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**The normal distribution**

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Normal density curves

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**Mean of the Normal Distribution**

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**Variance of the Normal Distribution**

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**Normal Distribution Computations 3 Graphs of 3 Normals**

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**Normal Distribution Computations 4 Graphs of Standard Normal pdf and cdf**

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**The standard normal distribution**

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**Normal Distribution Computation P(-.38<Z<1.25)**

Look up the probability in the 1.25 slot in the table. We get .8944 Look up the probability in the -.38 slot in the table. We get .3520 The desired probability is = .5424 If we use R note that pnorm(1.25) = and pnorm(-.38) = Subtraction gives

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**Nonstandard Normal Computations**

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**Normal Distribution Computations Standardizing**

If X is a N(μ,σ) random variable then has a standard normal, N(0,1) distribution. This is standardizing and counts standard deviations from the mean.

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Example There are two machines available for cutting corks intended for use in wine bottles. The first produces corks with diameters that are normally distributed with mean 3 cm and standard deviation 0.10 cm. The second machine produces corks with diameters that have a normal distribution with mean 3.04 cm and standard deviation 0.04 cm. Acceptable corks have diameters between 2.9 cm and 3.1 cm. a) What is the probability that the first machine produces an acceptable cork? b) What is the probability that the second machine produces an acceptable cork? c) Which machine is more likely to produce an acceptable cork?

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Example Spray drift is a constant concern for pesticide applicators and agricultural producers. The inverse relationship between droplet size and drift potential is well known. The paper "Effects of 2,4-D Formulation and Quinclorac on Spray Droplet Size and Deposition"† investigated the effects of herbicide formulation on spray atomization. A figure in a paper suggested the normal distribution with mean 1050 µm and standard deviation 150 µm was a reasonable model for droplet size for water (the "control treatment") sprayed through a 760 ml/min nozzle. (a) What is the probability that the size of a single droplet is less than 1350 µm? At least 1000 µm? (b) What is the probability that the size of a single droplet is between 1000 and 1350 µm? (c) How would you characterize the smallest 2% of all droplets? I.e. The smallest 2% of droplets are those smaller than µm in size… (d) If the sizes of five independently selected droplets are measured, what is the probability that at least one exceeds 1350 µm?

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§ 5.3 Normal Distributions: Finding Values. Probability and Normal Distributions If a random variable, x, is normally distributed, you can find the probability.

§ 5.3 Normal Distributions: Finding Values. Probability and Normal Distributions If a random variable, x, is normally distributed, you can find the probability.

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