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Standardizing to z © Christine Crisp Teach A Level Maths Statistics 1

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Standardizing to Z "Certain images and/or photos on this presentation are the copyrighted property of JupiterImages and are being used with permission under license. These images and/or photos may not be copied or downloaded without permission from JupiterImages" S1: Standardizing to z AQA Edexcel Normal Distribution diagrams in this presentation have been drawn using FX Draw ( available from Efofex at )www.efofex.com

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Standardizing to Z We cant have a table of probabilities for every possible mean and variance ( as we would need an infinite number of tables! ) So, we always standardise to. ( Mean = 0, variance = 1 ). This is easy to do. If X is a random variable with distribution then, if The rule is subtract the mean and divide by the standard deviation Since this formula holds for X, it also holds for all the values of X, given by x.

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Standardizing to Z So, Tables only give 2 d.p. for z so this is all we need. Solution: (a) x = 400, so e.g.1 If X is a random variable with distribution find (a) (b)

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Standardizing to Z e.g.1 If X is a random variable with distribution find (a) (b) Solution: (b) So, There are 2 values to convert so we use subscripts for z. N.B. This is left of the mean so the z value will be negative.

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Standardizing to Z e.g.1 If X is a random variable with distribution Solution: (b) find (a) (b)

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Standardizing to Z Tip: The diagrams for X and Z show the same areas so I dont always draw both. If the question is straightforward I draw only the Z diagram but if Im not sure what to do Ill draw the X diagram ( and maybe the Z one as well ). SUMMARY To use tables to solve problems, we convert the values of the random variable X to values of the standardised normal variable using We need to be careful not to confuse standard deviation and variance. e.g. means = 4.

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Standardizing to Z e.g.2 A batch of batteries is claimed to last for 24 hours. In fact their running time has a normal distribution with mean time of 29 hours and standard deviation 6 hours. What proportion of batteries do not last for the claimed time? Solution: Let X be the random variable life of battery ( hours )

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Standardizing to Z e.g.2 A batch of batteries is claimed to last for 24 hours. In fact their running time has a normal distribution with mean time of 29 hours and standard deviation 6 hours. What proportion of batteries do not last for the claimed time? Solution: Let X be the random variable life of battery ( hours ) We want to find

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Standardizing to Z e.g.2 A batch of batteries is claimed to last for 24 hours. In fact their running time has a normal distribution with mean time of 29 hours and standard deviation 6 hours. What proportion of batteries do not last for the claimed time? Solution: Let X be the random variable life of battery ( hours ) We want to find So, Approximately 20 % do not last for 24 hours.

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Standardizing to Z Exercise 1. If X is a random variable with distribution 2. A shop sells curtain rails labelled 90 cm. In fact the lengths are normally distributed with mean 90·2 cm. and standard deviation 0·4 cm. What percentage of the rails are shorter than 90 cm ? find (a) (b) (c)

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Standardizing to Z Solutions: (a) 1. ( This is 1 standard deviation above the mean. )

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Standardizing to Z Solutions: 1. (b)

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Standardizing to Z Solutions: 1. (c)

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Standardizing to Z 2. A shop sells curtain rails labelled 90 cm. In fact the lengths are normally distributed with mean 90·2 cm. and standard deviation 0·4 cm. What percentage of the rails are shorter than 90 cm ? Solution: Let X be the random variable length of rail (cm) We want to find Approximately 31 % are shorter than 90 cm.

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Standardizing to Z The following slides contain repeats of information on earlier slides, shown without colour, so that they can be printed and photocopied. For most purposes the slides can be printed as Handouts with up to 6 slides per sheet.

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Standardizing to Z We cant have a table of probabilities for every possible mean and variance ( as we would need an infinite number of tables! ) So, we always standardise to. ( Mean = 0, variance = 1 ). This is easy to do. If X is a random variable with distribution then, if The rule is subtract the mean and divide by the standard deviation Since this formula holds for X, it also holds for all the values of X, given by x.

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Standardizing to Z SUMMARY We need to be careful not to confuse standard deviation and variance. e.g. means = 4. To use tables to solve problems, we convert the values of the random variable X to values of the standardised normal variable using

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Standardizing to Z So, Tables only give 2 d.p. for z so this is all we need. Solution: (a) x = 400, so e.g.1 If X is a random variable with distribution find (a) (b)

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Standardizing to Z Solution: (b)

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Standardizing to Z e.g.2 A batch of batteries is claimed to last for 24 hours. In fact their running time has a normal distribution with mean time of 29 hours and standard deviation 6 hours. What proportion of batteries do not last for the claimed time? Solution: Let X be the random variable life of battery ( hours ) We want to find So, Approximately 20% do not last for 24 hours.

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