Question

Apples are most widely planted and are commercially the most important

fruit crop in Jammu and Kashmir. The cultivation of apple crop in Jammu

and Kashmir shows particular interest for a number of reasons. In terms of

both area and production, apple is very beneficial fruit crop. This provides

a major source of income and employment in Jammu and Kashmir.

Horticultural department has tasked their statistical officer to create a model for

farmers to be able to predict their produce output based on various factors. A

box containing 250 apples was opened and each apple was weighed. The

distribution of the masses of the apples is given in the following table:
Mass (in

grams)

80-100 100-120 120-140- 140-160 160-180

Frequency

20

60

70

x

60

On the basis of the above information, answer any four of the following

questions:

(i)

How many apples are in the range 140-160 mass?

(a) 40 b) 50(c) 60 d) 70

(ii)

What is the mean mass of the apples?

(a) 131 grams b) 135 grams

(c) 150 grams d) 156 grams

(iii)

What is the upper limit of the median class?

(a) 80 b) 100

(c) 120 d) 140

(iv)

What is the modal mass of the apples?

(a) 122 b) 125

(c) 128 d) 132

(v) What is the median mass of the apples?

(a) 122.33 grams b) 128.67 grams​

Answer 1

i) a)40

ii)134.8g

iii)140

iv)125g

Answer 2

Answer:

(i) There are 40 apples in the range 140 - 160 mass.

(ii) The mean mass of the apples is 135 gram

(ii) The upper limit of the median class = 140

(iv) The modal mass is 125 gram

(v) The median mass is 141.67 gram

Step-by-step explanation:

The given data:

weight               frequency(f)            CF         Class mark(x)                      fx

80 - 100                           20             20            90                               1800

100 - 120                           60            80             110                              6600

120 - 140                           70            150            130                              9100

140 - 160                       x =40           190             150                             6000

160 - 180                           60           250            170                             10200

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                                       250                                                              33700

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(i) Given that the total apples are 250

=> 20 + 60 + 70 + x + 60 = 250

=> 210 + x = 250

=> x = 250 - 210

=> x = 40

Therefore, there are 40 apples in the range 140 - 160 mass.

(ii) We know that Mean  = $\frac{\sum{fx}}{\sum f}$

                                        = $\frac{33700}{250}$

                                        = 134.80

                                        ≈ 135

Therefore, the mean mass of the apples is 135 gm

(iii) To find the upper class, we write the cumulative frequency of the data.

As $\sum f$ = 250 = even,

we calculate  $\sum f$/2 = 250/2 = 125

As 125 will lie in the interval 120 - 140, it will be the median class.

Therefore, the upper limit of the median class = 140

(iv) The modal class is the interval with highest frequency

=> here it is 120 - 140 with frequency 70

We know that Mode = l + $\frac{f - f_1}{2f - f_1-f_2}$ * h

where, l = lower limit of the modal class

          f = frequency of the modal class

          f₁ = frequency of the class before the modal class

          f₂ = frequency of the class after the modal class

           h = size of the class interval

Therefore, here,

Mode = 120 + $\frac{70-60}{2*70-60-40}$ * 20

          = 120 + $\frac{10}{140-100}$ * 20

          = 120 + $\frac{10}{40}$ * 20

          = 120 + 5

          = 125

Therefore, the modal mass is 125 gram

(v) We know that Mode = 3 Median - 2 Mean

Therefore, on substitution,

=> 125 = 3 Median - 2 * 135

=> 125 = 3 Median - 270

=> 3 Median = 125 + 270

=> 3 Median = 425

=> Median = 425/3

=> Median = 141. 67

Therefore the median mass is 141.67 gram