The normal distribution

Normal density curves

Mean of the Normal Distribution

Variance of the Normal Distribution

Normal Distribution Computations 3Graphs of 3 Normals

Normal Distribution Computations 4Graphs of Standard Normal pdf and cdf

The standard normal distribution

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 .8944 - .3520 = .5424

If we use R note that pnorm(1.25) = 0.8943502 andpnorm(-.38) = 0.3519727. Subtraction gives 0.5423775

Nonstandard Normal Computations

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.

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?

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?

Example