The operating lifespans (in hours) of a sample of 60 newly manufactured micro-components t… — JAMB UTME

The operating lifespans (in hours) of a sample of $60$ newly manufactured micro-components tested under laboratory conditions are presented in the frequency table below:

Lifespan (hours)	Frequency ( $f$ )
$10 - 19$	$5$
$20 - 29$	$8$
$30 - 39$	$12$
$40 - 49$	$20$
$50 - 59$	$15$

Using linear interpolation on class boundaries, calculate the median lifespan of the components in hours.

Cevap: 42 hours

Cevap

The median lifespan of the micro-components is 42 hours.

To calculate the median of grouped data, first determine cumulative frequencies: 5, 13, 25, 45, 60. Total frequency is

N = 60

, placing the median at

N/2 = 30

. The median class is

40 - 49

with lower class boundary

L = 39.5

, class frequency

f = 20

, cumulative frequency prior to median class

c.f. = 25

, and class width

c = 10

. Substituting into

\text{Median} = L + \left(\frac{N/2 - c.f.}{f}\right) \times c

gives

39.5 + \left(\frac{30 - 25}{20}\right) \times 10 = 39.5 + 2.5 = 42

Adım Adım Çözüm

Calculate the total frequency

N

and the median rank

N/2

N = 5 + 8 + 12 + 20 + 15 = 60

, so

N/2 = 30

The median corresponds to the value at position 30 in the ordered cumulative frequency distribution.

Determine cumulative frequencies to locate the median class.

Cumulative frequencies are 5, 13, 25, 45, 60. The 30th value falls in the interval

40 - 49

The cumulative frequency first reaches or exceeds 30 at the

40 - 49

class.

Identify boundary parameters for the median class.

Lower boundary

L = 39.5

, class width

c = 10

, class frequency

f = 20

, and cumulative frequency of preceding class

c.f. = 25

Continuous data analysis requires using lower class boundaries rather than class limits.

Apply the grouped median linear interpolation formula.

\text{Median} = 39.5 + \left(\frac{30 - 25}{20}\right) \times 10 = 39.5 + 2.5 = 42

Linear interpolation within the median class yields the precise median value.

Anahtar Kavram

Grouped Data Median via Class Boundary Interpolation

Tahmini Süre:2m 0s

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