What are Confidence Intervals?The BasicsCI's: InterpretationCI's for PercentagesProcedureExamples and LimitationsCI's for Sample Average (Ch. 23)OverviewExampleSummaryConfidenceIntervalsWhat areConfidenceIntervals?The BasicsCI’s:InterpretationCI’s forPercentagesProcedureExamples andLimitationsCI’s forSampleAverage (Ch.23)OverviewExampleSummaryParts VI-VIISampling and EstimationLecture 19Confidence Intervals (Chapters 21, 23)ConfidenceIntervalsWhat areConfidenceIntervals?The BasicsCI’s:InterpretationCI’s forPercentagesProcedureExamples andLimitationsCI’s forSampleAverage (Ch.23)OverviewExampleSummaryMotivationLast lecture I mentioned that estimates are really r.v.’s. Why?Because they are just calculations made from the data (e.g.,average). Since the data has randomness in it, the randomnesscarries through with the calculations.In general, making calculations from the data does noteliminate the randomness.ConfidenceIntervalsWhat areConfidenceIntervals?The BasicsCI’s:InterpretationCI’s forPercentagesProcedureExamples andLimitationsCI’s forSampleAverage (Ch.23)OverviewExampleSummaryMotivationImportant Note: The randomness is in the connectionbetween the data and the population from which it wasdrawn. There is no randomness in the numbersthemselves. The sample average is exactly the sampleaverage; no uncertainty here! It is the connection to thepopulation average that is random and uncertain.But how random is an estimate? In particular, the pointestimate? And why do we need to know this?ConfidenceIntervalsWhat areConfidenceIntervals?The BasicsCI’s:InterpretationCI’s forPercentagesProcedureExamples andLimitationsCI’s forSampleAverage (Ch.23)OverviewExampleSummaryMotivationHere in America we almost always usesports terms to explain the day-to-daymeaning of things (even the scatter-plot was “football-shaped”!)For Confidence Intervals (CI’s), themost relevant such term is ball-park.The point estimate is “in the ball-park” of the true value. But how bigis this ball-park?In statistical terms, how precise is our point estimate?ConfidenceIntervalsWhat areConfidenceIntervals?The BasicsCI’s:InterpretationCI’s forPercentagesProcedureExamples andLimitationsCI’s forSampleAverage (Ch.23)OverviewExampleSummaryDefinitionThe Standard Error (SE) gave a partial answer to this question.The CI is the more practical and commonly-used answer.Confidence Interval (CI): DefinitionA “P”% Confidence Interval (CI) is an interval built aroundthe point estimate of some quantity, in such a way that itcontains the true value of that quantity with probability “P”%. The number “P” is known as the confidence level.ConfidenceIntervalsWhat areConfidenceIntervals?The BasicsCI’s:InterpretationCI’s forPercentagesProcedureExamples andLimitationsCI’s forSampleAverage (Ch.23)OverviewExampleSummaryExample 3 from Last Lecture(continued)In Example 3, 53% of the voters in the sample were in favor ofthe candidate. The SE for the percentage was estimated as 1%.Since the sample was large enough (2500 voters) we can applythe CLT and assume that the point estimate has approximatelynormal behavior. Meaning that a chance error of more than 2SEs has only 5% probability.So let’s go 2 SEs in each direction: (51%, 55%) is a 95% CI forthe population percentage.One legitimate interpretation: we are about 95% confident thatthis interval captures the percentage of voters in the populationwho favor the candidate.ConfidenceIntervalsWhat areConfidenceIntervals?The BasicsCI’s:InterpretationCI’s forPercentagesProcedureExamples andLimitationsCI’s forSampleAverage (Ch.23)OverviewExampleSummaryCI Trick Question # 1Which of the following is completely wrong?(a) We are 95% confident that the interval captures thesample percentage(b) We are 95% confident that the interval captures thepopulation percentageConfidenceIntervalsWhat areConfidenceIntervals?The BasicsCI’s:InterpretationCI’s forPercentagesProcedureExamples andLimitationsCI’s forSampleAverage (Ch.23)OverviewExampleSummaryConfidence LevelsWe can make confidence intervals with any confidence level.Some common levels when using the normal approximation are•90% CI: Point estimate ± 1.65 SEs•95% CI: Point estimate ± 2.0 SEs•99% CI: Point estimate ± 2.6 SEs•What confidence level to choose? This is trade-off between“being safe” and “being meaningful”.ConfidenceIntervalsWhat areConfidenceIntervals?The BasicsCI’s:InterpretationCI’s forPercentagesProcedureExamples andLimitationsCI’s forSampleAverage (Ch.23)OverviewExampleSummaryThe Interpretation ProblemWhat does it mean that we are about “95% confident” that theinterval captures the population parameter?Warning: this could be a treacherous subject. There’sdisagreement even among statisticians here.What is controversial? The role of probability. We definitelyused probability to calculate the CI. But why do we call this“confidence” and not probability?ConfidenceIntervalsWhat areConfidenceIntervals?The BasicsCI’s:InterpretationCI’s forPercentagesProcedureExamples andLimitationsCI’s forSampleAverage (Ch.23)OverviewExampleSummaryInterpretation: ExampleHere’s a simple run-down. Suppose that 100 pollingorganizations surveyed the same population with the samequestion on the same day, using proper probability methods.Then, each of them calculated a 90% CI for their estimate.Due to sampling variability, each of them would get a differentCI for the population percentage. But the true percentage isone and the same for all of them! It is a fixed number (at leastfor this example).First lesson: the randomness is mostly in the intervalboundaries, not in the number we are trying to estimate.ConfidenceIntervalsWhat areConfidenceIntervals?The BasicsCI’s:InterpretationCI’s forPercentagesProcedureExamples andLimitationsCI’s forSampleAverage (Ch.23)OverviewExampleSummaryThe Connection between the CI’sand RandomnessHow is the CI’s randomness related to confidence, and to thetrue value?The CI as a Coin TossCalculating a confidence interval is like tossing a coin. “Heads”means we caught the true value inside the interval; “Tails” wemissed. The coin is not fair: the “Heads” probability is equal tothe confidence level.In reality, we (almost) never see the outcome of this “toss”.After computing a confidence interval, we don’t know whetherit contains the true parameter (if we did know the answer, theconfidence level would immediately change