Question

Given an approximately normal distribution with a mean of 175 and a standard deviation of 37.

(a) What percent of values outside the interval (138, 212)?

(b) What percent of values are outside the interval (101, 249)?

(c) What percent of values are outside the interval (64, 286)?

Answer 1

The distribution of random variable X is normal with mean 175 and standard deviation 37.

By central limit theorem

a) Required Percentage = P ( X < 138 ) + P ( X > 212)

= P ( Z < -1 ) + P ( Z > 1)

Since normal distribution is symmetric

P ( X < 138 ) + P ( X > 212) = 2 * P ( Z >1)

From normal probability table

P ( Z >1) = 0.1587

Required Percentage = 0.3174

b) Required Percentage = P ( X < 101) + P ( X > 249)

= P ( Z < -2 ) + P ( Z > 2)

Since normal distribution is symmetric

P ( X < 101 ) + P ( X > 249) = 2 * P ( Z >2)

From normal probability table

P ( Z >2) = 0.02275

Required Percentage = 0.0455

c) Required percent = P ( X < 64) + P ( X > 286)

= P ( Z < -3 ) + P ( Z > 3)

Since normal distribution is symmetric

P ( X < 64) + P ( X > 249) = 2 * P ( Z >3)

From normal probability table

P ( Z >3) = 0.0013

Required Percentage = 0.0026