Qualls - 6. Confidence Interval of Population Proportion Objectives Lecture 6 At the end of this section you should be able to answer questions concerning point and interval estimates of a population proportion, and determining the requisite sample size for a given confidence level. Confidence Intervals and Sample Sizes Part 2: Proportions Specifically, you should understand: • the difference between a point estimate and an interval estimate • how to calculate a confidence interval for a population proportion • how to determine the requisite sample size given a desired margin of error and confidence level. DePaul University Bill Qualls 1 2 Point Estimate of a Population Proportion • We used the following example when we introduced the binomial distribution: Assume my free-throw average is 40%. If I throw 3 free-throws, what is the probability that I will miss all three? Hit 1? Hit 2? Hit 3? Confidence Intervals about a Population Proportion • In our problems dealing with binomial probabilities, we have been given p. But what if we would like to estimate p with a known level of confidence? 3 4 Point Estimate of a Population Proportion Point Estimate of a Population Variance • A point estimate is a single value used to • Given that the variance for a binomial approximate a population parameter. distribution is defined as σ²=npq, where q = 1-p, and that p-hat is the best estimate for the population • The best point estimate ("p-hat") of a population parameter p, it follows then that the best estimate proportion (p) is the sample proportion. for the population variance is: successes pˆ = trials σˆ 2 = npˆ qˆ where qˆ = 1 − pˆ • The use of a carat symbol (^) over a letter is read as "hat", and indicates it is an estimated value. 5 Updated 3/15/2014 6 Qualls - 6. Confidence Interval of Population Proportion Interval Estimates The Problem with Point Estimates • If I sink 4 free-throws out of 10, then my point estimate for p is .4. • We can, however, assign a level of confidence to an interval estimate. • Likewise, if I sink 40 free-throws out of 100, then my point estimate for p is still .4. • We would intuitively have more confidence in the second statistic than in the first. • If you were asked to come up with a 95% confidence interval for the first case (4 free-throws out of 10), you might say you were 95% confident that the true proportion is between .3 and .5. • But these are both point estimates, and the problem with a point estimate is that we cannot assign any statistical level of confidence to it. • But in the second case (40 free-throws out of 100), you might say you were 95% confident that the true proportion is between .35 and .45. (Numbers used above are "guesses" only, for illustrative purposes.) 7 CI for Population Proportion 8 90% Confidence Interval • The formula for the confidence interval (CI) for a population proportion is usually shown as: p = pˆ ± zα / 2 pˆ qˆ n • Some texts prefer the notation: p = pˆ ± E where E is the margin of error and is calculated as: E = zα / 2 pˆ qˆ n • These formulas require np ≥ 15 and nq ≥ 15 (or else the distribution is too skewed; not normal.) 9 95% Confidence Interval 10 99% Confidence Interval 11 Updated 3/15/2014 12 Qualls - 6. Confidence Interval of Population Proportion Together Calculating Confidence Intervals • I attempt 100 free throws, and make a basket 40 times. Calculate a 95% confidence interval for my true free throw percentage. • Solution: pˆ qˆ n (.4)(.6) = .4 ± 1.96 100 = .4 ± .096 p = pˆ ± zα / 2 = [.304, .496] 13 14 Interpretation Interpretation So what does it mean? A miss like this will occur 5% of the time. Wrong: We are 95% confident that the true population proportion is between .304 and .496. Correct: If the sampling process were repeated many times, and the interval calculated each time, 95% of those intervals would capture the true population proportion. 15 16 Using the TI-83 Plus Together • Press [STAT] [TESTS] [1-PropZInt] In a survey of 1002 people, 701 said that they voted in a recent presidential election (based on data from ICR Research Group). Voting records show that 61% of eligible voters actually did vote. a. Find a 99% confidence interval estimate of the proportion of people who say that they voted. b. Are the survey results consistent with the actual voter turnout of 61%? Why or why not? • These are always "z", never "t". • Careful! Don't choose 1-PropZTest (yet). (Source: Triola, Page 333, Section 7-2, #34) 17 Updated 3/15/2014 18 Qualls - 6. Confidence Interval of Population Proportion Together Margin of Error Assume that a sample is used to estimate the population proportion p. Find the margin of error E that corresponds to the given statistics and confidence level: n = 1200, x = 800, 99% confidence. Given a confidence interval of [0.25, 0.39]. • What is p-hat? (Answer: 0.32) • What is the margin of error? (Answer: 0.07) E E 0.25 0.39 • What is the margin of error for the previous problem? (Source: Triola, Page 333, Section 7-2, #18) 20 19 Together Find the margin of error: Determining the Proper Sample Size 21 22 Sample Size Together • How large does sample need to be to get an estimate of p, with an acceptable margin of error? • My earlier attempts indicate that my free throw percentage is around 40%. But I would like a more narrow confidence interval than the ±9.6% I got with n=100. How many free throws should I attempt in order to get a 95% confidence interval with a 3% margin of error? E = zα / 2 pˆ qˆ [z ]2 pˆ qˆ → solve for n → n = α / 2 2 n E • In the above formula, E might be, for example, .03 for a 3% margin of error. • If no prior estimate of p is known then use .5 as .5 will always give you the maximum sample size. 23 Updated 3/15/2014 24 Qualls - 6. Confidence Interval of Population Proportion What about Population Size? Together "Many people incorrectly believe that the sample size should be some percentage of the population, but (the above formula) shows that the population size is irrelevant. (In reality, the population size is sometimes used, but only in cases in which we sample without replacement from a relatively small population.) Polls commonly use sample sizes in the range of 1000 to 2000 and, even though such polls may involve a very small percentage of the total population, they can provide results that are quite good." (Triola, page 330) Use the given data to find the minimum sample size required to estimate a population proportion or percentage. Margin of error: four percentage points; confidence level: 95%; no prior estimate of p-hat is available. 25 Together 26 Effect of Sample Size on C.I. Width Toyota provides an option of a sunroof and side air bag package for its Corolla model. This package costs $1400 ($1159 invoice price). Assume that prior to offering this option package, Toyota wants to determine the percentage of Corolla buyers who would pay $1400 extra for the sunroof and side air bags. How many Corolla buyers must be surveyed if we want to be 95% confident that the sample percentage is within four percentage points of the true percentage for all Corolla buyers? (Source: Triola, Page 333, Section 7-2, #44) Do parts of this problem sound familiar ? ? ? 27 Gut check Estimate of margin of error given sample size: E≈ 1 n Estimate of sample size for given margin of error: n≈ 1 E2 29 Updated 3/15/2014 28