A random variable follows a normal distribution with a mean

A random variable follows a normal distribution with a mean of 126 and a standard deviation of 26. What is the probability that a randomly selected value from this population is between 117 and 185? What is the probability that a randomly selected value from this population is between 105 and 140?

Solution

Answer to the question)

M = 126

s = 26

P(117 < x < 185) = P(x < 185) - P(x < 117)

P(x < 185) = P(z < (185-126)/26) = P(z < 2.27) = 0.9884

P(x < 117) = P(z < (117-126)/26) = P( z< -0.35) = 0.3632

P(117 < x < 185) = 0.9884 - 0.3632

P(117 < x < 185) = 0.6252

.

P(105 < x < 140) = P( x< 140) - P(x < 105)

P(x < 140) = P(z < (140-126)/26) = P(z < 0.54) = 0.7054

P(x < 105) = P( z < (105-126)/26) = P(z < -0.81) = 0.2090

P(105 < x < 140) = 0.7054 - 0.2090 = 0.4964