Question

The monthly returns for a financial advisory service can be modeled by a Normal distribution with a mean of $145 and standard deviation of $79, per $10,000 invested. Find the following boundaries: (use 5 decimals for all answers) PLEASE SHOW ALL WORK AND CALCULATIONS PLEASE AND THANK YOU

1. the highest 20% of monthly returns is _____

2. the lowest 20% of monthly returns is ______

3. the highest 10% of monthly returns is _______

4. the middle 20% of monthly returns. is ______ and ______

Answer 1

part 1)

X ~ N ( µ = 145 , σ = 79 )
P ( X > x ) = 1 - P ( X < x ) = 1 - 0.2 = 0.8
To find the value of x
Looking for the probability 0.8 in standard normal table to calculate Z score = 0.8416
Z = ( X - µ ) / σ
0.8416 = ( X - 145 ) / 79
X = 211.48798

P ( X > 211.48798 ) = 0.2

Part 2)

X ~ N ( µ = 145 , σ = 79 )
P ( X < x ) = 20% = 0.2
To find the value of x
Looking for the probability 0.2 in standard normal table to calculate Z score = -0.8416
Z = ( X - µ ) / σ
-0.8416 = ( X - 145 ) / 79
X = 78.51202
P ( X < 78.51202 ) = 0.2

Part 3)

X ~ N ( µ = 145 , σ = 79 )
P ( X > x ) = 1 - P ( X < x ) = 1 - 0.1 = 0.9
To find the value of x
Looking for the probability 0.9 in standard normal table to calculate Z score = 1.28155
Z = ( X - µ ) / σ
1.28155 = ( X - 145 ) / 79
X = 246.24245
P ( X > 246.24245 ) = 0.1

Part 4)

X ~ N ( µ = 145 , σ = 79 )
P ( a < X < b ) = 0.2
Dividing the area 0.2 in two parts we get 0.2/2 = 0.1
since 0.5 area in normal curve is above and below the mean
Area below the mean is a = 0.5 - 0.1
Area above the mean is b = 0.5 + 0.1
Looking for the probability 0.4 in standard normal table to calculate Z score = -0.25335
Looking for the probability 0.6 in standard normal table to calculate Z score = 0.25335
Z = ( X - µ ) / σ
-0.25335 = ( X - 145 ) / 79
a = 124.98535
0.25335 = ( X - 145 ) / 79
b = 165.01465
P ( 124.98535 < X < 165.01465 ) = 0.2