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2.2 Standard Normal Calculations

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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 ... – PowerPoint PPT presentation

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Title: 2.2 Standard Normal Calculations


1
2.2 Standard Normal Calculations
2
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.

3
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.

4
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.

5
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.

6
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?
7
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.
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