STA3032 Chapter 5 and 6 Inference for one population

# STA3032 Chapter 5 and 6 Inference for one population -...

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Unformatted text preview: Revised on 6/7/2011 18:04 a6/p6 Statistical Inferences about one population (Sections 5.1, 5.2, 5.3, 6.1, 6.2, 6.3, 6.4) Methods of Statistical Inference: • Point Estimation • Interval Estimation • Significance Tests (Chapter 6) Point Estimation A point estimator of a parameter is the sample statistic that predicts the value of the parameter. • A point estimator of the population mean is the sample mean: X ˆ X μ = • A point estimator of the population variance is the sample variance: 2 2 X X ˆ S σ = • A point estimator of the population proportion is the sample proportion ˆ p π = Desirable properties of Point Estimators: • Efficiency • Unbiasedness • Normality 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). STA3032 Inf. on One Pop., Page 1 of 15 5.1 And 5.3 Confidence interval for μ 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. Since we are interested in the population mean, μ, its estimator is the mean of a random sample, , X , and hence a confidence interval for the population mean, μ, is ( X ± ME). Since the standard error of the sample mean is ( ) X X SE X n σ σ = = , we have, /2 X ME z n α σ = if the population standard deviation, σ is KNOWN. In the above 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. When σ is unknown , we replace it with its point estimate, S, to obtain an estimate of the standard error of the sample mean, ( ) . S se X n = Then, we will use the t-distribution and write, ( , /2) df S ME t n α = if the population standard deviation is UNKNOWN In this formula, t (df, α/2) is the value of t from the tables of the t-distribution with df = degrees of freedom = n – 1, that gives an upper tail probability of α/2, i.e., P(T (n-1) ≥ t (n-1, α/2) ) = α/2. Finally, (1 – α)×100% is the confidence level. STA3032 Inf. on One Pop., Page 2 of 15 One-sided Confidence intervals Note that the above confidence interval is bound on both ends. In some problems we may be interested in the upper or the lower end of the interval only. In such cases we put all of α in that tail. Hence, 100(1 – α) % lower-bound CI for μ: , X X z n α σ & & &- 100(1 – α) % upper-bound CI for μ: , X X z n α σ...
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STA3032 Chapter 5 and 6 Inference for one population -...

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