Transcript Normal distribution 1

Normal Distribution
To
understand the normal distribution
To be able to find probabilities given the
Z score
To be able to find the Z score given the
probability
•Most commonly observed probability
distribution
•1800s, German mathematician and
physicist Karl Gauss used it to analyse
astronomical data
•Sometimes called the Gaussian
distribution in science.
Normal Distribution
• Occurs naturally(e.g. height, weight,..)
• Often called a “bell curve”
• Centres around the mean 
Normal Distribution
• Spread depends on standard deviation 
• Percentage of distribution included
depends on number of standard
deviations from the mean
Properties of Normal Distribution
• Symmetrical
• Area under
curve = 1
Standard Normal Distribution
• Mean (=0
• Standard
deviation
()=1
Standard Normal Distribution
•Tables are provided to
help us to calculate the
probability for the
standard normal
distribution , Z
• Z-scores are a means of answering
the question ``how many standard
deviations away from the mean is this
observation?''
Tables give us P(Z<z)
It is vital that you always sketch a graph
Find P(Z<1.25)
P(Z<1.25) = 0.8944
Tables give us P(Z<z)
It is vital that you always sketch a graph
Find P(Z>1.25)
P(Z>1.25) = 1- 0.8944 = 0.1056
It is vital that you always sketch a graph
a) Find P(Z < 1.52)
b) Find P(Z > 2.60)
c) Find P(Z < -0.75)
d) Find P(-1.18 < Z < 1.43)
SOLUTIONS
a) Find P(Z < 1.52)
P(Z < 1.52) = 0.9357
SOLUTIONS
b) Find P(Z > 2.60)
P(Z > 2.60) = 1 - 0.9053 = 0.0047
SOLUTIONS
c) Find P(Z < -0.75)
P(Z < -0.75) = P(Z > 0.75)
P(Z > 0.75) = 1 – P(Z < 0.75)
P(Z > 0.75) = 1 – 0.7734 = 0.2266
SOLUTIONS
d) Find P(-1.18 < Z < 1.43)
P(Z<1.43) = 0.9236
P(Z>1.18) = 1-0.881
P(Z>1.18) = 0.119
P(-1.18<Z<1.43) = 0.9236 - 0.119 =
0.8046
Reversing the process
Given the probability find the value of a in
P(Z<a)
P(Z<1.25) = 0.8944
P(Z<-0.25) = 0.4013
If the probability is >0.5 then a is positive
If the probability is <0.5 then a is negative
It is vital that you always sketch a graph
a) P(Z < a) = 0.7611
b) P(Z > a) = 0.0287
c) P(Z < a) = 0.0170
d) P(Z > a) = 0.01
ASK ABOUT THIS ONE
SOLUTIONS
a) P(Z < a) = 0.7611
0.7611
a = 0.71
SOLUTIONS
b) P(Z > a) = 0.0287
0.9713
a = 1.9
0.0287
SOLUTIONS
c) P(Z < a) = 0.0170
0.0170 < 0.5 so a is negative
0.9830
z = 2.12
so
a = -2.12
0.0170
SOLUTIONS
d) P(Z > a) = 0.01
Use percentage points of normal
distribution table which gives P(Z>z)
a = 2.3263
Normal distribution calculator