Question

5. Given x is approximately normal with a mean of 85 and standard deviation of 25

Find P(x > 60)

Find P(x < 110)

Find P(60 < x < 110)

Find P(x > 140)What is the value of x that is larger then 75% of the x values?

What value of x is greater than 14% of the x values?

What are the values of x that contain 60% of the distribution?

Answer 1

Part a)

P ( X > 60 ) = 1 - P ( X < 60 )
Standardizing the value

Z = ( 60 - 85 ) / 25
Z = -1

P ( Z > -1 )
P ( X > 60 ) = 1 - P ( Z < -1 )
P ( X > 60 ) = 1 - 0.1587
P ( X > 60 ) = 0.8413

Part b)

P ( X < 110 )
Standardizing the value

Z = ( 110 - 85 ) / 25
Z = 1

P ( X < 110 ) = P ( Z < 1 )
P ( X < 110 ) = 0.8413

Part c)

P ( 60 < X < 110 )
Standardizing the value

Z = ( 60 - 85 ) / 25
Z = -1
Z = ( 110 - 85 ) / 25
Z = 1
P ( -1 < Z < 1 )
P ( 60 < X < 110 ) = P ( Z < 1 ) - P ( Z < -1 )
P ( 60 < X < 110 ) = 0.8413 - 0.1587
P ( 60 < X < 110 ) = 0.6827

Part d)

P ( X > 140 ) = 1 - P ( X < 140 )
Standardizing the value

Z = ( 140 - 85 ) / 25
Z = 2.2

P ( Z > 2.2 )
P ( X > 140 ) = 1 - P ( Z < 2.2 )
P ( X > 140 ) = 1 - 0.9861
P ( X > 140 ) = 0.0139

part e)

P ( X > ? ) = 1 - P ( X < ? ) = 1 - 0.75 = 0.25
Looking for the probability 0.25 in standard normal table to calculate critical value Z = -0.67

-0.67 = ( X - 85 ) / 25
X = 68.25
P ( X > 68.25 ) = 0.75

part f)

P ( X > ? ) = 1 - P ( X < ? ) = 1 - 0.14 = 0.86
Looking for the probability 0.86 in standard normal table to calculate critical value Z = 1.08

1.08 = ( X - 85 ) / 25
X = 112
P ( X > 112 ) = 0.14

Part g)

P ( a < X < b ) = 0.6
Dividing the area 0.6 in two parts we get 0.6/2 = 0.3
since 0.5 area in normal curve is above and below the mean
Area below the mean is a = 0.5 - 0.3
Area above the mean is b = 0.5 + 0.3
Looking for the probability 0.2 in standard normal table to calculate critical value Z = -0.84
Looking for the probability 0.8 in standard normal table to calculate critical value Z = 0.84

-0.84 = ( X - 85 ) / 25
a = 64
0.84 = ( X - 85 ) / 25
b = 106
P ( 64 < X < 106 ) = 0.6