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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

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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) 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 process Given the probability find the value of a in P(Z<a) P(Z<1.25) = 0.8944P(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

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