Question:
Suppose weekly income of worker in a locality is normally distributed with mean Rs. 500 and standard deviation Rs. 100. What is probability of falling the workers within one, two and three standard deviations from their mean income?
Solution:
Set x is a normal random variable having mean and standard deviation. Thus according to area properties.
a. Probability of falling the worker within one standard deviation of mean income means the probability of finding the worker having income between 400 i.e. (500-100) and 600 i.e. (500+100) is
P(400 ≤ x ≤ 600) = .6826
It means there are 68.26% chances that worker will have income between 400 and 600.
b) The probability of finding a worker having income between two standard deviation is
p[500-2100 ≤ x ≤ 500 + 2 x 100] = .9545
or p[300 ≤ x ≤ 700] = .9545
c) The probability of finding a worker having income between three standard deviation is
p [(500-3 x 100) ≤ x ≤ (500 + 3 x 100)] = .9973
p [200 ≤ x ≤ 800] = .9973