Chapter 7: Statistics Exercise – 7.3

Question: 1

The following table gives the distribution of total household expenditure (in rupees) of manual workers in a city.

Find the average expenditure (in rupees) per household

Solution:

Let the assumed mean (A) = 275

We have A = 275, h = 50
Mean = A + h * sum/N
= 275 + 50 * - 35/200
= 275 – 8.75
= 266.25

Question: 2

A survey was conducted by a group of students as a part of their environmental awareness program, in which they collected the following data regarding the number of plants in 200 houses in a locality. Find the mean number of plants per house.

Which method did you use for finding the mean, and why?

Solution:

Let us find class marks (x_i­) = (upper class limit + lower class limit)/2. Now we may compute x_i and f_ix_i as following.

From the table we may observe that N = 20, Sum = 162

So mean number of plants per house is 8.1. We have used for the direct method values Xi and fi are very small

Question: 3

Consider the following distribution of daily wages of workers of a factory

Find the mean daily wages of the workers of the factory by using an appropriate method.

Solution:

Let the assume mean (A) = 150

We have N = 50, h = 20

= 150 - 4.8
= 145.2

Question: 4

Thirty women were examined in a hospital by a doctor and the number of heart beats per minute recorded and summarized as follows. Find the mean heart beats per minute for these women, choosing a suitable method. Number of heart

Solution:

We may find marks of each interval (xi) by using the relation (xi¬) = (upper class limit + lower class limit)/2 Class size of this data = 3 Now talking 75.5 as assumed mean (a) We may calculate di, ui, fiui as following

Now we may observe from table that N = 30, sum = 4

= 75.5 + 0.4
= 75.9 
So mean heart beats per minute for those women are 75.9 beats per minute

Question: 5

Find the mean of each of the following frequency distributions: (5 - 14)

Solution:

Let us assume mean be 15

A = 15, h = 6, N = 40

= 15 + 0.45

= 15.47

Question: 6

Solution:

Let us assumed mean be 100

A = 100, h = 20

= 100 + 12.2
= 122.2

Question: 7

Solution:

Let the assumed mean (A) = 20

We have A = 20, h = 8 
Mean = A + h (sum/N) 
= 20 + 8 (7/40) 
= 20 + 1.4 
= 21.4 

Question: 8

Solution:

Let the assumed mean be (A) = 15

We have A = 15, h = 6 
Mean = A + h(sum/N) 
= 15 + 6 (5/40) 
= 15 + 0.75 
= 15.75

Question: 9

Solution:

Let the assumed mean (A) = 25

We have A = 25, h = 10 
Mean = A + h (sum/N) 
= 25 + 19 (8/60) 
= 25 + (4/3) 
= 26.333 

Question: 10

Solution:

Let the assumed mean (A) = 20

We have, A = 20, h = 8 
Mean = A + h (sum/N) 
= 20 + 8 (5/ 40) 
= 20 + 1
= 21

Question: 11

Solution:

Let the assumed mean (A) = 20

We have, A = 20, h = 8 
Mean = A + h (sum/N) 
= 20 + 8 (-9/ 20) 
= 20 – (72/20) 
= 20 – 3.6 
= 16.4 

Question: 12

Solution:

Let the assumed mean (A) = 60

We have A = 60, h = 20 
Mean = A + h (sum/N) 
= 60 + 20 (14/ 5) 
= 60 + 5.6 
= 65.6

Question: 13

Solution:

Let the assumed mean (A) = 50

We have A = 50, h = 10 
Mean = A + h (sum/N) 
= 50 + 10 (-2/ 40) 
= 50 - 0.5 
= 49.5 

Question: 14

Solution:

Let the assumed mean (A) = 42

We have A = 42, h = 5 
Mean = A + h (sum/N) 
= 42 + 5 (-79/70) 
= 42 – 79/14 
= 36.357

Question: 15

For the following distribution, calculate mean using all suitable methods:

Solution:

By direct method

Mean = (sum/N) 
= 848/ 64= 13.25 

By assuming mean method Let the assumed mean (A) = 65

Mean = A + sum/N 
= 6.5 + 6.75 
= 13.25

Question: 16

The weekly observation on cost of living index in a certain city for the year 2004 – 2005 are given below. Compute the weekly cost of living index.

Solution:

Let the assumed mean (A) = 1650

We have A = 16, h = 100 
Mean = A + h (sum/N)
= 1650 + 100 (7/52)
= 1650 + (175/13) 
= 21625/13 
= 1663.46

Question: 17

The following table shows the marks scored by 140 students in an examination of a certain paper:

Solution:

(i) Direct method:

Mean = sum/ N 
= 3620/ 140 
= 25.857

(ii) Assumed mean method: Let the assumed mean = 25 Mean = A + (sum/ N)

Mean = A + (sum/ N)
= 25 + (120/ 140)
= 25 + 0.857
= 25.857

(iii) Step deviation method: Let the assumed mean (A) = 25

Mean = A + h (sum/ N)
= 25 + 10(12/140)
= 25 + 0.857
= 25.857

Question: 18

The mean of the following frequency distribution is 62.8 and the sum of all the frequencies is 50. Compute the miss frequency f₁ and f₂.

