ECON206_1011_01_Handout_04[1]

ECON206_1011_01_Handout_04[1] - ECON 206 METU- Department...

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ECON 206 November 08, 2010 METU- Department of Economics Instructor: H. Ozan ERUYGUR e-mail: oeruygur@gmail.com 1 LECTURE 04 SAMPLING DISTRIBUTIONS AND CENTRAL LIMIT THEOREM - III Outline of today’s lecture: C. Student’s t Distribution . ................................................................................................ 1 D. F Distribution . ............................................................................................................... 5 III. Normal Distribution Functions in Excel . ....................................................................... 10 A. NORMDIST Function. ................................................................................................ 10 B. NORMINV Function. .................................................................................................. 13 C. Calculating a Random Number from a Normal Distribution . ..................................... 14 C. Student’s t Distribution Theorem 7.1 tells us that ( ) / nY μ σ has a standard normal distribution. When is unknown, can be estimated by 2 SS = , and the quantity Y n S ⎛⎞ ⎜⎟ ⎝⎠ provides the basis for developing methods for inferences about population mean, μ . Definition 7.2 Let Z be a standard normal variable, and let W be a chi-square distributed variable with v degrees of freedom. Then, if Z and W are independent: / Z T Wv = is said to have a t distribution with v degrees of freedom (df).
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ECON 206 November 08, 2010 METU- Department of Economics Instructor: H. Ozan ERUYGUR e-mail: oeruygur@gmail.com 2 By Theorem 7.1, we know that ( ) / Zn Y μ σ =− has a standard normal distribution. By Theorem 7.3, we know that 2 2 (1 ) nS W = has a chi-square distribution with v =n-1 degrees of freedom (Z and W are independent since Y and 2 S are independent). Thus, we can write: / Z T Wv = () 22 / )/ / nY T v μσ = ⎡⎤ ⎣⎦ / ) / ) ) T S n n μμ σσ −− == = = S T S = has a t distribution with ( n-1 ) degrees of freedom. We will not give the equation for t density function here. But note that like the standard normal density function, the t density function is symmetric about zero.
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ECON 206 November 08, 2010 METU- Department of Economics Instructor: H. Ozan ERUYGUR e-mail: oeruygur@gmail.com 3 Comparison of the standard normal and t density functions. Example 7.5 The tensile strength for a type of wire is normally distributed with unknown mean μ and unknown variance σ 2 . Six pieces of wire were randomly selected from a large roll; and Y i , the tensile strength for portion i , is measured for i = 1, 2, . . . , 6. The population mean and variance 2 can be estimated by Y and 2 S , respectively. o Because 2 2 Y n = , it follows that 2 Y can be estimated by 2 S n . Find the approximate probability that Y will be within 2 2 Y S n = of the true population mean μ . Solution () [] 22 2 2 SS Y PY P n P T S nn ⎡⎤ ⎛⎞ −≤ = = ⎜⎟ ⎢⎥ ⎣⎦ ⎝⎠
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ECON 206 November 08, 2010 METU- Department of Economics Instructor: H. Ozan ERUYGUR e-mail: oeruygur@gmail.com 4 where T has a t distribution with n-1=5 degrees of freedom ( df ). If we look at the t distribution table , we see that, for 5 df, the upper- tail area to the right of 2.015 is 0.05. Hence, [ ] 2.015 2.015 0.90 PT −≤ = . So, the probability that Y will be within two estimated standard deviations of μ is slightly less than 0.90.
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ECON206_1011_01_Handout_04[1] - ECON 206 METU- Department...

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