Lecture24_FinalReview

Lecture24_FinalReview - ECE 340 Probabilistic Methods in...

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ECE 340 Probabilistic Methods in Engineering M/W 3-4:15 Prof. Vince Calhoun Prof. Vince Calhoun Lecture 19: Sums of RV Lecture 19: Sums of RV ’s, Sample s, Sample Mean, Laws of Large Numbers Mean, Laws of Large Numbers
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ECE 340 Probabilistic Methods in Engineering M/W 3-4:15 Prof. Vince Calhoun Prof. Vince Calhoun Lecture 20: Central Limit Theorem Lecture 20: Central Limit Theorem
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ECE 340 Probabilistic Methods in Engineering M/W 3-4:15 Prof. Vince Calhoun Prof. Vince Calhoun Lecture 21: Hypothesis Testing 1 Lecture 21: Hypothesis Testing 1
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Confidence Interval Confidence interval on mean, variance known If: random sample of size n: X 1 , …, X n X i ~ N( μ , σ 2 ) and X ~ N( μ , σ 2 /n) Then the test statistic: n X Z / 2 σ μ = ~ N(0, 1) by CLT With a CI, we want some range on μ , P[-Z α /2 Z Z α /2 ] = α , P[-Z α /2 n X / Z α /2 ] = 1- α probability test statistic between 2 points is 1- α
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Confidence Interval P[-Z α /2 n / σ μ X Z α /2 n / ] = 1- α want a range on μ P[- X + (-Z α /2 n / ) - μ - X + (Z α /2 n / )] = 1- α P[ X + Z α /2 n / μ X - Z α /2 n / ] = 1- α P[ X - Z α /2 n / μ X + Z α /2 n / ] = 1- α a 100(1- α )% CI (2-sided) on μ is: X - Z α /2 n / μ X + Z α /2 n / or X ± Z α /2 n / A CI is a statistic ± (table value) x standard error.
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Confidence Interval CI on mean, variance unknown Up to now, we have known σ . But typically we do not know, so what do we do? 1. If n 30, we can replace σ in the CI for the mean with the sample SD, S= () 2 1 1 1 N j j X N μ = . 2. if n < 30, then if X 1 , …, X n ~ N( μ , σ 2 ) the the test statistic t = n S X / ~ t-distribution with (n-1) degrees of freedom.
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Confidence Interval Get a CI for μ P [-t α /2, n-1 t t α /2, n-1 ] = 1- α P [-t α /2, n-1 n S X / μ t α /2, n-1 ] = 1- α 100(1- α )% CI on μ is: X - (t α /2, n-1 n S / ) μ X + (t α /2, n-1 n S / ) or X ± t α /2, n-1 n S /
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ECE 340 Probabilistic Methods in Engineering M/W 3-4:15 Prof. Vince Calhoun Prof. Vince Calhoun Lecture 22: Hypothesis Testing II Lecture 22: Hypothesis Testing II
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Hypothesis Tests • If the population variance is unknown, use s of the sample to approximate population variance, since under central limit theorem, s = σ when n > 30. Thus solve the problem as before, using s • With smaller sample sizes, we have a different problem. But it is solved in the same manner. Instead of using the z distribution, we use the t distribution
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Hypothesis Tests • Using t distribution when: • Sample is small (<30) • Parent population is essentially normal • Population variance ( σ ) is unknown • As n decreases, variation within the sample increases, so distribution becomes flatter.
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Tests of Hypotheses Two types of hypotheses: Null (H 0 )and alternative (H 1 )
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μ CP_1 CP_2
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Sample Size Determination
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Student’s t-distribution •S u p p o s e t h a t X 1 , X 2 , …, X n are n random samples from a normal distribution with mean μ and standard deviation s. Then the PDF of • is given by n s X T / μ = , , ] 1 ) / [( 1 ) 2 / ( ] 2 / ) 1 [( ) ( 2 / ) 1 ( 2 < < + Γ + Γ = + t k t k k k t f k π . , ) ( 0 1 k number positive any for dx e x k x k = Γ
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This note was uploaded on 02/10/2010 for the course TBE 2300 taught by Professor Cudeback during the Spring '10 term at Webber.

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Lecture24_FinalReview - ECE 340 Probabilistic Methods in...

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