Question

In a certain distribution of​ numbers, the mean is 60​, with a standard deviation of 5. Use​ Chebyshev's Theorem to tell what percent of the numbers are less than 50 or more than 70.  

Answer 1

Solution:

Given, the distribution have

mean = 60 and SD = 5

According to Chebyshevs Theorem ,  

P[( - k) < X < ( + k) ] =  

Here , observe that

50 = 60 - (2 * 5) = ( - 2)

70 = 60 + (2 * 5) = ( - 2​​​​​​​)

So , k = 2

P(Between 50 and 70)

= P[ ( - 2​​​​​​​) < X   ( + 2​​​​​​​)]

=  

= 1 - [1/4]

= 3/4

= 0.75

Now ,

P[ less than 50 or more than 70 ]

= P[X < 50 OR X > 70]

= 1 - { P[Between 50 and 70] }

= 1 - 0.75

= 0.25

= 25%

Answer : 25% of the numbers are less than 50 or more than 70.