Question

Scores on a certain intelligence test for children between ages 13 and 15 years are approximately normally distributed with μ=96 and σ=27.

(a) What proportion of children aged 13 to 15 years old have scores on this test above 78 ? (NOTE: Please enter your answer in decimal form. For example, 45.23% should be entered as 0.4523.)
Answer:

(b) Enter the score which marks the lowest 25 percent of the distribution.
Answer:

(c) Enter the score which marks the highest 5 percent of the distribution.
Answer:

Answer 1

Solution:

Given that,

mean =  = 96

standard deviation =   = 27

a ) p ( x > 78 )

= 1 - p (x < 78 )

= 1 - p ( x -  / ) < ( 78 - 96 / 27)

= 1 - p ( z < -18 / 27 )

= 1 - p ( z < -0.67)

Using z table

= 1 - 0.2514

= 0.7486

Probability = 0.7486

b ) Using standard normal table,

P(Z < z) = 25%

P(Z < z) = 0.25

P(Z < - 0.6745) = 0.99

z = - 0.6745

Using z-score formula,

x = z * +

x = - 0.6745 * 27 + 96

= 77.7885

The distribution = 78

c ) Using standard normal table,

P(Z > z) = 5%

1 - P(Z < z) = 0.05

P(Z < z) = 1 - 0.01 = 0.95

P(Z < 1.645 ) = 0.95

z = 1.645

Using z-score formula,

x = z * +

x = 1.645 * 27 + 96

= 140.415

The distribution = 140