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2.2 Standard Normal Calculations

Empirical vs. Standardizing

- Since all normal distributions share common

properties, we have been able to use the

68-95-99.7 Rule to describe the distribution.

All normal distributions are the same if measured

in units of size ? about the mean, µ, as the

center. If units arent measured like this, we

can change the unit, which is called

standardizing.

To Standardize or z - score

- Let x an observation from a distribution.
- The standardized value of x is defined by the

formula, , which is known as the

z-score. - The z score tells us how many standard

deviations the observation falls from the mean

and in what direction.

Return to the Giraffe Problem

- The mean, µ 204 inches with a standard

deviation, ? 5.5. - So for a giraffe that is 215 inches tall, his z

score, standardized height, would be

z(215-204)/5.52. - Interpreted as two standard deviations to the

right of the mean. If we wanted to know the

probability/percentages of giraffes that are up

to 215 inches tall, we refer to the standard

normal probabilities table, first page (back and

front) of your text or yellow packet. We look

for 2.000 and we get .9772 or 97.72.

Giraffe Problem continued

- Using the calculator, we go to 2nd VARS

(DISTR). We select normalcdf(0, 215, 204, 5.5)

or normal cdf(0, 215, 2(z-score)). The calc

returns with a proportion of .9772, again roughly

97.7. - On average, about 97.7 of giraffes are 215

inches tall or shorter.

Giraffes Giraffes

- For a giraffe that is 187.5 inches tall, his z

score is ________, which means that this height

is _____ standard deviations to the _________ of

the mean. What is the probability of giraffes

being at most this tall? _________.

What percentages of giraffes are less than

192.285 inches? What percentages are taller than

192.285 inches?

Summary

- If we standardize a normal distribution, it makes

the distribution into a single distribution that

is still normal called the standard normal

distribution. In this distribution the mean 0

and the standard deviation is 1, N(0,1). - If a variable x has any normal distribution

N(µ,?), then the standardized variable is z. - Classwork The Length of Pregnancy worksheet.

Give each answer as a one-liner in context.