Question

Given the following information of first year students of a particular school.
​​​​​​​
Age (Years)
23
24
26
28
29
Number of students
8




Probability
1.5a
2.5 a
0.4
0.25
0.1
​
Find
i. a
ii. Interquartile range

Answer 1

i. In a probability distribution, the sum of all $p_i$ must be equal to 1. So

$1.5 a +2.5a+0.4+0.25+0.1 = 1$

$4a +0.75=1$

$4a = 0.25$

$a=0.0625$

ii. Interquartile range, by definition, is $Q_3 - Q_1$ or $x_{75} - x_{25}$

5 values are given: 23 24 26 28 29

Given an odd 2n+1 number of values

first quartile Q₁ = median of the n smallest values

third quartile Q₃ = median of the n largest values

In out case, n=2, so $Q_1 =23.5$ (median of 23 and 24) and $Q_3 = 28.5$ (median of 28 and 29).

$IQR = 28.5 - 23.5 = 5$