Estimation with confidence Intervals - Lecture Slides | STAT 515, Study notes of Data Analysis & Statistical Methods

Download Estimation with confidence Intervals - Lecture Slides | STAT 515 and more Study notes Data Analysis & Statistical Methods in PDF only on Docsity! 10/8/2007 1 Chapter 7 Inferences Based on a Single Sample: Estimation with Confidence Intervals Identifying the Target Parameter Target parameter – the unknown population parameter of interest for estimating P t K W d Ph T f D t 2 arame er ey or s or rases ype o a a Mean; average Quantitative p Proportion; percentage; fraction; rate Qualitative μ Large-Sample Confidence Interval for a Population Mean How to estimate the population mean and assess the estimate’s reliability? is an estimate of , and we use CLT to μx 3 assess how accurate that estimate is According to CLT, 95% of all from sample size n lie within of the mean We can use this to assess accuracy of as an estimate of x xσ96.1± μ x Large-Sample Confidence Interval for a Population Mean 22 .95xx x n σσ± = ± ≈ 4 We are about 95% confident, for any from sample size n, that will lie in the interval 2x n σ ±μ x We usually don’t know , but with a large sample s is a good estimator of . We can calculate confidence intervals for different confidence coefficients. Large-Sample Confidence Interval for a Population Mean σ σ 5 Interval Estimator (or confidence interval) – a formula used to calculate an interval estimate from sample data Confidence coefficient – probability that a randomly selected confidence interval encloses the population parameter Confidence level – Confidence coefficient expressed as a percentage Large-Sample Confidence Interval for a Population Mean The confidence coefficient is equal to 1- , and is split between the two tails of the distribution α 6 10/8/2007 2 Large-Sample Confidence Interval for a Population Mean The Confidence Interval is expressed more generally as zxzx σσ αα 22 ±=± 7 For samples of size > 30, the confidence interval is expressed as Requires that the sample used be random nx ⎟ ⎠ ⎞ ⎜ ⎝ ⎛± n szx 2α Large-Sample Confidence Interval for a Population Mean 8 Small-Sample Confidence Interval for a Population Mean 2 problems presented by sample sizes of less than 30 –CLT no longer applies 9 –Population standard deviation is almost always unknown, and s may provide a poor estimation when n is small Small-Sample Confidence Interval for a Population Mean If we can assume that the sampled population is approximately normal, then the sampling distribution of can be assumed t b i t l l x 10 o e approx ma e y norma Instead of using we use This t is referred to as the t-statistic n xz σ μ− = ns xt μ−= Small-Sample Confidence Interval for a Population Mean The t-statistic has: a sampling distribution very similar to z 11 Variability dependent on n, or sample size. Variability is expressed as (n-1) degrees of freedom (df). As (df) gets smaller, variability increases Small-Sample Confidence Interval for a Population Mean •Table for t- distribution contains t-value for various bi ti f 12 com na ons o degrees of freedom and tα •Partial table here shows components of table