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8.1 Confidence Intervals: The Basics Estimation: The assignment of values to a population parameter based on a value of the corresponding sample statistic. 1. Estimator or Point Estimator: A sample statistic used to estimate a population parameter. Parameter: ?? We don't know what this is a) Maybe it's too expensive to find, or impossible to collect. b) So we take a sample and find a statistic! Ex: If μ = population mean then the Estimator is x(bar) the sample mean If we collect data and find x(bar) = 5 then x(bar) = 5 becomes an Estimate or Point Estimate. Steps for Estimation 1. Select a Sample 2. Collect the required measurement from the sample (height, weight, etc. ) 3. Calculate the value of the sample statistic 4. Assign value or values to the corresponding population parameter. These values are Unbiased but not consistent xxx xxx xxx xxx μ These values are unbiased (on target) and consistent (together) they are estimating what we want to estimate! These values are Consistent but Biased (off target) 1 Definition: Point Estimator and Point Estimate A point estimator is a statistic that provides an estimate of a population parameter. The value of that statistic from a sample is called a point estimate. Ideally, a point estimate is our "best guess" at the value of an unknown parameter. Example: x(bar) = 5 s (sample standard deviation)= 1.4 phat (sample proportion) = .75 Take a statistic and set it equal to one value. That is a point estimate! 2 Example: P470 From Batteries to Smoking 3 THE IDEA OF A CONFIDENCE INTERVAL Example: Mystery Mean p471 4 Definition: Confidence Interval, Margin of Error, Confidence Level A confidence interval for a parameter has two parts: 1. An interval calculated from the data, which has the form estimate ± margin of error The margin of error tells how close the estimate tends to be to the unknown parameter in repeated random sampling. 2. A confidence level C, which gives the overall success rate of the method for calculating the confidence interval. That is, in C% of all possible samples, the method would yield an interval that captures the true parameter value. There are several ways to write a confindence interval: 240.79 ±10 or 230.79≤μ≤250.79 or (230.79, 250.79) Other Definitions: Interval Estimate: An interval constructed around the point estimate. It is stated that this interval is likely to contain the population parameter. Margin of Error The # added to and subtracted from the point estimate. Point Estimate ± Margin of Error Confidence Interval How much confidence we have that this interval contains the true population parameter. We usually choose a confindence interval of above 90% b/c we want to sure of our conclusions. 95% is the most common confindence interval. 5 Example: 95% Confidence Interval There is a Probability of .95 that a statistic will differ from the parameter by less than the margin of error. a) Wrong Interpretation: The parameter has a 95% chance of falling inside the confidence interval. b) Right Interpretation: The methods used to produce the confidence intervals, in the long run, will contain the parameter in 95% of the random samples collected. c) We Say: "With 95% confidence, the parameter falls inside the lower and upper confidence limits." This is the most common confidence interval! 6 Activity: p473 The Confidence Interval Applet 7 Interpreting Confidence Levels and Confidence Intervals Confidence Levels: To say that we are 95% confident is shorthand for "95% of all possible samples of a given size from this population will result in an interval that captures the unknown parameter. Confidence Interval: to interpret a C% confidence interval for an unknown parameter say, "We are C% confident that the interval from _____ to _____captures the actual value of the [population parameter in context]. Plausible does not mean the same thing as possible!! Some would argue that just about any value of a parameter is possible. A plausible value of a parameter is a reasonable or believable value based on the data. Caution: The confidence level does not tell us the chance that a particular confidence interval captures the population parameter. Instead, the confidence interval gives us a set of plausible values for the parameter. 8 Example: the Mystery Mean p475 9 Think About It! p475 What's the probability that our 95% confidence interval captures the parameter? 10 Example: Do You Use Twitter? p476 AP Exam Tip: On a given problem, you may be asked to interpret the confidence interval, the confidence level, or both. Be sure you understand the difference: the confidence level describes the longrun capture rate of the method, and the confidence interval gives a set of plausible values for the parameter. 11 Check Your Understanding p476 Activity: The confidence interval applet p477 12 CONSTRUCTING A CONFIDENCE INTERVAL Book Defn: Calculating a Confidence Interval The confidence interval for estimating a population parameter has the form margin of error statistic ± (critical value) x (standard deviation of statistic) where the statistic we use is the point estimator for the parameter. margin of error Black Book Defn: LOGIC OF A CONFIDENCE INTERVAL 1. Point Estimate ±Confidence Level x Standard Error (standard deviation of statistic) Confidence Interval for μ standard error x(bar) ± (Confidence) x (σ/√n) margin of error 1. As a Confidence Level Increases, the margin of error increases result: A wider Confidence Interval 2. As Sample size Increases, the Margin of Error will decrease result: A smaller Confidence Interval b/c √n is increasing Read p478 Recall that σx = σ/√n and σphat = √p(1p)/n So as the sample size n increases, the standard deviation of the statistic decreases. 13 USING CONFIDENCE INTERVALS WISELY 3 IMPORTANT CONDITIONS FOR CONSTRUCTING CONFIDENCE INTERVALS 1. Random: The data come from a welldesigned random sample or randomized experiment. 2. Normal: The sampling distribution of the statistic is approximately Normal. 3. Independent: Individual observations are independent. Wehn sampling without replacement, the sample size n should be no more than 10% of the population size N (the 10% condition) to use the formula for the standard deviation of the statistic. Two Important Reminders: 1. Our method of calculation assumes that the data come from an SRS of size n from the population of interest. 2. The margin of error in a confidence interval covers only chance variation due to random sampling or random assignment. 14 Homework: p481 513odd p481 and 496 17, 1924, 27, 31, 33 15