Question

1. Given a normal distribution with μ=102 and σ=25, if you select a sample of n=12, what is the probability that ?̅ is

a. less than 90 ?
b. between 90 and 92.5 ?
c. above 103.6 ?

2. Given a normal distribution with μ=101 and σ=15, if you select a sample of n=9, what is the probability that ?̅ is
a. less than 95 ?
b. between 90 and 92.5 ?
c. above 101.8 ?

Answer 1

Question 1

Part a)

P ( X < 90 )
Standardizing the value


Z = -1.66

P ( X < 90 ) = P ( Z < -1.66 )
P ( X < 90 ) = 0.0482

Part b)

P ( 90 < X < 92.5 )
Standardizing the value


Z = -1.66

Z = -1.32
P ( -1.66 < Z < -1.32 )
P ( 90 < X < 92.5 ) = P ( Z < -1.32 ) - P ( Z < -1.66 )
P ( 90 < X < 92.5 ) = 0.094 - 0.0482
P ( 90 < X < 92.5 ) = 0.0458

Part c)

P ( X > 103.6 ) = 1 - P ( X < 103.6 )
Standardizing the value


Z = 0.22

P ( Z > 0.22 )
P ( X > 103.6 ) = 1 - P ( Z < 0.22 )
P ( X > 103.6 ) = 1 - 0.5877
P ( X > 103.6 ) = 0.4123

Question 2

Part a)

P ( X < 95 )
Standardizing the value


Z = -1.2

P ( X < 95 ) = P ( Z < -1.2 )
P ( X < 95 ) = 0.1151

Part b)

P ( 90 < X < 92.5 )
Standardizing the value


Z = -2.2

Z = -1.7
P ( -2.2 < Z < -1.7 )
P ( 90 < X < 92.5 ) = P ( Z < -1.7 ) - P ( Z < -2.2 )
P ( 90 < X < 92.5 ) = 0.0446 - 0.0139
P ( 90 < X < 92.5 ) = 0.0307

Part c)

P ( X > 101.8 ) = 1 - P ( X < 101.8 )
Standardizing the value


Z = 0.16

P ( Z > 0.16 )
P ( X > 101.8 ) = 1 - P ( Z < 0.16 )
P ( X > 101.8 ) = 1 - 0.5636
P ( X > 101.8 ) = 0.4364