The cumulative frequency distribution table below shows the completion times (in minutes) … — JAMB UTME

The cumulative frequency distribution table below shows the completion times (in minutes) for $120$ software engineers during a technical skill test:

Using linear interpolation from the cumulative frequency data, what is the $65^{\text{th}}$ percentile completion time?

39.0\text{ minutes}

To find the

65^{\text{th}}

percentile (

P_{65}

), we calculate

\frac{65}{100} \times 120 = 78

. The cumulative frequency table shows that rank

78

falls within the class interval

30 - 39

. The true lower boundary for this class is

L = 29.5

, the cumulative frequency of the preceding class is

F = 40

, the frequency of the percentile class is

f = 40

, and the class width is

c = 10

. Substituting into

P_{65} = L + \left(\frac{78 - F}{f}\right)c

yields

29.5 + \left(\frac{38}{40}\right) \times 10 = 39.0\text{ minutes}

Determine the rank of the

65^{\text{th}}

percentile (

P_{65}

Rank

= \frac{65}{100} \times 120 = 78

The percentile rank identifies the position of the data point within the total frequency

N = 120

Identify the percentile class and its boundaries.

Percentile class is

30 - 39

, with lower boundary

L = 29.5

, upper boundary

= 39.5

, and class width

c = 10

The cumulative frequency increases from

40

80

across this interval, containing rank

78

Apply the linear interpolation formula for percentiles on grouped data.

P_{65} = L + \left(\frac{\frac{65N}{100} - F}{f}\right) \times c = 29.5 + \left(\frac{78 - 40}{40}\right) \times 10 = 29.5 + 9.5 = 39.0\text{ minutes}

Where

F = 40

is the cumulative frequency prior to the class and

f = 40

is the frequency of the class.

Linear Interpolation of Percentiles from Cumulative Frequency Distributions

Question