45-733: lecture 10 (chapter 8)

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45-733 lecture 10 (chapter 8)

• Interval Estimation

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CI for difference in means, N()

• Suppose we have two populations we are interested
  in
• Sample from population X X1, X2,, Xn
• X has mean
• X has variance
• Sample from population Y Y1, Y2,, Ym
• Y has mean
• Y has variance

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CI for difference in means, N()

• Suppose we have two populations we are interested
  in
• We are interested in a CI for
• E(X)-E(Y)
• ?X- ?Y
• Example How much lower are defect rates at the
  Cleveland plan compared to the Pittsburgh plant?

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CI for difference in means, N()

• Case 1 matched pairs
• When we do our sampling, we sample an equal
  number from populations X and Y
• When we do our sampling, we carefully match our
  choices from each population to try to make sure
  all characteristics other than the one we are
  interested in are the same
• Example We select 100 items from Clv and Pgh
  plants, matching by type of product, day of week,
  hour of day, week of year, etc

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CI for difference in means, N()

• Case 1 matched pairs
• We are interested in E(X)-E(Y)
• Notice, that we can essentially sample from X-Y
  by looking at dixi-yi
• E(D)E(X)-E(Y) by rules of expectation
• V(D)V(X)V(Y) by rules of variance
• So, we can just use everything we know about
  making CI for means!

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CI for difference in means, N()

• Case 1 matched pairs
• Notice, if the sample size is large, we do not
  need to assume normal distributions
• If the sample size is large, then the CLT will
  assure

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CI for difference in means, N()

• Case 1 matched pairs
• Example problem pg 311, number 34

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CI for difference in means, N()

• Case 2 independent samples
• In this case, we just literally have two totally
  independent samples.
• Number of obs may be different
• Not carefully matched
• Example What is the difference between our
  companys mean compensation of salespeople and
  other companies in our industry?
• Not likely to be carefully matched because too
  expensive
• Much effort to find employees of other companies
  with the same education, experience, past
  performance, etc as ours

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CI for difference in means, N()

• Case 2 independent samples
• We are interested in E(X)-E(Y)
• But the populations may also differ in variance,
  so that we must also be concerned with V(X) and
  V(Y)
• Well, we can still calculate

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CI for difference in means, N()

• Case 2 independent samples
• We are interested in ?X- ?Y
• It would be natural, therefore, to look at
• Well,

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CI for difference in means, N()

• Case 2 independent samples
• If X and Y are from a normally distributed
  population, then X-bar and Y-bar are normally
  distributed
• Since the sum or difference of normals is normal

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CI for difference in means, N()

• Case 2 independent samples
• We can easily calculate the variance
• So,

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CI for difference in means, N()

• Case 2 independent samples
• Normalizing

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CI for difference in means, N()

• Case 2 independent samples
• Of course, we dont know the variance
• If the sample sizes are large, the CLT comes to
  the rescue

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CI for difference in means, N()

• Case 2 independent samples
• Of course, we dont know the variance
• If the sample sizes are large, the CLT comes to
  the rescue
• Notice, if we are using the CLT, we do NOT need
  to assume that X and Y are normal in this case!

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CI for difference in means, N()

• Case 2 independent samples
• Example pg 312, number 36

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CI for difference in means, N()

• Case 2 independent samples
• Of course, we dont know the variance
• If we know that the population variances are the
  same

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CI for difference in means, N()

• Case 2 independent samples
• If we know that the population variances are the
  same

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CI for difference in means, N()

• Case 2 independent samples
• Population variances the same
• Then, as with our previous arguments

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CI for difference in proportions
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CI for difference in proportions

• Suppose we have two populations we are interested
  in
• Sample from population X X1, X2,, Xn
• X is Bernoulli
• X has parameter px
• Sample from population Y Y1, Y2,, Ym
• Y is Bernoulli
• Y has parameter pY

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CI for difference in proportions

• Suppose we have two populations we are interested
  in
• We are interested in a CI for
• E(X)-E(Y)
• pX- pY
• Example How much less likely are women than men
  to vote for George W Bush?

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CI for difference in proportions

• Since we are interested in
• We will want to examine
• This has mean and variance

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CI for difference in proportions

• Obviously, the two sample proportions are sample
  means
• So, for large samples, we can apply a CLT

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CI for difference in proportions

• Example pg 313, number 44

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CI for difference in variance

• Suppose we have two populations we are interested
  in
• Sample from population X X1, X2,, Xn
• X has mean
• X has variance
• Sample from population Y Y1, Y2,, Ym
• Y has mean
• Y has variance

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CI for difference in variance

• Suppose we have two populations we are interested
  in
• We are interested in a CI for
• V(X)/V(Y)
• ?X/ ? Y
• Example How much lower is the variance in
  income in Sweden compared to the US?

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CI for difference in variance

• Suppose we have two populations we are interested
  in
• We are interested in a CI for
• V(X)/V(Y)
• ?X/ ? Y
• Example How much lower is the variance in
  income in Sweden compared to the US?

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CI for difference in variance

• Since we are interested in
• We will examine
• We know that

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CI for difference in variance

• The F-distribution
• The ratio of two independent chi-squared random
  variables (each divided by their respective
  degrees of freedom) is distributed with the
  F-distribution

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CI for difference in variance

• The F-distribution

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CI for difference in variance

• The F-distribution
• ?1 is called numerator degrees of freedom
• ?2 is called denominator degrees of freedom
• The F-distribution is compiled in an F-table in
  our book

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CI for difference in variance

• The F-distribution
• We know that
• So that means

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CI for difference in means, N()

• The F-distribution
• Then that means

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CI for difference in variance

• Making the CI

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CI for difference in variance

• Making the CI
• Our table is limited
• Our table has tabulations for PFgta?
• For ?0.05
• For ?0.01
• And for various values of numerator and
  denominator degrees of freedom
• (Picture)

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CI for difference in variance

• Making a 90 CI (picture)

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CI for difference in variance

• Making a 90 CI (picture)
• This can be had from the table

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CI for difference in variance

• Making a 90 CI (picture)
• This can be had from the table

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CI for difference in variance

• Example
• Two classes
• Class 1 has 30 students
• Class 2 has 22 students
• Variance
• Class 1 std dev on a test is 10
• Class 2 std dev on a test is 15
• Question assuming normality, is there a
  difference in the variance of the test scores?