Question

The monthly returns for a financial advisory service can be modeled by a Normal distribution with a mean of $119 and standard deviation of $91, per $10,000 invested. Find the following boundaries: (use 4 decimals for all answers)

(a) the highest 10% of monthly returns:_______
(b) the lowest 10% of monthly returns: _______
(c) the highest 20% of monthly returns: _________
(d) the middle 60% of monthly returns: _______ and________ (Enter the lower value first.)

Answer 1

Solution :

Given that,

mean = = 119

standard deviation = = 91

Using standard normal table,

a ) P( Z > z) = 10%

P(Z > z) = 0.10

1 - P( Z < z) = 0.10

P(Z < z) = 1 - 0.10

P(Z < z) = 0.90

z = 1.282

Using z-score formula,

x = z * +

x = 1.28 * 91 + 119

x = 235.4800

b ) P( Z < z) = 10%

P(Z < z) = 0.10

z = -1.28

Using z-score formula,

x = z * +

x = -1.28* 91 + 119

x = 2.5200

c ) P( Z > z) = 20%

P(Z > z) = 0.20

1 - P( Z < z) = 0.20

P(Z < z) = 1 - 0.20

P(Z < z) = 0.80

z = 0.84

Using z-score formula,

x = z * +

x = 0.84 * 91 + 119

x = 195.4400

d ) P(-z < Z < z) = 60%
P(Z < z) - P(Z < z) = 0.60
2P(Z < z) - 1 = 0.60
2P(Z < z ) = 1 + 0.60
2P(Z < z) = 1.60
P(Z < z) = 1.60 / 2
P(Z < z) = 0.80
z = -0.84 and z = 0.84

sing z-score formula,

x = z * +

x = -0.84 * 91 + 119

x = 42.5600

Using z-score formula,

x = z * +

x = 0.84 * 91 + 119

x = 195.4400