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# Chapter_4 - Decision Making for a Single Sample Chapter 4...

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Decision Making for a Single Sample Chapter 4

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2 Sections 4-1 through 4-4 I will not be covering sections 4-1 through 4-4 in strict order
3 Statistical Inference Statistical Inference uses methods to determine information about a population based upon data taken from samples of that population Two categories of Statistical Inference : 1. Parameter estimation 2. Hypothesis testing

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4 Terminology A parameter of a population is generically represented as θ in most statistics and probability texts A statistic is a random variable that can assume many values. Random variable are always in upper case. A point estimate is a specific numerical value that the statistic can assume. Point estimates are always in lower case Example: X is a random variable because it can assume different values is a numerical value computed from a sample of size n. It is just one value that X can assume. x is a point estimate of the population mean μ = = n i i n x x 1
5 Parameter Estimation The population mean μ is estimated by the point estimate = = n i i n x x 1 ( 29 1 1 2 2 - - = = n x x s n i i n X p = = = - = - 2 1 1 2 2 1 1 1 2 1 n i i n i i n x n x x x 2 2 1 1 2 1 n X n X p p - = -

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6 Other Notation Sometimes the point estimate of a parameter is represented as the parameter but with a “hat” on it: x is the point estimate of the population mean, μ is the point estimate of the population mean, μ Same idea: μ ˆ 2 ˆ σ p ˆ 2 1 ˆ ˆ μ μ -
7 Why do we need to know the terminology and notation? Notation is sometimes the means by which you can tell what is being asked for by a. your boss b. your teacher c. an exam question d. the government Minimize confusion. Learn the proper notation. www.mytakeonthings.com

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8 “Good” Estimator Properties Unbiased Example for sample mean Low Mean Square Error Mean Square Error (MSE) = If is unbiased, then the MSE is just the variance of the estimator: The smaller MSE of an estimator, the better ( 29 Θ = Θ ˆ E ( 29 ( 29 2 ˆ ˆ Θ - Θ = Θ E MSE μ = ) ( Recall X E ( 29 ( 29 ( 29 ( 29 { } { } { } μ μ μ μ μ = = + + + = + + = = = n n n X E X E X E n n X E X E n n i i 1 1 1 2 1 1 Recall that E(X) = μ θ ˆ ( 29 ( 29 Θ = Θ ˆ ˆ V MSE Θ ˆ n X 2 2 Recall σ σ =
Hypothesis Testing The method of determining if a statement about a population parameter is (likely) true or not

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10 Example Population of men’s heights – H 0 : μ = 71 inches – H 1 : μ 71 inches • H 0 is called the null hypothesis . In English, the one above says, The mean of the population of men’s heights is 71 inches – H 0 is always a statement of an equality • H 1 is called the alternative hypothesis . In English, the one above says, The mean of the population of men’s heights is NOT 71 inches.
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Chapter_4 - Decision Making for a Single Sample Chapter 4...

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