Download Probability Framework, Confidence Intervals - Lecture Notes | BUSSTAT 207 and more Study notes Introduction to Business Management in PDF only on Docsity! 10/24/2010 1 Probability framework Probability Distributions Discrete Continuous Sampling Distributions Confidence Intervals Rules Confidence intervals Confidence Interval Estimate Point Estimate Lower Confidence Limit Upper Confidence Limit Width of confidence interval The general formula for all confidence intervals is: Point Estimate (Critical Value)(Standard Error) Confidence Interval for μ (σ Known) • Assumptions −Population standard deviation σ is known −Population is normally distributed −If population is not normal, use large sample • Confidence interval estimate n σ zx Finding the Critical Value • Consider a 95% confidence interval: 1.96z Common Levels of Confidence • Commonly used confidence levels are 90%, 95%, and 99% Confidence Level Confidence Coefficient, Critical value, z 1.28 1.645 1.96 2.33 2.58 3.08 3.27 .80 .90 .95 .98 .99 .998 .999 80% 90% 95% 98% 99% 99.8% 99.9% 1 10/24/2010 2 Margin of Error • Data variation, σ : e as σ • Sample size, n : e as n • Level of confidence, 1 - : e if 1 - n σ zx n σ ze Example: Margin of error for estimating μ, σ known: • Assumptions −Population standard deviation is unknown −Population is normally distributed − If population is not normal, use large sample • Use Student’s t Distribution Confidence Interval for μ (σ Unknown) n s tx 1-n )x(x s n 1i 2 i Student’s t Distribution • The t is a family of distributions • The t value depends on degrees of freedom (d.f.) − Number of observations that are free to vary after sample mean has been calculated d.f. = n - 1 Determining Sample Size • The required sample size can be found to reach a desired margin of error (e) and level of confidence (1 - ) − Required sample size, σ known: 2 222 e σz e σz n Required Sample Size Example If = 45, what sample size is needed to be 90% confident of being correct within ± 5? (Always round up) 219.19 5 (45)1.645 e σz n 2 22 2 22 So the required sample size is n = 220 Confidence Intervals for the Population Proportion, π • Recall that the distribution of the sample proportion is approximately normal if the sample size is large, with standard deviation • We will estimate this with sample data: n p)p(1 sp n π)π(1 σπ
