Question

Heart Rate : The heart rate is one of the 'vital signs' of health in the human body. It measures the number of times per minute that the heart contracts or beats. While a normal heart rate does not guarantee that a person is free of health problems, it is a useful benchmark for identifying a range of health issues.

Thirty women were examined by doctors of AIIMS and the number of heart beats per minute were recorded and summarized as follows:

Number of heart beats per minute	Number of Women
65 − 68	2
68 − 71	4
71 − 74	3
74 − 77	8
77 − 80	7
80 − 83	4
83 − 86	2

Based on the above information, answer the following questions:

1. How many women are having heart beat in the range 68 − 77? (1)
2. What is the median class of heart beats per minute for these women? (1)
3. 1. Find the modal value of heart beats per minute for these women. (2)
      OR
   2. Find the median value of heart beats per minute for these women. (2)
      

Answer 1

Number of heart beats per minute	Number of Women	Cumulative frequency
65 − 68	2	2
68 − 71	4	6
71 − 74	3	9
74 − 77	8	17
77 − 80	7	24
80 − 83	4	28
83 − 86	2	30
	Σ(f) = 30

(i) Number ot women having a heartbeat in range 68 − 77

⇒ 4 +3 + 8

= 15   ...(i)

(ii) Total frequency = 30

∴ `"N"/2`

= `30/2`

= 15

Since Cumulative Frequency greater than or equal to 15 is lies in class Interval 74 − 77

∴ Median class of heartbeats = 74 − 77.

(iii) (a) Mode = `l + (("f"_1 - "f"_0)/(2"f"_1 - "f"_0 - "f"_2)) xx "h"`

where,

Modal class having highest frequency = 74 − 77

l = 74

h = 3

f₁ = 8

f₀ = 3

f₂ = 7

Thus, Mode = `74 + ((8 - 3)/(2 xx 8 - 3 - 7)) xx 3`

= `74 + 5/6 xx 3`

= `74 + 5/2`

= `(148 + 5)/2`

= `153/2`

= 76.5

OR

(iii) (b) Here N = Σf = 30.

∴ `"N"/2`

= `30/2`

= 15

So, the median class is 74 − 77.

lower limit of median class, l = 74

Class size h = 3

Cumulative frequency Preceding class, Cf = 9

Frequency of medias class = f = 8

Median = `l + (("N"/2 - "Cf"))/"f" xx "h"`

= `74 + ((30/2 - 9))/8 xx 3`

= `74 + ((15 - 9))/8 xx 3`

= `74 + 6/8 xx 3`

= `74 + 18/8`

= 74 + 2.25

= 76.25.