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Chapter 9 & 10 One sample case

Chapter 9 & 10 One sample case - Revised...

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Revised: November 12, 2011 Statistical Inferences from One Sample (Sections 9.1, 9.2, 10.1, 10.2 and 10.7) Methods of Statistical Inference: Point Estimation Interval Estimation Significance Tests Point Estimation A point estimator is a statistic that specifies how to use the sample data to estimate an unknown parameter of a population. Some commonly used parameters and their estimators: Parameter Estimator μ = Population mean X ˆ X μ = = Sample Mean 2 σ = Population variance 2 2 X X ˆ S σ = = Sample variance p = Population proportion ˆ / p X n = = Sample proportion Desirable properties of Point Estimators ( ˆ θ ): Small variance [ 2 ˆ θ σ = Var( ˆ θ ) ] and small MSE = Var + Bias 2 Unbiasedness: ˆ ( ) E θ θ = Bias = ˆ ( ) E θ θ - Normality 2 ˆ ˆ ~ ( , ) N θ θ θ σ The above estimators have the following properties: 1. They are efficient , i.e. one cannot find other estimators that have smaller standard errors and these estimators are closer to the true parameter values. 2. They are unbiased . In repeated sampling the estimates average out to give the true values of the parameters. (S is not an unbiased estimator of σ but its bias is small and decreases as the sample size increases.) 3. The sample mean and the sample proportion have approximate normal distributions (but not the sample variance). General formula for a confidence interval CI = (Estimate ± Margin of Error) ME = Margin of Error = (table value) × (Standard Error of Estimate) The width of a CI Increases as the confidence level increases Decreases as the sample size increases. STA3032 Chapters 9 & 10, Page 1 of 14
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Inferences about population Mean, μ Assumptions that must be satisfied For making inferences about μ 1. SRS 2. Quantitative random variable 3. Normal population or large n. Point Estimation: The sample mean is a point estimator of the population mean, μ, i.e., X ˆ X μ = Interval Estimation: Confidence interval for the mean of a population, μ Since we are interested in the population mean, μ, its estimator is the sample mean, X , and the standard error of the sample mean is ( ) / X X SE X n σ σ = = . We want to construct a confidence interval with level of confidence = 1 – α. The formula for the margin of error (ME) depends on what is known about the population: IF the population standard deviation, σ is KNOWN and the population has a normal distribution or n ≥ 30, THEN /2 / X ME z n α σ = × . In this formula z α/2 is the value of z from the tables of the standard normal distribution that gives an upper tail probability of α/2, i.e., P(Z ≥ z α/2 ) = α/2. IF the population standard deviation is UNKNOWN and the population has a normal distribution THEN ( 1, /2) / n ME t S n α - = × . In this formula t α/2 is the value of t from the tables of the t-distribution with df = n – 1, that gives an upper tail probability of α/2, i.e., P(T (n-1) ≥ t (n-1, α/2) ) = α/2.
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