Question

Find the following probabilities for test scores X, for which the mean is 400 and the standard deviation is 120. Assume that the test scores are described by a normal curve. (Round all answers to four decimal places.)

(a) P(X ≤ 400).

(b) P(X ≤ 580).

(c) P(X ≥ 640).

(d) P(400 ≤ X ≤ 640).

Answer 1

Solution :

Given that ,

mean = = 400

standard deviation = = 120

a) P(x 400)

= P[(x - ) / (400 - 400) / 120]

= P(z 0 )

Using z table,

= 0.5

b) P(x 580)

= P[(x - ) / (580 - 400) / 120]

= P(z 1.5 )

Using z table,

= 0.9332

c) P(x 640 ) = 1 - P(x   640)

= 1 - P[(x - ) / (640 - 400) /120 ]

= 1 -  P(z 2.0)   

  Using z table,

= 1 - 0.9772

= 0.0228

d) P(400 x 640)

= P[(400 - 400 / 120) (x - ) / (640 - 400 / 120 ) ]

= P(0 z 2.0)

= P(z 2.0) - P(z 0)

Using z table,  

= 0.9772 - 0.5

= 0.4772