Solution:

Given, sum of frequency
50 = f₁ + f₂ + 30
f₁ + f₂ = 50 – 30
f₁ +f₂ = 20
3f₁ + 3f₂ = 60 ---- (1)       [multiply both side by 3] 
And mean = 62.8 Sum/ N 
= 62.8 = (30f₁+ 70f₂+ 2060)/50 
= 3140 = 30f₁ + 70f₂ + 2060
30f₁+ 70f₂ = 3140 – 2060
30f₁ + 70f₂ = 1080
3f₁ + 7f₂ = 108 ---- (2) [divide it by 10]
Subtract equation (1) from equation (2)
3f₁ + 7f₂ - 3f₁ - 3f₂ = 108 – 60
4f₂ = 48 = f₂ = 12

Put value of f₂ in equation (1)
3f₁ + 3(12) = 60
f₁ = 24/3 = 8
 f₁ = 8, f₂ = 12

Question: 19

The following distribution shows the daily pocket allowance given to the children of a multistory building. The average pocket allowance is Rs 18.00. Find out the missing frequency.

Solution:

Given mean = 18, Let the missing frequency be v

792 + 18x = 752 + 20x
20x – 18x = 792 – 752
x = 40/2
x = 20

Question: 20

If the mean of the following distribution is 27. Find the value of p.

Solution:

Given Mean = 27
Mean = sum/ N

1161 + 27p = 1245 + 15p
27p – 15p = 1245 – 1161
12p = 84
p = 84/12
p = 7 

Question: 21

In a retail market, fruit vendors were selling mangoes kept in packing boxes. These boxes contain varying number of mangoes. The following was the distribution of mangoes according to the number of boxes.

Find the mean number of mangoes kept in packing box. Which method of finding the mean did you choose?

Solution:

We may observe that class internals are not continuous There is a gap between two class intervals. So we have to add 1/2 from lower class limit of each interval and class mark (xi) may be obtained by using the relation xi = (upper limit + lower class limit)/2. Class size (h) of this data = 3 Now taking 57 as assumed mean (a) we may calculated di , ui, fiui as follows

Now we have N = 400, Sum = 25, 
Mean = A + h (sum/ N) 
= 57 + 3 (25/400) 
= 57 + 3/16 
= 57+ 0.1875 
= 57.19
Clearly mean number of mangoes kept in packing box is 57.19 

Question: 22

The table below shows the daily expenditure on food of 25 households in a locality

Find the mean daily expenditure on food by a suitable method.

Solution:

We may calculate class mark (xi) for each interval by using the relation xi = (upper limit + lower class limit)/2. Class size = 50 Now, talking 225 as assumed mean (xi) we may calculate di ,ui, fiui as follows:

Now we may observe that N = 25 Sum = -7

225 = 50 + (-7/ 25) × 225 
225 – 14 
= 211 
So, mean daily expenditure on food is Rs 211

Question: 23

To find out the concentration of SO2 in the air (in parts per million i. e ppm) the data was collected for localities for 30 localities in a certain city and is presented below:

Find the mean concentration of SO₂ in the air 

Solution:

We may find class marks for each interval by using the relation xi = (upper limit + lower class limit)/2. Class size of this data = 0.04 Now taking 0.04 assumed mean (xi) we may calculate di, ui, fiui as follows:

From the table we may observe that N = 30, Sum = - 31

= 0.04 + (-31/30) × (0.04) 
= 0.099 ppm 
So mean concentration of SO2 ¬in the air is 0.099 ppm 

Question: 24

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.

Solution:

We may find class mark of each interval by using the relation xi = (upper limit + lower class limit)/2 Now, taking 16 as assumed mean (a) we may Calculate di and fi di  as follows

Now we may observe that N = 40, Sum = -113

16 + (-113/ 40)
= 16 – 2.825
= 13.75
So mean number of days is 13.75 days, for which student was absent

Question: 25

The following table gives the literacy rate (in percentage) of 35 cities. Find the mean literacy rate.

Solution:

We may find class marks by using the relation x_i = (upper limit + lower class limit)/2 Class size (h) for this data = 10 Now taking 70 as assumed mean (a) wrong Calculate d_i ,u_i, f_iu_i as follows

Now we may observe that N = 35, Sum = -2

= 70 + (-2/35) × 10 
= 70 – 4/7 
= 70 – 0.57 
= 69.43 
So, mean literacy rate is 69.43 %