Hypothesis test examples, Correlation and Covariance

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Hypothesis test examples, Correlation and
Covariance

• Home Education case
• Hypothesis tests examples
• Correlation
• Covariance

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Home Education

• What financial reserve should be set aside for
  the TV kits?
• After presenting your estimates to the auditors
  you are surprised to find that the auditors dont
  think the estimate is accurate enough. In
  particular, the auditors insist that the
  financial reserve be within 10,000 of the actual
  liability. What do you do now?

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Colmar Brunton Opinion Poll

• Are population means or proportions being
  estimated?
• How do you calculate a margin of error for a
  proportion?
• How do you think this study did it?
• So, is the reporter correct?

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Hypothesis Testing
Make decision, compute p-value
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Funding program

• A Victorian state funding program defines a
  low-income suburb as one where the mean
  household income is below 25,000 per year. Any
  low-income suburb is qualified for funding. Mean
  income is determined by taking a sample of
  households from a suburb and calculating the mean
  of that sample. The state wishes to have only a
  1 chance of refusing funding to a qualified
  suburb.

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Funding program Statpro

• Suppose a sample of 30 households in a suburb was
  taken and found to have a mean of 27,586 and a
  standard deviation of 8256.
• Should the state fund this suburb?
• Download funding.xls from web to do this with
  Statpro

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Apartment Rental

• The monthly rent for a 2-bedroom apartment in a
  particular country town is reported to average
  350 per month. Assume we would like to test this
  with H0 m 350 and H1 m ? 350. A sample of
  36 two-bedroom apartments is selected. The sample
  mean turns out to be 338 with a sample standard
  deviation of 40.
• Use a 0.05 significance level to conduct this
  hypothesis test. What is the p-value?

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Apartment rental by hand

• What is a confidence interval centred around the
  null value?
• Null value 350
• Standard error 40/6 6.67
• Margin of error 2 x 6.67 13.33
• Interval around null is 336.67 to 363.33
• Does the sample fall within this?
• YES. So must accept the null hypothesis.

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Apartment rental template

• Using the Confidence Intervals and Hypothesis
  Tests Excel template you should get the same
  results as you did by hand.

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Correlation

• Correlation coefficient describes how two
  variables are related.
• Doesnt imply causality.

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Covariance and Correlation
y
x
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Correlation

• Correlation is standardised. (Between -1 and
  1.)
• Correlation near zero implies no relationship
• Correlation near 1 strong positive relationship
• Correlation near -1 strong negative relationship

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Example Baggage Handling

• Data
• Airport Handling time Number of flights
• A 18 23
• B 23 48
• C 17 39
• D 27 4
• E 31 9

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Baggage Handling

• Mean handling time 23.2 minutes
• Standard dev. of handling time 5.93 mins
• Mean number of flights 24.6
• Standard dev. of number of flights 18.88
• Covariance -71.4
• Correlation -0.64
• What does this mean?

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Try the Guessing Correlations game on the class
web page.
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Correlations using Excel

• Choose Data Analysis from the Tools menu.
• Choose Correlation
• Enter cell range of variables into Input Range

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Covariance

• When might you use covariance?
• Some equations
• Var(aX) a2Var(X)
• Var(aXbY) a2Var(X) b2Var(Y) 2ab Cov(X,Y)
• When variables are independent then Covariance
  0.
• Applications?
• Implications?

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Example Share portfolio

• Share A average return 8
• variance 1.5
• so stddev 12.25
• Share B average return 5
• variance 1
• so stddev 10
• Cov(A,B) -0.8 (why might this be negative?)

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Share portfolio

• Put equal dollars in A and B (equal weighted
  portfolio)
• Then Var(A B) 0.015 0.01 0.016 0.009
  0.9
• Standard deviation (portfolio) 9.5
• By contrast put all the money in B (lowest risk)
  Var(2B) 0.04 4
• Standard deviation (all money in B) 20

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Family car is killing us

• Do you see correlations being used in this
  article?
• What evidence does Athol Robinson give for his
  conclusions?
• Do you think the reporter did a good job of
  writing this article? Why or why not?

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Next class

• Read supplementary materials on Simple Regression
  and More Regression
• Read Quality Kitchens
• Download qualkit.xls and bring on laptop
• If you hand in a one page summary of the Quality
  Kitchens case I will give you feedback on it
  (this is optional)
• Try the Guessing Correlations on the class web
  site.