Normal Distribution

To understand the normal distribution To be able to find probabilities given the Z scoreTo 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,..)

• Centres around the mean

• Often called a “bell curve”

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

• Z-scores are a means of answering the question ``how many standard deviations away from the mean is this observation?''

•Tables are provided to help us to calculate the probability for the standard normal distribution , Z

Find P(Z<1.25)

Tables give us P(Z<z)

It is vital that you always sketch a graph

P(Z<1.25) = 0.8944

Find P(Z>1.25)

Tables give us P(Z<z)

It is vital that you always sketch a graph

P(Z>1.25) = 1- 0.8944 = 0.1056

a) Find P(Z < 1.52)

It is vital that you always sketch a graph

b) Find P(Z > 2.60)

c) Find P(Z < -0.75)

d) Find P(-1.18 < Z < 1.43)

a) Find P(Z < 1.52)

SOLUTIONS

P(Z < 1.52) = 0.9357

SOLUTIONS

P(Z > 2.60) = 1 - 0.9053 = 0.0047

b) Find P(Z > 2.60)

SOLUTIONS

P(Z < -0.75) = P(Z > 0.75)

c) Find P(Z < -0.75)

P(Z > 0.75) = 1 – P(Z < 0.75)

P(Z > 0.75) = 1 – 0.7734 = 0.2266

SOLUTIONS

P(Z<1.43) = 0.9236

d) Find P(-1.18 < Z < 1.43)

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 processGiven 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

a) P(Z < a) = 0.7611

It is vital that you always sketch a graph

b) P(Z > a) = 0.0287

c) P(Z < a) = 0.0170

d) P(Z > a) = 0.01 ASK ABOUT THIS ONE

SOLUTIONS

a = 0.71

a) P(Z < a) = 0.7611

0.7611

SOLUTIONS

a = 1.9

b) P(Z > a) = 0.0287

0.02870.9713

SOLUTIONS

z = 2.12 so a = -2.12

c) P(Z < a) = 0.0170

0.0170 < 0.5 so a is negative

0.01700.9830

SOLUTIONS

Use percentage points of normal distribution table which gives P(Z>z)

d) P(Z > a) = 0.01

a = 2.3263

Normal distribution calculator