normal distribution to understand the normal distribution to be able to find probabilities given the...
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![Page 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](https://reader035.vdocuments.mx/reader035/viewer/2022081515/5516063f550346d46f8b5dac/html5/thumbnails/1.jpg)
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
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•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.
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Normal Distribution• Occurs naturally(e.g. height, weight,..)
• Centres around the mean
• Often called a “bell curve”
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Normal Distribution• Spread depends on standard deviation
• Percentage of distribution included depends on number of standard deviations from the mean
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Properties of Normal Distribution
• Symmetrical
• Area under curve = 1
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Standard Normal Distribution
• Mean (=0
• Standard deviation ()=1
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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
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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
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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
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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)
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a) Find P(Z < 1.52)
SOLUTIONS
P(Z < 1.52) = 0.9357
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SOLUTIONS
P(Z > 2.60) = 1 - 0.9053 = 0.0047
b) Find P(Z > 2.60)
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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
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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
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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
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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
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SOLUTIONS
a = 0.71
a) P(Z < a) = 0.7611
0.7611
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SOLUTIONS
a = 1.9
b) P(Z > a) = 0.0287
0.02870.9713
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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
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SOLUTIONS
Use percentage points of normal distribution table which gives P(Z>z)
d) P(Z > a) = 0.01
a = 2.3263
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Normal distribution calculator