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ECON206_1011_01_Handout_06[1] - ECON 206 METU Department of...

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ECON 206 November 29, 2010 METU- Department of Economics Instructor: H. Ozan ERUYGUR e-mail: [email protected] 1 LECTURE 06 CONFIDENCE INTERVALS Outline of today’s lecture: I. Introduction ........................................................................................................................ 1 II. Large Sample Confidence Intervals .................................................................................. 2 A. Interpretation of (1- α ) Confidence Intervals ................................................................. 7 B. Simulation to Show that a Confidence Interval is a Random Variable ......................... 8 III. Selecting the Sample Size ................................................................................................ 9 IV. Small Sample Confidence Intervals ............................................................................... 11 A. Confidence Interval for μ ............................................................................................ 11 B. Confidence Interval for μ 1 - μ 2 ...................................................................................... 12 V. Confidence Intervals for σ 2 ............................................................................................. 18 Appendix Constructing Confidence interval for the mean .................................................. 20 I. Introduction Confidence intervals (CI) is an important part of statistical inference. It refers to obtaining statements such as [ ] 1 1 ( ,..., ) ( ,..., ) 1 n n P a X X b X X θ α = where θ is the parameter of interest and a , b are quantities computed based on the iid sample X 1 , . . . ,X n . The probability 1 α is called the confidence coefficient . It is generally taken to be 0.9, 0.95 or 0.99.
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ECON 206 November 29, 2010 METU- Department of Economics Instructor: H. Ozan ERUYGUR e-mail: [email protected] 2 In contrast to point estimators ˆ θ which give us a specific guess for θ , CIs provide an interval - which is less accurate than a specific number. The advantage of confidence intervals is that we can characterize the confidence of our statement [ ] , a b θ . CIs of the form [ ] , b −∞ or [ ] , a are called one-sided CIs (lower or upper). In general, to construct a CI, we need to know some partial information concerning the unknown distribution - for example that it is a normal distribution. o Such CIs are called small sample confidence intervals . If we can not make such an assumption we can still construct CIs by appealing to the central limit theorem . However, in this case, the CI will be only approximately correct - with the approximation improving in its quality as the sample size increases n → ∞ . o Such CIs are called large sample confidence intervals . II. Large Sample Confidence Intervals If the target parameter θ is μ or μ 1 - μ 2 , then for large samples
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ECON 206 November 29, 2010 METU- Department of Economics Instructor: H. Ozan ERUYGUR e-mail: [email protected] 3 ˆ ˆ Z θ θ θ σ = possesses approximately a standard normal distribution. Consequently, ˆ ˆ Z θ θ θ σ = forms (at least approximately) a pivotal quantity, and hence the pivotal method can be employed to develop intervals for the target parameter θ . Example 1 Let ˆ θ be a statistic that is normally distributed with mean θ and standard error ˆ θ σ . Find a confidence interval for θ that possesses a confidence coefficient equal to (1- α ). Solution We know that ˆ ˆ Z θ θ θ σ = has a standard normal distribution. Now, we select two values in the tails of this distribution, namely, z α /2 and -z α /2 , such that (see figure below) [ ] /2 /2 1 P z Z z α α α =
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ECON 206 November 29, 2010 METU- Department of Economics Instructor: H. Ozan ERUYGUR e-mail: [email protected] 4 Figure 1 Location of - z α /2 and z α /2 Substituting for Z in the probability statement yields /2 /2 ˆ ˆ 1
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