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Lecture21_HypothesisTest1

# Lecture21_HypothesisTest1 - ECE 340 Probabilistic Methods...

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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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Homework Assignment 7: From the text: problems 7.2, 7.18, 7.21, 7.22, 7.25 Due date: Mon, Apr 28 at the beginning of class
Parameter estimation Point estimation A parameter is a point estimation (numeric fact) of a population. A statistic is a point estimation (numeric fact) of a sample. A statistic is used to estimate a parameter. Common point estimation of a parameter: μ is estimated by X , σ 2 is estimated by S 2 . There may be different point estimations for a parameter. How do we decide which point estimation is best for estimating a particular population parameter? Point estimation must be “close” to the true value of the unknown parameter being estimated. [bias 0 (or = 0)] if ( ) θ θ = ˆ E , then θ ˆ (statistic) is an unbiased estimation of θ (parameter). ( ) θ θ = ˆ E says that the expected/ mean/ average value of the test statistic is the parameter.

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Confidence Interval Confidence Interval (CI) estimation for the mean In many cases, a point estimate is not enough, a range of allowable values is better. An interval of the form: Lower (l) μ Upper (u) may be useful. To construct a CI for a parameter μ , we need to calculate 2 statistics l and u such that: P(l μ u) = 1 - α . This interval is a 100(1- α )% CI for parameter. l and u are lower and upper confidence limits. 1- α is called the confidence coefficient. α = level of significance for Type I error (rejecting valid hypotheses).
Confidence Interval Interpreting a confidence interval μ is covered by interval with confidence 100(1- α )%. If many samples are taken and a 100(1- α )% CI is calculated for each, then 100(1- α )% of them will contain/ cover the true value for μ . Note: the larger (wider) a CI, the more confident we are that the interval contains the true value of μ . But, the longer it is, the less we know about μ , due to variability or uncertainty need to balance

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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 / σ μ Z α /2 ] = 1- α probability test statistic between 2 points is 1- α
Confidence Interval P[-Z α /2 n / σ μ X Z α /2 n / σ ] = 1- α want a range on μ P[- X + (-Z α /2 n / σ ) - μ - X + (Z α /2 n / σ )] = 1- α P[ X + Z α /2 n / σ μ X - Z α /2 n / σ ] = 1- α P[ X - Z α /2 n / σ μ X + Z α /2 n / σ ] = 1- α a 100(1- α )% CI (2-sided) on μ is: X - Z α /2 n / σ μ X + Z α /2 n / σ or X ± Z α /2 n / σ A CI is a statistic ± (table value) x standard error.

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