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ch5_slides - INFERENCES ABOUT One of the major objectives...

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INFERENCES ABOUT μ One of the major objectives of statistics is to make inferences about the distribution of the elements in a population based on information contained in a sample . Numerical summaries that characterize the population distribution are called parameters . The population mean μ and population variance σ 2 are two important parameters. Others are median, range, mode, etc. 1
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Methods for making inferences are basically designed to answer one of two types of questions: (a) Approximately what is the value of the parameter? or (b) Is the value of the parameter less than (say) 6? Statisticians answer the first question by estimating the parameter using the sample. The second case might require a test of a hypothesis . 2
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Point and Interval Estimation of μ when σ is known and n is large Point estimation of μ does not require that σ be known or a large n The point estimate of μ is the sample mean ¯ y Point estimates by themselves do not tell how much ¯ y might differ from μ , that is, the accuracy or precision of the estimate A measure of accuracy is the difference between sample mean ¯ y and population mean μ is called sampling error 3
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A Confidence Interval for μ An interval estimate, called a confidence interval , incorporates information about the amount of sampling error in ¯ y A confidence interval for μ takes the form y - E, ¯ y + E ) , for a number E An associated number called the confidence coefficient helps assess how likely it is for μ to be in the interval. 4
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To derive the specific form of the confidence interval for μ , for the case when σ is known and n is large , the CLT result must be used. By the CLT, Z = ( ¯ Y - μ ) ¯ y is approximately N (0 , 1) . Let Z have a N (0 , 1) distribution (exactly). Let z α/ 2 denote the 1 - α/ 2 quantile of the standard normal distribution, for a given number α , 0 < α < 1 . Then the following probability statement is true: P - z α/ 2 ( ¯ Y - μ ) σ ¯ y z α/ 2 = 1 - α 5
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Manipulating the inequalities, without changing values, we have P ( σ ¯ y z α/ 2 μ - ¯ Y ≥ - σ ¯ y z α/ 2 ) = 1 - α P ( ¯ Y - σ ¯ y z α/ 2 μ ¯ Y + σ ¯ y z α/ 2 ) = 1 - α If, for example, α = 0.05 then P ( ¯ Y - σ n z . 025 μ ¯ Y + σ n z . 025 ) = 0 . 95 and, since z 0 . 025 = 1 . 96 , P ( ¯ Y - 1 . 96 σ n μ ¯ Y + 1 . 96 σ n ) = 0 . 95 6
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Using this statement, a (1- α )100% confidence interval for μ , with confidence coefficient 1 - α can be calculated using a random sample of data y 1 , y 2 , . . . , y n It is usually written in the form ¯ y - σ n z α/ 2 , ¯ y + σ n z α/ 2 . Example: Suppose n = 36, σ = 12 , ¯ y = 24 . 8 And for α = 0 . 05 , z α/ 2 z 0 . 025 = 1 . 96 7
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Thus a 95% C.I. for μ is 24 . 8 - 12 36 × 1 . 96 , 24 . 8 + 12 36 × 1 . 96 The interval for μ is: (20.88, 28.72) with confidence coefficient 0.95.
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