Question

Suppose you have a normal distribution with known mu = 5 and sigma =1.N(5,1). Use Z=scores to compute the probability that a value drawn from this distribution will be:

a.Greater than 4.5

b.Greater than 6.3

c.Less than 3

d.Between 3.5 and 6.5

e.Greater than 6 or less than 4

2)For each a---e above, plot an graph of the normal distribution and shade in the are a under the curve corresponding to the probability you reported.(NOTE: make a separate graph for each)

Answer 1

Part a)

P ( X > 4.5 ) = 1 - P ( X < 4.5 )
Standardizing the value

Z = ( 4.5 - 5 ) / 1
Z = -0.5

P ( Z > -0.5 )
P ( X > 4.5 ) = 1 - P ( Z < -0.5 )
P ( X > 4.5 ) = 1 - 0.3085
P ( X > 4.5 ) = 0.6915

Part b)

P ( X > 6.3 ) = 1 - P ( X < 6.3 )
Standardizing the value

Z = ( 6.3 - 5 ) / 1
Z = 1.3

P ( Z > 1.3 )
P ( X > 6.3 ) = 1 - P ( Z < 1.3 )
P ( X > 6.3 ) = 1 - 0.9032
P ( X > 6.3 ) = 0.0968

Part c)

P ( X < 3 )
Standardizing the value

Z = ( 3 - 5 ) / 1
Z = -2

P ( X < 3 ) = P ( Z < -2 )
P ( X < 3 ) = 0.0228

Part d)

P ( 3.5 < X < 6.5 )
Standardizing the value

Z = ( 3.5 - 5 ) / 1
Z = -1.5
Z = ( 6.5 - 5 ) / 1
Z = 1.5
P ( -1.5 < Z < 1.5 )
P ( 3.5 < X < 6.5 ) = P ( Z < 1.5 ) - P ( Z < -1.5 )
P ( 3.5 < X < 6.5 ) = 0.9332 - 0.0668
P ( 3.5 < X < 6.5 ) = 0.8664

e.Greater than 6 or less than 4

P ( 4 < X < 6 )
Standardizing the value

Z = ( 4 - 5 ) / 1
Z = -1
Z = ( 6 - 5 ) / 1
Z = 1
P ( -1 < Z < 1 )
P ( 4 < X < 6 ) = P ( Z < 1 ) - P ( Z < -1 )
P ( 4 < X < 6 ) = 0.8413 - 0.1587
P ( 4 < X < 6 ) = 0.6827
Required probability = 1 - 0.6827 = 0.3173