1STA 291 - Lecture 17 1STA 291Lecture 17• Chap. 10 Estimation– Estimating the Population Proportion p– We are not predicting the next outcome (which is random), but is estimating a fixed number --- the population parameter.STA 291 - Lecture 17 2Review: Population Distribution, and Sampling Distribution• Population Distribution– Unknown, distribution from which we select the sample– Want to make inference about its parameter , like p• Sampling Distribution– Probability distribution of astatistic (for example, sample mean/proportion)– Used to determine the probability that a statistic falls within a certain distance of the population parameter– For large n, the sampling distribution of the samplemean/proportion looks more and more like a normal distributionSTA 291 - Lecture 17 3Chapter 10• Estimation: Confidence interval– Inferential statistical methods provide estimates about characteristics of a population, based on information in a sample drawn from that population– For quantitative variables, we usually estimate the population mean (for example, mean household income) + (SD)– For qualitative variables, we usually estimate population proportions (for example, proportion of people voting for candidate A)2STA 291 - Lecture 17 4Two Types of Estimators• Point Estimate– A single number that is the best guess for the (unknown) parameter– For example, the sample proportion/mean is usually a good guess for the population proportion/mean• Interval Estimate– A range of numbers around the point estimate– To give an idea about the precision of the estimator – For example, “the proportion of people voting for A is between 67% and 73%”STA 291 - Lecture 17 5Point Estimator• A point estimator of a parameter is a sample statistic that estimates the value of that parameter• A good estimator is – Unbiased: Centered around the true parameter – Consistent: Gets closer to the true parameter as the sample size n gets larger– Efficient: Has a standard error that is as small as possibleSTA 291 - Lecture 17 6• Sample proportion, , is unbiased as an estimator of the population proportion p.It is also consistent and efficient.ˆp3STA 291 - Lecture 17 7New: Confidence Interval• An inferential statement about a parameter should always provide the accuracy of the estimate (error bound)• How close is the estimate likely to fall to the true parameter value?• Within 1 unit? 2 units? 10 units?• This can be determined using the sampling distribution of the estimator/sample statistic• In particular, we need the standard error to make a statement about accuracy of the estimator• How close?• How likely?STA 291 - Lecture 17 8STA 291 - Lecture 17 9New: Confidence Interval• Example: interview 1023 persons, selected by SRS from the entire USA population.• Out of the 1023 only 153 say “YES” to the question “economic condition in US is getting better”• Sample size n = 1023, = 153/1023=0.15ˆp4• The sampling distribution of is (very close to) normal, since we used SRS in selection of people to interview, and 1023 is large enough.• The sampling distribution has mean = p , and • SD = STA 291 - Lecture 17 10ˆp(1)0.15(10.15)0.0111023ppn−−==• The 95% confidence interval for the unknown p is• [ 0.15 – 2x0.011, 0.15 + 2x0.011]• Or [ 0.128, 0.172]• Or “15% with 95% margin of error 2.2%”STA 291 - Lecture 17 11STA 291 - Lecture 17 12Confidence Interval• A confidence interval for a parameter is a range of numbers that is likely to cover (or capture) the true parameter.• The probability that the confidence interval captures the true parameter is called the confidence coefficient/confidence level.• The confidence level is a chosen number close to 1, usually 95%, 90% or 99%5STA 291 - Lecture 17 13Confidence Interval• To calculate the confidence interval, we used the Central Limit Theorem• Therefore, we need sample sizes of at least moderately large, usually we require both np > 10 and n(1-p) > 10• Also, we need a that is determined by the confidence level• Let’s choose confidence level 0.95, then =1.96 (the refined version of “2”)/2zα/2zαSTA 291 - Lecture 17 14• confidence level 0.90, çè =1.645• confidence level 0.95 çè =1.96• confidence level 0.99 çè =2.575 /2zα/2zα/2zαSTA 291 - Lecture 17 15Confidence Interval• So, the random interval betweenWill capture the population proportion p with 95% probabilityµµ µµµ µ(1)(1)1.96 and 1.96ppppppnn−−−+• This is a confidence statement, and the interval is called a 95% confidence interval6STA 291 - Lecture 17 16Confidence Interval: Interpretation• “Probability” means that “in the long run, 95% of these intervals would contain the parameter”• If we repeatedly took random samples using the same method, then, in the long run, in 95% of the cases, the confidence interval will cover the true unknown parameter• For one given sample, we do not know whether the confidence interval covers the true parameter or not. • The 95% probability only refers to the method that we use, but not to this individual sampleSTA 291 - Lecture 17 17Confidence Interval: Interpretation• To avoid the misleading word “probability”, we say:“We are 95% confident that the true population p is within this interval”• Wrong statements: • 95% of the p’s are going to be within 12.8% and 17.2%Statements• 15% of all US population thought “YES”.• It is probably true that 15% of US population thought “YES”• We do not know exactly, but we know it is between 12.8% and 17.2%STA 291 - Lecture 17 187• We do not know, but it is probably within 12.8% and 17.2%• We are 95% confident that the true proportion (of the US population thought “YES”) is between 12.8% and 17.2%• You are never 100% sure, but 95% or 99% sureis quite close.STA 291 - Lecture 17 19STA 291 - Lecture 17 20Confidence Interval• If we change the confidence level from 0.95 to 0.99, the confidence interval changes• Increasing the probability that the interval contains the true parameter requires increasing the length of the interval• In order to achieve 100% probability to cover the true parameter, we would have to increase the length of the interval to infinite -- that would not be informative• There is a tradeoff between length of confidence interval and coverage probability. Ideally,