Question

Given that ?~?(15,1.252 ), find: a. ?(? ≤ 15) b. ?(? > 10) c. ?(|? − 15| < 3) d. Find the 95th percentile of X

Answer 1

Solution :

Given that ,

mean = = 15

standard deviation = = 1.25

(a)

P(x 15) = P((x - ) / (15 - 15) / 1.25) = P(z 0)

Using standard normal table,

P(x 15) = 0.5

(b)

P(x > 10) = 1 - P(x < 10)

= 1 - P((x - ) / < (10 - 15) / 1.25)

= 1 - P(z < -4)

= 1 - 0   

= 1

P(x > 10) = 1

(c) P(|X - 15| < 3) = P(|X - 15 / 1.25| < 3 - 15 / 1.25)

= P(|z| < -9.6)

= 2 * P(Z < -9.6)

= 2 * 0

P(|X - 15| < 3) = 0

(d)

P(Z < z) = 95%

P(Z < 1.645) = 0.95

z = 1.645

5Using z-score formula,

x = z * +

x = 1.645 * 1.25 + 15 = 17.05625

95th percentile = 17.06