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Unformatted text preview: 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 • 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 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 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 • 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 ) /σ ¯ 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 , < α < 1 . • Then the following probability statement is true: P z α/ 2 ≤ ( ¯ Y μ ) σ ¯ y ≤ z α/ 2 = 1 α 5 • 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 . 025 = 1 . 96 , P ( ¯ Y 1 . 96 σ √ n ≤ μ ≤ ¯ Y + 1 . 96 σ √ n ) = 0 . 95 6 • 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 ....
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This note was uploaded on 01/23/2010 for the course STAT 213 taught by Professor Hao during the Spring '10 term at Internet2.
 Spring '10
 hao
 Statistics

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