lesson7 - Estimation and Confidence Intervals

lesson7 - Estimation and Confidence Intervals - Lesson7:

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 Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson7-1 Lesson 7: Estimation and Confidence  Estimation and Confidence  Intervals Intervals 
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 Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson7-2 Outline Point and interval estimates Confidence intervals Student’s t-distribution Degree of freedom Confidence interval for population mean Confidence interval for a population proportion Selecting a sample size Summary
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 Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson7-3 Point and Interval Estimates point estimate  is a single value (statistic) used to estimate a  population value (parameter).  confidence interval  is a range of values within which the  population parameter is expected to occur.  
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 Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson7-4 Confidence Intervals The degree to which we can rely on the statistic  is as important as  the initial calculation.  Samples give us estimates of the population parameter –  only  estimates . Ultimately, we are concerned with the accuracy of the  estimate. 1. Confidence interval provides a range of values  Based on observations from 1 sample 2. Confidence interval gives information about closeness to  unknown population parameter Stated in terms of probability Exact closeness not known because knowing exact closeness  requires knowing unknown population parameter
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Areas Under the Normal Curve Between: ±  σ  - 68.26% ±  σ  - 95.44% ±  σ  - 99.74% µ µ-1 σ µ+1 σ µ-2 σ µ+2 σ µ+3 σ µ-3 σ If we draw an observation from  the normal distributed  population, the drawn value is  likely (a chance of 68.26%) to lie  inside the interval of  (µ-1 σ , µ+1 σ ). P((µ-1 σ  <x<µ+1 σ ) =0.6826.
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 Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson7-6 P(µ-1 σ  <x<µ+1 σ )  vs P(x-1 σ  <µ <x+1 σ ) P(µ-1 σ  <x<µ+1 σ )   is the probability that a randomly drawn  observation will lie between (µ-1 σ , µ+1 σ ). P(µ-1 σ  <x<µ+1 σ )   = P(µ-1 σ   -µ-x  <x  -µ-x   +1   σ -µ-x = P(-1 σ  -x <-µ<1   σ -x) = P(-(-1 σ  -x ) > -(-µ) > -(1   σ -x)) = P(1 σ  +x >µ>-1   σ +x) P(x - 1 σ  <µ <x+1 σ ) P(x-1 σ  <µ <x+1 σ ) is the probability that the population mean  will lie between (x-1 σ , x+1 σ ).
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 Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson7-7 P(m-1  σ  <µ <m+1  σ ) (m=sample mean) m m m m
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 Ka-fu Wong © 2007 ECON1003: Analysis of Economic Data Lesson7-8 P(µ-a <x<µ+b)  vs P(x-a<µ <x+b) P(µ-a <x<µ+b)   is the probability that a drawn observation will  lie between (µ-a, µ+b).
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lesson7 - Estimation and Confidence Intervals - Lesson7:

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