Confidence - Confidence Interval Problem: We want to...

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Confidence Interval Problem: We want to estimate the true mean of a random variable X accurately and economically. Approach 0: Estimating the mean with known variance Assumption 1) ( ) 2 ~, XN µσ 2) We know the variance 2 σ , but we do not know the mean µ . We want to estimate . Goodness of our assumption. Not practical, but a nice stepping stone toward a practical approach.
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Step 1) Sample Mean Have a sample mean of size n , 1 1 ˆ n j j XX n = = where j X ’s are independent identical samples of X . Then ˆ [] EX µ = and 2 ˆ () VAR X n σ = . Step 2) Normalize Gaussian Sample Mean 2 ˆ , XN n    . Normalize the sample mean: ˆ ~ (0,1) / X N n
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Step 3) Find the tail probability For an arbitrary 0 α > , find 2 z such that 2 2 2 2 1 2 2 u z Q z e du π   = =  . Then 2 ˆ P1 / X z n µ σ   ≤=   . 2 z
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Step 4) the confidence interval Get 1 α confidence interval as follows. 22 2 2 2 ˆˆ ˆˆ // XX z z z Xz n n nn αα µ σσ −− ⇒− ≤ ≤ + implies ˆˆ Pr 1 µα   ≤≤ + =   ˆˆ , is called the 1 confidence interval   −+  . For a larger n , we will have a tight bound. 2 However we have to know the variance σ
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Example. Assume ~ ( ,4) XN µ . We want to estimate . We will get the sample mean from 16 samples of X . Find the 80% confidence interval. For 80% confidence, 0.1 2 1 0.8 , 1.28 zz α −= = = . 2 Therefore 1.28 2/ 16 0.64 z n σ = ⋅= . We get ˆˆ Pr{ 0.64 0.64} 0.8 XX ≤≤ + = What does this mean? ˆ Go ahead to get 16 samples and the sample mean . ˆˆ It is true that 0.64 0.64 with probability 0.8 x xx ≤+
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Using Excel 2 1 returns 2 NORMSINV z α    ref. Excel file ‘normal-dist Confidence Interval Example’ Error Function, Complementary Error Function Many books and computer libraries have tables or routines for finding error function 2 2 () () 1 z t o erf z e dt erfc z erf z π ≡− erf(z) is an integral of N (0,1/2) from -z to z . erfc : complementary error function. 1 ( ) (0,1) ( ) 2 2 z z Q z N dt erfc = = So 2 z is found from solving 2 2 z erfc = .
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Review In many cases, it is safe to assume X is gaussian. However we can not expect to know the variance when we don’t know the mean
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Sample Variance Assume 2 ~ (, ) XN µσ We want to estimate µ and we do not know 2 σ . Define the sample variance as 22 1 1 ˆ () 1 n j j s XX n = = , where j X ’s are independent samples of X , and ˆ X is the sample mean defined as before.
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This note was uploaded on 11/23/2010 for the course EE EE528 taught by Professor Majungsoo during the Spring '10 term at Korea Advanced Institute of Science and Technology.

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Confidence - Confidence Interval Problem: We want to...

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