ECON206_1011_01_Handout_06[1]

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

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ECON 206 November 29, 2010 METU- Department of Economics Instructor: H. Ozan ERUYGUR e-mail: oeruygur@gmail.com 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 [ ] 11 ( ,..., ) ( ,..., ) 1 nn PaX X bX X θ α ≤≤ = where is the parameter of interest and a , b are quantities computed based on the iid sample X 1 , . . . ,X n . The probability 1 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: oeruygur@gmail.com 2 In contrast to point estimators ˆ θ which give us a specific guess for , 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 [ ] , ab . 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 is μ or 1 - 2 , then for large samples
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ECON 206 November 29, 2010 METU- Department of Economics Instructor: H. Ozan ERUYGUR e-mail: oeruygur@gmail.com 3 ˆ ˆ Z θ σ = possesses approximately a standard normal distribution. Consequently, ˆ ˆ Z = forms (at least approximately) a pivotal quantity, and hence the pivotal method can be employed to develop intervals for the target parameter . Example 1 Let ˆ be a statistic that is normally distributed with mean and standard error ˆ . Find a confidence interval for that possesses a confidence coefficient equal to (1- α ). Solution We know that ˆ ˆ Z = 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 1 Pz Zz αα −≤ =
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ECON 206 November 29, 2010 METU- Department of Economics Instructor: H. Ozan ERUYGUR e-mail: oeruygur@gmail.com 4 Figure 1 Location of - z α /2 and z /2 Substituting for Z in the probability statement yields /2 ˆ ˆ 1 Pz z αα θ θθ σ ⎡⎤ −≤ = ⎢⎥ ⎣⎦ Multiplying by ˆ , we obtain
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This note was uploaded on 03/22/2012 for the course ECON 106 taught by Professor Kücüksenel during the Spring '12 term at Middle East Technical University.

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

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