ECON206_1011_01_Handout_03[1]

ECON206_1011_01_Handout_03[1] - ECON 206 METU- Department...

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ECON 206 October 31, 2010 METU- Department of Economics Instructor: H. Ozan ERUYGUR e-mail: [email protected] 1 LECTURE 03 SAMPLING DISTRIBUTIONS AND CENTRAL LIMIT THEOREM - II Outline of today’s lecture: I. The Central Limit Theorem (CLT) . .................................................................................... 1 II. Sampling Distributions Related to Normal Distribution . .................................................. 3 A. Sampling Distribution of Mean. .................................................................................... 3 B. Chi-Square Distribution . ............................................................................................... 7 C. Student’s t Distribution . .............................................................................................. 16 D. F Distribution . ............................................................................................................. 20 III. Normal Distribution Functions in Excel . ....................................................................... 25 A. NORMDIST Function. ................................................................................................ 25 B. NORMINV Function. .................................................................................................. 28 C. Calculating a Random Number from a Normal Distribution . ..................................... 29 I. The Central Limit Theorem (CLT) The Central Limit Theorem (CLT) The mean of a random sample has a sampling distribution whose shape can be approximated by a Normal distribution. The larger the sample, the better the approximation will be. Theorem 7.4 Let Y 1 , Y 2 , …, Y n be independent and identically distributed random variables with E(Y i )= μ and V(Y i )= σ 2 < . Define / n Y U n ⎛⎞ = ⎜⎟ ⎝⎠ where 1 1 n i i YY n = =
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ECON 206 October 31, 2010 METU- Department of Economics Instructor: H. Ozan ERUYGUR e-mail: [email protected] 2 Then the distribution of n U converges to a standard normal distribution function as n Æ . That is, probability statements about n U can be approximated by corresponding probabilities for the standard normal random variable if n is large. o Usually a value of n greater than 30 will ensure that the distribution of n U can be closely approximated by a normal distribution. The conclusion of the central limit theorem is often replaced with the simpler statement that Y is asymptotically normally distributed with mean μ and variance σ 2 .
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ECON 206 October 31, 2010 METU- Department of Economics Instructor: H. Ozan ERUYGUR e-mail: [email protected] 3 II. Sampling Distributions Related to Normal Distribution A. Sampling Distribution of Mean Theorem 7.1 Let Y 1 , Y 2 , …, Y n be independent, normal random variables, each with mean μ and variance σ 2 . Then: 1 1 n i i YY n = = is normally distributed with mean Y μ = and variance 2 2 Y n σ = .
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ECON 206 October 31, 2010 METU- Department of Economics Instructor: H. Ozan ERUYGUR e-mail: [email protected] 4 Example 7.1 and 7.2 A bottling machine can be regulated so that it discharges an average of μ ounces per bottle. It has been observed that the amount of fill dispensed by the machine is normally distributed with σ = 1.0 ounce. A sample of n = 9 filled bottles is randomly selected from the output of the machine on a given day (all bottled with the same machine setting) and the ounces of fill measured for each. a) Find the probability that the sample mean will be within 0.3 ounce of the true mean μ for that particular setting. b) How many observations should be included in the sample if we wish Y to be within 0.3 ounce of μ with probability 0.95?
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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_03[1] - ECON 206 METU- Department...

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