Chapter 11- Confidence Intervals for Univariate Data Math 22 Introductory Statistics Introduction into Estimation Point Estimate – the value of a sample statistic used to estimate the population parameter. Interval Estimate – an interval bounded by two values calculated from the sample data, used to estimate a population parameter. Introduction into Estimation Level of Confidence – The probability that the sample to be selected yields an interval that includes the parameter being estimated. Confidence Interval – An interval estimate with a specified level of confidence. Assumption – a condition that needs to exist in order to properly apply a statistical procedure to be valid. Confidence Interval A confidence interval for a population parameter is an interval of possible values for the unknown parameter. The interval is computed in such a way that we have a high degree of confidence that the interval contains the true value of a parameter. Confidence Level The confidence, stated as a percent, is the confidence level. In practice, estimates of unknown parameters are given in the form: estimate margin of error Developing a Confidence Interval Three determinations must be made to develop a Confidence Interval: A good point estimator of the parameter. The sampling dist. (or approximate sampling dist.) of the point estimator. The desired confidence level, usually stated as a percentage. Standard Error of a Statistic The standard deviation of its sampling dist. when all unknown population parameters have been estimated. Interpreting Confidence Intervals Q: A: What does a 99% C.I. really mean? A 99% C.I. means that of 100 different intervals obtained from 100 different samples, it is likely 99 of those intervals will contain the true parameter and one will not. Validity and Precision of Confidence Levels Validity - Measured by the confidence level, which is the probability that the interval will contain the true value of the parameter. Precision - measured by the length of the interval Confidence Interval for the Population Proportion pˆ (1 pˆ ) pˆ z / 2 n pˆ - sample proportion Reducing the Margin of Error Two ways to reduce the margin of error: Decrease z (Problem - Reduces Validity) Increase n (No Problem) Calculating Sample Size for Proportions Margin of Error (ME) z / 2 2 pˆ (1 pˆ ) n z / 2 ˆ (1 p ˆ) n p ME Estimation of the Mean When the Standard Deviation is Known When the population standard deviation is known, a (1-)100% confidence interval for x based on m is given by the limits: x z / 2 n Estimation of the Mean When the Standard Deviation is Unknown We must make sure that the sampled population is normally distributed. Normal Plots Student-t Distribution Many times we do not know what is . In these cases, we use s as the standard deviation. The standard error of the sample mean is now s n Characteristics of the Studentt Distribution Bell shaped and symmetric, just like the normal distribution is bell shaped and symmetric. The t-distribution “looks” like the normal distribution but is not normal. The t-distribution is a family of distributions, each member being uniquely identified by its degrees of freedom (df) which is simply n1 where n is the sample size. Characteristics of the Studentt Distribution As the sample size increases the tdistribution becomes indistinguishable from the standard normal curve. The t-Interval x t( / 2) s n Using the t-Interval For small sample sizes: If the sample size is less than 30, construct a normal plot of your data. If your data appears to be from a normal distribution, then use the t-distribution. If the data does not appear to be normal, then use a non-parametric technique that will be introduced later. Using the t-Interval For large sample sizes: If the sample size is 30 or more, use the t-distribution citing the Central Limit theorem as justification for having satisfied the required assumption of normality. Sample Size for Inference Concerning the Mean z / 2 n Margin of Error ( ME ) z / 2 n ME 2 z / 2 n ME Confidence Interval for the Median Large Sample Confidence Interval for the Median: Sample size must be 20 or more. We can construct a confidence interval for q based on p. We can then produce a confidence interval for p with a sample proportion of .50 (this is used to represent the definition of the median, 50% below this mark, 50% above this mark.) Large Sample Confidence Interval for the Median Basic steps for conducting a large sample confidence interval for the median: Construct a normal plot to see if the data is normal. If the normal assumption is violated, construct a (1-)100% for p based on a sample proportion of .50. Multiply the upper and lower bound of the C.I. by n, the sample size. Round up the lower bound and round down the upper bound. Large Sample Confidence Interval for the Median Sort the data and identify the data values in those positions identified by the previous step. Small Sample Confidence Interval for the Median Sample size must be less than 20. The method we will explore is based strictly on the binomial distribution. Small Sample Confidence Interval for the Median Basic steps for conducting a small sample confidence interval for the median: Create a table that contains the discrete cumulative probability distribution for 0 to n for a binomial distribution where p = .50. Identify the position for the lower bound with a cumulative probability as near /2 as possible. Small Sample Confidence Interval for the Median Identify the position for the upper bound with a cumulative probability as near 1-/2 as possible. Sort the data and identify the data values corresponding to the position located in the last two steps. Report the actual confidence level by summing the tail probabilities associated with the positions chosen for the C.I. Bounds.