropriate method.
Answer:
To find the class mark for each interval, the following relation is used.
Class size (h) of this data = 20
Taking 150 as assured mean (a), d_{i}, u_{i}, and f_{i}u_{i} can be calculated as follows.
Daily wages (in Rs) | Number of workers (f_{i}) | x_{i} | d_{i} = x_{i }− 150 | f_{i}u_{i} | |
100 −120 | 12 | 110 | − 40 | − 2 | − 24 |
120 − 140 | 14 | 130 | − 20 | − 1 | − 14 |
140 − 160 | 8 | 150 | 0 | 0 | 0 |
160 −180 | 6 | 170 | 20 | 1 | 6 |
180 − 200 | 10 | 190 | 40 | 2 | 20 |
Total | 50 | − 12 |
From the table, it can be observed that
Therefore, the mean daily wage of the workers of the factory is Rs 145.20.
Video Solution
Question 3:
The following distribution shows the daily pocket allowance of children of a locality. The mean pocket allowance is Rs.18. Find the missing frequency f.
Daily pocket allowance (in Rs) | 11 − 13 | 13 − 15 | 15 −17 | 17 − 19 | 19 − 21 | 21 − 23 | 23 − 25 |
Number of workers | 7 | 6 | 9 | 13 | f | 5 | 4 |
Answer:
To find the class mark (x_{i}) for each interval, the following relation is used.
Given that, mean pocket allowance,
Taking 18 as assured mean (a), d_{i} and f_{i}d_{i} are calculated as follows.
Daily pocket allowance (in Rs) | Number of children f_{i} | Class mark x_{i} | d_{i} = x_{i }− 18 | f_{i}d_{i} |
11 −13 | 7 | 12 | − 6 | − 42 |
13 − 15 | 6 | 14 | − 4 | − 24 |
15 − 17 | 9 | 16 | − 2 | − 18 |
17 −19 | 13 | 18 | 0 | 0 |
19 − 21 | f | 20 | 2 | 2 f |
21 − 23 | 5 | 22 | 4 | 20 |
23 − 25 | 4 | 24 | 6 | 24 |
Total | 2f − 40 |
From the table, we obtain
Hence, the missing frequency, f, is 20.
Video Solution
Page No 271:
Question 4:
Thirty women were examined in a hospital by a doctor and the number of heart beats per minute were recorded and summarized as follows. Fine the mean heart beats per minute for these women, choosing a suitable method.
Number of heart beats per minute | 65 − 68 | 68 − 71 | 71 −74 | 74 − 77 | 77 − 80 | 80 − 83 | 83 − 86 |
Number of women | 2 | 4 | 3 | 8 | 7 | 4 | 2 |
Answer:
To find the class mark of each interval (x_{i}), the following relation is used.
Class size, h, of this data = 3
Taking 75.5 as assumed mean (a), di, u_{i}, f_{i}u_{i} are calculated as follows.
Number of heart beats per minute | Number of women f_{i} | x_{i} | d_{i} = x_{i }− 75.5 | f_{i}u_{i} | |
65 − 68 | 2 | 66.5 | − 9 | − 3 | − 6 |
68 − 71 | 4 | 69.5 | − 6 | − 2 | − 8 |
71 − 74 | 3 | 72.5 | − 3 | − 1 | − 3 |
74 − 77 | 8 | 75.5 | 0 | 0 | 0 |
77 − 80 | 7 | 78.5 | 3 | 1 | 7 |
80 − 83 | 4 | 81.5 | 6 | 2 | 8 |
83 − 86 | 2 | 84.5 | 9 | 3 | 6 |
Total | 30 | 4 |
From the table, we obtain
Therefore, mean hear beats per minute for these women are 75.9 beats per minute.
Video Solution
Question 5:
In a retail market, fruit vendors were selling mangoes kept in packing boxes. These boxes contained varying number of mangoes. The following was the distribution of mangoes according to the number of boxes.
Number of mangoes | 50 − 52 | 53 − 55 | 56 − 58 | 59 − 61 | 62 − 64 |
Number of boxes | 15 | 110 | 135 | 115 | 25 |
Find the mean number of mangoes kept in a packing box. Which method of finding the mean did you choose?
Answer:
Number of mangoes | Number of boxes f_{i} |
50 − 52 | 15 |
53 − 55 | 110 |
56 − 58 | 135 |
59 − 61 | 115 |
62 − 64 | 25 |
It can be observed that class intervals are not continuous. There is a gap of 1 between two class intervals. Therefore, has to be added to the upper class limit and has to be subtracted from the lower class limit of each interval.
Class mark (x_{i}) can be obtained by using the following relation.
Class size (h) of this data = 3
Taking 57 as assumed mean (a), d_{i}, u_{i}, f_{i}u_{i} are calculated as follows.
Class interval | f_{i} | x_{i} | d_{i} = x_{i} − 57 | f_{i}u_{i} | |
49.5 − 52.5 | 15 | 51 | − 6 | − 2 | − 30 |
52.5 − 55.5 | 110 | 54 | − 3 | − 1 | − 110 |
55.5 − 58.5 | 135 | 57 | 0 | 0 | 0 |
58.5 − 61.5 | 115 | 60 | 3 | 1 | 115 |
61.5 − 64.5 | 25 | 63 | 6 | 2 | 50 |
Total | 400 | 25 |
It can be observed that
Mean number of mangoes kept in a packing box is 57.19.
