Suppose X is a normally distributed random variable with a mean of 400 and a standard deviation of 60.

Question

P(380<X<420).?

Answer 1

The mean of random variable is 400

The standard deviation of randon variable is 60

P(380 < X < 420)

Z-score = (X - mean)/standard deviation

For X = 380 then Z1 = (380 - 400)/60 = -0.33

For X = 420 then Z2 = (420 - 400)/60 = 0.33

Hence P(380 < X < 420) = P(-0.33 < Z < 0.33)

= [Area of the region when Z = 0.33] - [Area of the region when Z = -0.33]

From the Z-score table,

If Z = -0.33 then area = 0.3707

If Z = 0.33 then area = 0.6293

Then P(380 < X < 420) = 0.6293 - 0.3707 = 0.2586

Therefore P(380 < X < 420) = 0.2586.