Question

A population is normally distributed with muequals200 and sigmaequals20. a. nbsp Find the probability that a value randomly selected from this population will have a value greater than 240. b. Find the probability that a value randomly selected from this population will have a value less than 190. c. Find the probability that a value randomly selected from this population will have a value between 190 and 240.

Answer 1

Given,

Population : X is normally distributed with = 200 and =20

a. Probability that a value randomly selected from this population will have a value greater than 240 = P(X>240)

P(X>240) = 1 - P(X<240)

P(X<240)

Z-score for 240 = (240-)/ = (240-200)/20 = 40/20 = 2

Z-score for 240 = 2

From Standard normal tables, P(Z<2) = 0.9772

P(X<240)=P(Z<2) = 0.9772

P(X>240) = 1 - P(X<240) = 1 - 0.9972=0.0228

Probability that a value randomly selected from this population will have a value greater than 240 = 0.0228

b. Probability that a value randomly selected from this population will have a value less than 190 = P(X<190)

Z-score for 190 = (190-200)/20 = -10/20 =-0.5

From standard normal tables ,

P(Z < - 0.5) = 0.3085

P(X<190) = P(Z < - 0.5) = 0.3085

Probability that a value randomly selected from this population will have a value less than 190 = 0.3085

c. Probability that a value randomly selected from this population will have a value between 190 and 240 = P(190<X<240)

P(190<X<240) = P(X<240) - P(X<190)

From a. P(X<240) = 0.9772

From b. P(X<190) = 0.3085

P(190<X<240) = P(X<240) - P(X<190) = 0.9772 - 0.3085 = 0.6687

Probability that a value randomly selected from this population will have a value between 190 and 240 = 0.6687