Question

let z be a random variable with a standard normal distrubtion. find the indicated probability (round your answer to four decimal places).

a) P(z ≥ -1.53)=

b) P(z ≤ -2.04)=

c) P(-2.00 ≤ z ≤ 1.05)=

d) P(0 ≤ z ≤ 0.50)=

Answer 1

Solution:

Z is a standard normal distribution.

a) We have to obtain P(Z ≥ -1.53)

P(Z ≥ -1.53) = 1 - P(Z < 1.53)

Using "pnorm" function of R we get, P(Z < -1.53) = 0.9370

P(Z ≥ -1.53) = 1 - 0.0630 = 0.9370

P(Z ≥ -1.53) = 0.9370

b) We have to obtain P(Z ≤ -2.04).

Using "pnorm" function of R we get, P(Z ≤ -2.04) = 0.0207

P(Z ≤ -2.04) = 0.0207

c) We have to obtain P(-2.00 ≤ Z ≤ 1.05).

P(-2.00 ≤ Z ≤ 1.05) = P(Z ≤ 1.05) - P(Z < -2.00)

Using "pnorm" function of R we get,

P(Z ≤ 1.05) = 0.8531 and P(Z < -2.00) = 0.0227

Hence, P(-2.00 ≤ Z ≤ 1.05) = 0.8531 - 0.0227

P(-2.00 ≤ Z ≤ 1.05) = 0.8304

d) We have to obtain P(0 ≤ z ≤ 0.50).

P(0 ≤ Z ≤ 0.50) = P(Z ≤ 0.50) - P(Z < 0)

Using "pnorm" function of R we get,

P(Z ≤ 0.50) = 0.6915 and P(Z < 0) = 0.5000

Hence, P(0 ≤ Z ≤ 0.50) = 0.6915 - 0.5000

P(0 ≤ Z ≤ 0.50) = 0.1915

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