The frequency distribution table below shows the daily rainfall (in mm) recorded across 80 weather monitoring stations during a storm: | Daily Rainfall (mm) | Frequency (f) | |---|---| | 10 - 19 | 8 | | 20 - 29 | 14 | | 30 - 39 | 26 | | 40 - 49 | 20 | | 50 - 59 | 12 | Using linear interpolation from

The frequency distribution table below shows the daily rainfall (in mm) recorded across 80… — JAMB UTME

The frequency distribution table below shows the daily rainfall (in mm) recorded across $80$ weather monitoring stations during a storm:

Daily Rainfall (mm)	Frequency ( $f$ )
$10 - 19$	$8$
$20 - 29$	$14$
$30 - 39$	$26$
$40 - 49$	$20$
$50 - 59$	$12$

Using linear interpolation from the cumulative frequency distribution (ogive), what is the 75th percentile ( $P_{75}$ ) of the daily rainfall in mm?

Cevap: 45.5 mm

Cevap

The 75th percentile of the daily rainfall distribution is

45.5\text{ mm}

To find the 75th percentile (

P_{75}

), determine the

60^{\text{th}}

cumulative frequency position (

0.75 \times 80 = 60

). The value lies within the

40 - 49

class interval. Applying the lower class boundary

L = 39.5

, preceding cumulative frequency

cf_b = 48

, frequency

f = 20

, and class width

c = 10

, linear interpolation yields

P_{75} = 39.5 + \frac{60 - 48}{20} \times 10 = 45.5\text{ mm}

Adım Adım Çözüm

Calculate cumulative frequencies across all class intervals.

Cumulative frequencies are

8

for

10-19

22

for

20-29

48

for

30-39

68

for

40-49

, and

80

for

50-59

. Total frequency

N = 80

Cumulative frequencies are necessary to locate percentile positions on an ogive.

Determine the position corresponding to the 75th percentile.

Position

= 0.75 \times 80 = 60^{\text{th}}

cumulative frequency item.

The 75th percentile represents

75\%

of the total sample size.

Identify the target class interval parameters containing the 60th observation.

The interval

40 - 49

contains cumulative frequencies from

49

68

. Parameters:

L = 39.5

c = 10

f = 20

cf_b = 48

Linear interpolation requires the exact boundaries and frequencies of the container class.

Apply the percentile interpolation formula

P_{75} = L + \left(\frac{60 - cf_b}{f}\right) \times c

P_{75} = 39.5 + \left(\frac{60 - 48}{20}\right) \times 10 = 39.5 + 6 = 45.5\text{ mm}

Calculates the exact rainfall value corresponding to the 75th percentile.

Anahtar Kavram

Linear interpolation for percentiles using cumulative frequency distribution (ogive)

Tahmini Süre:1m 30s

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