45 The Poisson Distribution

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Chapter 5Normal Probability Distributions

• 5-1 Overview
• 5-2 The Standard Normal Distribution
• 5-3 Applications of Normal Distributions
• 5-4 Sampling Distributions and Estimators
• 5-5 The Central Limit Theorem

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Chapter 5 Continuous Probability Distributions
and the Standard Normal Distribution
Created by Erin Hodgess, Houston, Texas
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LAPTOP3
Continuous Random Variables

• Continuous random variables take on an infinite
  number of values
• Continous Probability Distributions are shown by
• Probability function, f(x), or
• Probability Curves

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Probability Density Curves

• Probability Density Curve is the graph of a
  continuous probability distribution.
• Characteristics
• The total area under the curve must equal 1.
• Every point on the curve must have a vertical
  height that is 0 or greater.

Because the total area under the density curve is
equal to 1,there is a correspondence between area
and probability!
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Common Probability Curves

• Uniform
• Gamma
• Normal

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LAPTOP3
Normal Probability Distributions

• A continuous random variable x has a normal
  distribution if graph is bell-shaped and
  symmetric about its mean, µ.
• Using s, it also satisfies the 68-95-99.7 rule.
  

Notation xN(µ,s)
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Heights of Adult Men and Women
Figure 5-4
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Section 5-2 The Standard Normal Distribution
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Standard Normal Distribution

• Standard Normal Distribution a normal
  probability distribution that has a mean of 0 and
  a standard deviation of 1.
• x N(0,1) implies these are z-scores!
• z N(0,1)

Figure 5-5
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Area and the Standard Normal Curve

• Two types of questions
• Given z-scores, find area (probability)
• Given area (probability), find z-score

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Section 5-3 Applications of Normal Distributions
Created by Erin Hodgess, Houston, Texas
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Nonstandard Normal Distributions

• Given xN(µ,s), there are 2 applied questions
• Find the probability associated with given data
  values, x
• Find the data value associated with a given
  probability

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Normal Distributions Finding Probabilities

• Given data values, x, find probability
• Sketch xN(µ,s). Shade area of interest. Write
  a probability statement in terms of x.
• Find z-scores for x boundaries
• Write a probability statement in terms of z.
• Use Table 2 to find the area under the standard
  normal curve.

x? z ? P
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Given data values, x, find probability

x? z ? P
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Normal Distributions Finding Data Values

• Given probabilities, find data values, x
• Sketch xN(µ,s). Shade the given area.
• Use Table 2 to find the z-score associated with
  the above area.
• Convert the z-score to a data value.

P? z ? x
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Sections 5.4-5.5 Sampling Distributions and the
Central Limit Theorem
Created by Erin Hodgess, Houston, Texas
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Sampling Distributions

• Suppose you take repeated samples of the same
  size from a population.
• For each sample, you calculate the same
  statistic. The sample statistics change from
  sample to sample (this is called sampling
  variability)
• If you plotted all those statistics, what would
  be their shape, center, and spread?

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Sampling Distributions of Sample Means

• Given a population with mean µ and a standard
  deviation s
• If repeated samples of the same size n are drawn
  from the population, and x is computed from each
  sample, then the xs have the following
  probability distribution

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Sampling Distributions of Sample Means

• Mean of the sample means
• Standard deviation of the sample means
  
• (often called the standard error of the mean)
  (Note averages vary less than individuals)
• Shape of the sample means (Central Limit
  Theorem)
• If the population is normal, the xs will be
  normal (for any n)
• If the population is not normal, but n 30 ,
  then the xs will be normal anyway.

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Finding Probabilities for Sample Means

• For population, x xN(µ,s)
• The z-score for x is
• For sample means, x xN(µx,sx)
• The z-score for x is

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Definition
The Sampling Distribution of the Proportion is
the probability distribution of sample
proportions, with all samples having the same
sample size n.