Question

Given a standardized normal distribution (with a mean of 0 and a standard deviation of 1) complete parts (a) through (d)
a. The probability that Z is less than -1.59 is _______
b. The probability that Z is greater than 1.81 is________
c. The probability that Z is between -1.59 and 1.81 is______
d. The probability that Z is less than -1.59 or greater than 1.81______

Answer 1

a)

Here, μ = 0, σ = 1 and x = -1.59. We need to compute P(X <= -1.59). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (-1.59 - 0)/1 = -1.59

Therefore,
P(X <= -1.59) = P(z <= (-1.59 - 0)/1)
= P(z <= -1.59)
= 0.0559

b)

Here, μ = 0, σ = 1 and x = 1.81. We need to compute P(X >= 1.81). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (1.81 - 0)/1 = 1.81

Therefore,
P(X >= 1.81) = P(z <= (1.81 - 0)/1)
= P(z >= 1.81)
= 1 - 0.9649 = 0.0351

c)

Here, μ = 0, σ = 1, x1 = -1.59 and x2 = 1.81. We need to compute P(-1.59<= X <= 1.81). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z1 = (-1.59 - 0)/1 = -1.59
z2 = (1.81 - 0)/1 = 1.81

Therefore, we get
P(-1.59 <= X <= 1.81) = P((1.81 - 0)/1) <= z <= (1.81 - 0)/1)
= P(-1.59 <= z <= 1.81) = P(z <= 1.81) - P(z <= -1.59)
= 0.9649 - 0.0559
= 0.9090

d)
P(z < -1.59 or z > 1.81) = 0.0559 + 0.0351
= 0.0910