Step deviation method is used here as the values of f_{i, }d_{i} are big and also, there is a common multiple between all d_{i}.
Video Solution
Question 6:
The table below shows the daily expenditure on food of 25 households in a locality.
Daily expenditure (in Rs) | 100 − 150 | 150 − 200 | 200 − 250 | 250 − 300 | 300 − 350 |
Number of households | 4 | 5 | 12 | 2 | 2 |
Find the mean daily expenditure on food by a suitable method.
Answer:
To find the class mark (x_{i}) for each interval, the following relation is used.
Class size = 50
Taking 225 as assumed mean (a), d_{i}, u_{i}, f_{i}u_{i} are calculated as follows.
Daily expenditure (in Rs) | f_{i} | x_{i} | d_{i} = x_{i} − 225 | f_{i}u_{i} | |
100 − 150 | 4 | 125 | − 100 | − 2 | − 8 |
150 − 200 | 5 | 175 | − 50 | − 1 | − 5 |
200 − 250 | 12 | 225 | 0 | 0 | 0 |
250 − 300 | 2 | 275 | 50 | 1 | 2 |
300 − 350 | 2 | 325 | 100 | 2 | 4 |
Total | 25 | − 7 |
From the table, we obtain
Therefore, mean daily expenditure on food is Rs 211.
Video Solution
Question 7:
To find out the concentration of SO_{2} in the air (in parts per million, i.e., ppm), the data was collected for 30 localities in a certain city and is presented below:
concentration of SO_{2} (in ppm) | Frequency |
0.00 − 0.04 | 4 |
0.04 − 0.08 | 9 |
0.08 − 0.12 | 9 |
0.12 − 0.16 | 2 |
0.16 − 0.20 | 4 |
0.20 − 0.24 | 2 |
Find the mean concentration of SO_{2} in the air.
Answer:
To find the class marks for each interval, the following relation is used.
Class size of this data = 0.04
Taking 0.14 as assumed mean (a), d_{i}, u_{i}, f_{i}u_{i} are calculated as follows.
Concentration of SO_{2} (in ppm) | Frequency f_{i} | Class mark x_{i} | d_{i} = x_{i }− 0.14 | f_{i}u_{i} | |
0.00 − 0.04 | 4 | 0.02 | − 0.12 | − 3 | − 12 |
0.04 − 0.08 | 9 | 0.06 | − 0.08 | − 2 | − 18 |
0.08 − 0.12 | 9 | 0.10 | − 0.04 | − 1 | − 9 |
0.12 − 0.16 | 2 | 0.14 | 0 | 0 | 0 |
0.16 − 0.20 | 4 | 0.18 | 0.04 | 1 | 4 |
0.20 − 0.24 | 2 | 0.22 | 0.08 | 2 | 4 |
Total | 30 | − 31 |
From the table, we obtain
Therefore, mean concentration of SO_{2} in the air is 0.099 ppm.
Video Solution
Page No 272:
Question 8:
A class teacher has the following absentee record of 40 students of a class for the whole term. Find the mean number of days a student was absent.
Number of days | 0 − 6 | 6 − 10 | 10 − 14 | 14 − 20 | 20 − 28 | 28 − 38 | 38 − 40 |
Number of students | 11 | 10 | 7 | 4 | 4 | 3 | 1 |
Answer:
To find the class mark of each interval, the following relation is used.
Taking 17 as assumed mean (a), d_{i} and f_{i}d_{i} are calculated as follows.
Number of days | Number of students f_{i} | x_{i} | d_{i} = x_{i} − 17 | f_{i}d_{i} |
0 − 6 | 11 | 3 | − 14 | − 154 |
6 − 10 | 10 | 8 | − 9 | − 90 |
10 − 14 | 7 | 12 | − 5 | − 35 |
14 − 20 | 4 | 17 | 0 | 0 |
20 − 28 | 4 | 24 | 7 | 28 |
28 − 38 | 3 | 33 | 16 | 48 |
38 − 40 | 1 | 39 | 22 | 22 |
Total | 40 | − 181 |
From the table, we obtain
Therefore, the mean number of days is 12.48 days for which a student was absent.
Question 9:
The following table gives the literacy rate (in percentage) of 35 cities. Find the mean literacy rate.
Literacy rate (in %) | 45 − 55 | 55 − 65 | 65 − 75 | 75 − 85 | 85 − 95 |
Number of cities | 3 | 10 | 11 | 8 | 3 |
Answer:
To find the class marks, the following relation is used.
Class size (h) for this data = 10
Taking 70 as assumed mean (a), d_{i}, u_{i}, and f_{i}u_{i} are calculated as follows.
Literacy rate (in %) | Number of cities f_{i} | x_{i} | d_{i} = x_{i} − 70 | f_{i}u_{i} | |
45 − 55 | 3 | 50 | − 20 | − 2 | − 6 |
55 − 65 | 10 | 60 | − 10 | − 1 | − 10 |
65 − 75 | 11 | 70 | 0 | 0 | 0 |
75 − 85 | 8 | 80 | 10 | 1 | 8 |
85 − 95 | 3 | 90 | 20 | 2 | 6 |
Total | 35 | − 2 |
From the table, we obtain
Therefore, mean literacy rate is 69.43%.