Question

For a normal distribution, find the probability of being

(a) Within 2.7 standard deviations of the mean.

(b) Between 2 standard deviations below the mean and 2.5 standard deviations above the mean (

c) Less than μ−2.5σ

Answer 1

The number of standard deviations by which a value is away from the mean is the Z score of that value. Probability of occurence of an event corresponding to its Z score can be obtained from the standard normal distribution table

a) P(within 2.7 standard deviations of the mean) = P(-2.7 < Z < 2.7)

= P(Z < 2.7) - P(Z < -2.7)

= 0.9965 - 0.0035

= 0.9930

b) P(between 2 standard deviations below the mean and 2.5 standard deviations above the mean) = P(2.5 < Z < -2)

= P(Z < 2.5) - P(Z < -2)

= 0.9938 - 0.0228

= 0.9710

c) P(less than μ−2.5σ) = P(Z < -2.5)

= 0.0062