Question

The following questions ask you to find the area under a standard normal distribution (i.e., z distribution). You may use the table in your textbook or an online application such as StatKey. (PLEASE SHOW ME HOW TO WORK, NOT ONLY ANSWERS, If it is handwriting, please don't write cursive script)

A. What proportion of the z distribution is above z = 1.5? [3 points]

B. What proportion of the z distribution is below z = 1.5? [3 points]

C. What proportion of the z distribution is between z = 0 and z = 1.5? [3 points]

D. What proportion of the z distribution is between z = -1.5 and z = +1.5? [3 points]

E. What z score separates the top 5% of the z distribution from the bottom 95% of the z distribution? [3 points]

F. What z score separates the bottom 1% of the z distribution from the top 99% of the z distribution? [3 points]

G. What z scores separate the middle 92% of the z distribution from the outer 8% (i.e., 4% on the left and 4% on the right)? [3 points]

H. What z scores separate the middle 95% of the z distribution from the outer 5% (i.e., 2.5% on the left and 2.5% on the right)? [3 points]

Answer 1

Part a)

P ( Z > 1.5 ) = 1 - P ( Z < 1.5 )
P ( Z > 1.5 ) = 1 - 0.9332
P ( Z > 1.5 ) = 0.0668

Part b)

P ( Z < 1.5 ) = 0.9332

Part c)

P ( 0 < Z < 1.5 ) = P ( Z < 1.5 ) - P ( Z < 0 )
P ( 0 < Z < 1.5 ) = 0.9332 - 0.5
P ( 0 < Z < 1.5 ) = 0.4332

Part d)

P ( -1.5 < Z < 1.5 ) = P ( Z < 1.5 ) - P ( Z < -1.5 )
P ( -1.5 < Z < 1.5 ) = 0.9332 - 0.0668
P ( -1.5 < Z < 1.5 ) = 0.8664

Part e)

X ~ N ( µ = 0 , σ = 1 )
P ( Z > z ) = 1 - P ( Z < z ) = 1 - 0.05 = 0.95
To find the value of z
Looking for the probability 0.95 in standard normal table to calculate Z score = 1.6449
Z = 1.64

part f)

X ~ N ( µ = 0 , σ = 1 )
P ( Z < z ) = 1% = 0.01
To find the value of z
Looking for the probability 0.01 in standard normal table to calculate Z score = -2.3263

Z = -2.33

Part g)

P ( a < X < b ) = 0.92
Dividing the area 0.92 in two parts we get 0.92/2 = 0.46
since 0.5 area in normal curve is above and below the mean
Area below the mean is a = 0.5 - 0.46
Area above the mean is b = 0.5 + 0.46
Looking for the probability 0.04 in standard normal table to calculate Z score = -1.7507
Looking for the probability 0.96 in standard normal table to calculate Z score = 1.7507
Z1 = -1.75

Z2 = 1.75

Part h)
P ( a < X < b ) = 0.95
Dividing the area 0.95 in two parts we get 0.95/2 = 0.475
since 0.5 area in normal curve is above and below the mean
Area below the mean is a = 0.5 - 0.475
Area above the mean is b = 0.5 + 0.475
Looking for the probability 0.025 in standard normal table to calculate Z score = -1.96
Looking for the probability 0.975 in standard normal table to calculate Z score = 1.96
Z1 = -1.96

Z2 = 1.96