Question

Let X be normally distributed with mean μ = 12 and standard deviation σ = 6. [You may find it useful to reference the z table.]

a. Find P(X ≤ 0). (Round "z" value to 2 decimal places and final answer to 4 decimal places.)

b. Find P(X > 3). (Round "z" value to 2 decimal places and final answer to 4 decimal places.)

c. Find P(6 ≤ X ≤ 12). (Round "z" value to 2 decimal places and final answer to 4 decimal places.)

d. Find P(9 ≤ X ≤ 18). (Round "z" value to 2 decimal places and final answer to 4 decimal places.)

Answer 1

Solution :

Given that ,

mean = = 12

standard deviation = = 6

a)

P(x 0) = P((x - ) / (0-12) / 6)

= P(z -2.00)

= 0.0228 Using standard normal table

Probability = 0.0228

b)

P(x > 3) = 1 - P(x < 3)

= 1 - P((x - ) / < (3-12) / 6)

= 1 - P(z < -1.50)

= 1 - 0.0668   

= 0.9332

Probability = 0.9332

c)

P( 6 x 12 ) = P((6-12 / 6) (x - ) / (12-12 / 6) )

P(6 x 12)  = P(-1.00 z 0)

P(6 x 12) = P(z 0 ) - P(z -1.00 )

P(6 x 12) = 0.5000 - 0.1587

Probability = 0.3413

d)

P(9 x 18) = P((9-12 / 6) (x - ) / (18-12 / 6) )

P(9 x 18) = P(-0.50 z 1.00)

P(9 x 18) = P(z 1.00 ) - P(z -0.50)

P(9 x 18) = 0.8413 - 0.3085

Probability = 0.5328