Download Lecture 15 - Statistics

Today
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Today: Chapter 8
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Assignment: 5-R11, 5-R16, 6-3, 6-5, 8-2, 8-8
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Recommended Questions: 6-1, 6-2, 6-4, 6-10
8-1, 8-3, 8-5, 8-7
Reading:
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Sections 8.1, 8.2, 8.7, 8.8, 8.10
Will not do “suffiency”
Omit 8.4
Random Sampling
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A random sample of size n is a sequence of independent
observations from the population
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Random sampling has a good chance of producing a representative
sample (i.e., its accuracy reflects the population characteristic of
interest)
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The joint p.d.f of the observations is:
Random Sampling
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We will only consider random samples from infinite distributions or
with replacement
Example
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Suppose a random sample of size 5 is taken from a U(-1,1)
distribution. What is the joint pdf of the data?
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Suppose a random sample of size 5 is taken from a N(5,9)
distribution. What is the joint pdf of the data?
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Suppose a random sample of size n is taken from a N(μ,σ)
distribution. What is the joint pdf of the data?
Models and Parameters
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In statistics, likelihood has a very specific meaning
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We shall deal mainly with models that have that are defined by a
parameter, say, θ
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The probability model is written as f(x| θ)
Example
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Suppose a random sample of size 5 is taken from a N(5,9)
distribution.
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What is the joint pdf of the data?
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The parameter that defines this model is:
Likelihood
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Let f(x| θ) be the joint pdf of the sample X=(X1, X2,…,Xn)
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The function L(θ)
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Note: Terms that do not contain θ can be ignored in defining the
likelihood
 f(x| θ) is called the likelihood function
Example
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Write out the likelihood function for a random sample of size 10
from a (truncated exponential) distribution with pdf
f (x |  ) 
ex
1 e

; for 0  x  
Example
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Write out the likelihood function for a random sample of size n from
a N(μ,σ2) distribution
Example
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Write out the likelihood function for a random sample of size 10
from a Bernoulli distribution where 6 successes are observed,
followed by 4 failures
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Write out the likelihood function for a random sample of size 10
from a Bernoulli distribution where 6 successes are observed
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How are these different?
Likelihood principle
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If different experiments based on a model defined by θ result in the
same likelihood, one should draw the same conclusions
Properties of Sample Means and Proportions (8.7)
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Let X=(X1, X2,…,Xn) denote a random sample from some population
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The sample mean is:
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If This is a sample from a Bernoulli population, can denote this as:
Properties of Sample Means and Proportions
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When random sampling is performed, the sample observations are
indentically distributed
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Thus, each Xi come from a distribution with the same mean and
same variance
Properties of Sample Means and Proportions
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If X=(X1, X2,…,Xn) represents random sample then the expected
value of the sample mean is:
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The variance of the sample mean is:
Properties of Sample Means and Proportions
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If X=(X1, X2,…,Xn) represents random sample from a Bernoulli
population, then the expected value of the sample proportion (sample
mean) is:
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The variance of the sample proportion (sample mean) is:
Example
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A population of males has mean height of 70 inches and standard
deviation of 3 inches
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For a random sample of size 4, what is the mean and variance of the
sample mean
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For a random sample of size 40, what is the mean and variance of the
sample mean
Properties of Sample Means from Normal Populations
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Suppose X=(X1, X2,…,Xn) represents random sample from a Normal
population
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The distribution of the sample mean is:
Example
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A population of males has mean height of 70 inches and standard
deviation of 3 inches
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For a random sample of size 4, what is the mean and variance of the
sample mean
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For a random sample of size 40, what is the mean and variance of the
sample mean