Question

Suppose x is a normally distributed random variable with mu equals 43 and sigma equals 5. Find a value x 0 of the random variable x that satisfies the following equations or statements. a. ​P(x less than or equals x 0​)equals0.8413 b. ​P(x greater thanx 0​)equals0.025 c. ​P(x greater thanx 0​) equals 0.95 d.​ P(28 less than or equals x less thanx 0​) equals 0.8630 e.​ 10% of the values of x are less than x 0. f.​ 1% of the values of x are greater than x 0.

Answer 1

Solution :

mean = = 43

standard deviation = = 5

Using standard normal table,

(a)

P(Z < z) = 0.8413

P(Z < 1) = 0.8413

z = 1

Using z-score formula,

x = z * +

x= 1 * 5 + 43 = 48

Value = 48

(b)

P(Z > z) = 0.025

1 - P(Z < z) = 0.025

P(Z < z) = 1 - 0.0250 = 0.975

P(Z < 1.96) = 0.975

z = 1.96

Using z-score formula,

x = z * +

x= 1.96 * 5 + 43 = 52.8

Value = 52.8

(c)

P(Z > z) = 0.95

1 - P(Z < z) = 0.95

P(Z < z) = 1 - 0.95 = 0.05

P(Z < -1.645) = 0.05

z = -1.645

Using z-score formula,

x = z * +

x= -1.645 * 5 + 43 = 34.775 = 34.8

Value = 34.8

(e)

P(Z < z) = 0.10

P(Z < -1.28) = 0.10

z = -1.28

Using z-score formula,

x = z * +

x= -1.28 * 5 + 43 = 36.6

Value = 36.6