Lect18_L6_L7_handout-1

Lect18_L6_L7_handout-1 - lecture 18, Confidence Intervals...

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Unformatted text preview: lecture 18, Confidence Intervals II Review t-Based Confidence Intervals for a Population Mean: σ Unknown t Distribution t-Based Confidence Intervals Sample Size Determina- tion lecture 18, Confidence Intervals II Outline 1 Review 2 t-Based Confidence Intervals for a Population Mean: σ Unknown t Distribution t-Based Confidence Intervals 3 Sample Size Determination lecture 18, Confidence Intervals II Review t-Based Confidence Intervals for a Population Mean: σ Unknown t Distribution t-Based Confidence Intervals Sample Size Determina- tion lecture 18, Confidence Intervals II Review z-Based Confidence Intervals for a Population Mean: σ Known • If a population has standard deviation σ (known) • If the population is normal or if sample size is large ( n ≥ 30 ) , then • A ( 1- α ) 100 % confidence interval for the population mean μ is ¯ x ± z α/ 2 σ √ n • Built based on the sampling distribution of sample mean, Z = ¯ X- μ σ/ √ n is a standard normal lecture 18, Confidence Intervals II Review t-Based Confidence Intervals for a Population Mean: σ Unknown t Distribution t-Based Confidence Intervals Sample Size Determina- tion lecture 18, Confidence Intervals II Review General Formula Confidence interval is Point Estimate ± (Critical Value)(Standard Error) lecture 18, Confidence Intervals II Review t-Based Confidence Intervals for a Population Mean: σ Unknown t Distribution t-Based Confidence Intervals Sample Size Determina- tion lecture 18, Confidence Intervals II t-Based Confidence Intervals for a Population Mean: σ Unknown t-Based Confidence Intervals for a Mean: σ Unknown Usually, σ is unknown. How to build confidence intervals? σ is estimated by S . Define t = ¯ X- μ S / √ n , then (a) If the population is normal, then t has a t distribution with degrees of freedom n- 1 (b) If the population is not normal but the sample size n is large, then t has approximately a t distribution with degrees of freedom n- 1 lecture 18, Confidence Intervals II Review t-Based Confidence Intervals for a Population Mean: σ Unknown t Distribution t-Based Confidence Intervals Sample Size Determina- tion lecture 18, Confidence Intervals II t-Based Confidence Intervals for a Population Mean: σ Unknown t Distribution The t Distribution The curve of the t distribution is similar to that of the standard normal curve: • Symmetric around 0, bell-shaped • More spread out than the standard normal distribution • The spread of the t is given by the degrees of freedom • Denoted by df • The sampling distribution of t has a df that equals the sample size minus 1. That is, when the sample size is n , df = n- 1 . lecture 18, Confidence Intervals II Review t-Based Confidence Intervals for a Population Mean: σ Unknown t Distribution t-Based Confidence Intervals Sample Size Determina- tion lecture 18, Confidence Intervals II t-Based Confidence Intervals for a Population Mean: σ Unknown t Distribution The t Distribution and Degrees of Freedom As the degrees of freedom increases,...
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Lect18_L6_L7_handout-1 - lecture 18, Confidence Intervals...

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