Download Statistical Inference: Confidence Intervals for Population Parameters and more Study notes Statistics in PDF only on Docsity! Chapter 7 Statistical Inference: Confidence Intervals 7.1 What are Point and Interval Estimates of Population Parameters? A. Point Estimate of a Population Parameter • Definition: A single number that is our “best guess” for the unknown population parameter • Example: When sampling from a categorical (S/F) population, the sample proportion of S’s is a point estimate of the population proportion p • Example: When sampling from a quantitative population, the sample mean is a point estimate of a population mean • Properties of Good Point Estimators – The estimator is unbiased: the mean of the sampling distribution is equal to the parame- ter being estimated – The estimator has a small standard error compared to other estimators. 1 2 B. (Confidence) Interval Estimate of a Population Pa- rameter • Definition: A interval of numbers within which the parameter is believed to fall with a certain amount of confidence. • Margin of Error = multiple of standard error of point estimate, used to indicate how accurate point estimate is likely be in estimating parame- ter. • Left Endpoint = Point estimate - margin of error Right Endpoint = Point estimate + margin of error • Confidence Level of Interval Estimate = proba- bility that method produces an interval that con- tains parameter. Usually chosen close to 1, such as 0.95 or 0.90. 7.2 How Can We Construct a Confidence Inter- val to Estimate a Population Proportion? A. Finding the 95% Confidence Interval for a Popula- tion Proportion • Population proportion is symbolized by p • Sample proportion is symbolized by p̂ p̂ is point estimate of p 5 • Let n be the sample size • Let x be the sample mean • Let s be the sample standard deviation • The ratio x−µs/√n has a probability distribution called the t distribution • Properties of t distributions: page 336 2. Finding percentiles from a t distribution, Table B 3 Example. A study of the ability of individuals to walk in a straigth line. An article reported the fol- lowing data on cadence (strides per second) for a sample of n = 20 randomly selected men: 0.95 0.85 0.92 0.95 0.93 0.86 1.00 0.92 0.85 0.81 0.78 0.93 0.93 1.05 0.93 1.06 1.06 0.96 0.81 0.96 Calculate a 99% confidence interval for the popula- tion mean cadence. From SPSS, x = 0.926, s = 0.0809 Stem and Leaf display of data shows approximate bell shape From Table B, 0.995 percentile from t distribution with df = 20 − 1 = 19 is 2.861 Endpoints of confidence interval: x ± 2.861 s√ 20 After calcuation, endpoints are: 0.926 ± 0.052 99% confidence interval for population mean cadence is (0.874, 0.978 6 4 General Form of Confidence Interval for Population Mean µ Endpoints are: x ± t s√n t is appropriate percentile from t distribution (Table B) Interval is valid: (1) population is normal, n any size or (2) non-normal populations, n large (at least 30)
