lect12 - Chapter 8 CONFIDENCE INTERVALS 8.1 Introduction:...

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Chapter 8 CONFIDENCE INTERVALS 8.1 Introduction: The one-sample problem for means. The Z interval. 8.2 Confidence interval for a proportion. 8.3 Comparing means of two independent samples. 8.4 Comparing two proportions. For lecture 13: 8.5 The one-sample problem for means with unknown variance and small samples. Student’s t -distribution, the T-interval. 8.6 Two independent samples, small sample sizes, variances unknown. The two-sample t statistic. 8.7 Paired samples. 8.8 Confidence intervals for variances. The chi-squared distribution. 8.9 Comparing two variances. The F -distribution. 1
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8.1 INTRODUCTION: THE ONE-SAMPLE PROBLEM FOR MEANS Ref: Devore 6e, pages 281–289. Consider a random sample X 1 ,X 2 ,...,X n . The { X i } are independent from a distribution F which depends on an unknown parameter, θ say. We wish to give an interval estimate for θ based on the value of some statistic or point estimate ˆ θ for θ . For example, if θ is a mean, we might use ˆ θ = X . If θ is a variance we might consider using the sample variance ˆ θ = S 2 . The confidence interval for θ is then based on the sampling distribution of ˆ θ . Example: One-sample normal mean problem. Suppose X 1 ,X 2 ,...,X n are a random sample from N ( μ,σ 2 ) where μ is unknown and σ is known. We want to get a confidence interval for μ . We use X and note that the sampling distribution of X is N ( μ, σ 2 n ). Hence by the 68, 95, 99.7% rule: P [ - 2 < X - μ σ/ n < 2] 0 . 95 (Actually, 2 should be replaced by 1.96 if we want to be exact).
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lect12 - Chapter 8 CONFIDENCE INTERVALS 8.1 Introduction:...

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