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Chapter 7 Statistical Inference: Confidence Intervals Learn …. How to Estimate a Population Parameter Using Sample Data Agresti/Franklin Statistics, 1 of 87 Section 7.1 What Are Point and Interval Estimates of Population Parameters? Agresti/Franklin Statistics, 2 of 87 Point Estimate A point estimate is a single number that is our “best guess” for the parameter Agresti/Franklin Statistics, 3 of 87 Interval Estimate An interval estimate is an interval of numbers within which the parameter value is believed to fall. Agresti/Franklin Statistics, 4 of 87 Point Estimate vs Interval Estimate Agresti/Franklin Statistics, 5 of 87 Point Estimate vs Interval Estimate A point estimate doesn’t tell us how close the estimate is likely to be to the parameter An interval estimate is more useful • It incorporates a margin of error which helps us to gauge the accuracy of the point estimate Agresti/Franklin Statistics, 6 of 87 Point Estimation: How Do We Make a Best Guess for a Population Parameter? Use an appropriate sample statistic: • For the population mean, use the sample • mean For the population proportion, use the sample proportion Agresti/Franklin Statistics, 7 of 87 Point Estimation: How Do We Make a Best Guess for a Population Parameter? Point estimates are the most common form of inference reported by the mass media Agresti/Franklin Statistics, 8 of 87 Properties of Point Estimators Property 1: A good estimator has a sampling distribution that is centered at the parameter • An estimator with this property is unbiased • The sample mean is an unbiased estimator of the population mean • The sample proportion is an unbiased estimator of the population proportion Agresti/Franklin Statistics, 9 of 87 Properties of Point Estimators Property 2: A good estimator has a small standard error compared to other estimators • This means it tends to fall closer than other estimates to the parameter Agresti/Franklin Statistics, 10 of 87 Interval Estimation: Constructing an Interval that Contains the Parameter (We Hope!) Inference about a parameter should provide not only a point estimate but should also indicate its likely precision Agresti/Franklin Statistics, 11 of 87 Confidence Interval A confidence interval is an interval containing the most believable values for a parameter The probability that this method produces an interval that contains the parameter is called the confidence level • This is a number chosen to be close to 1, most commonly 0.95 Agresti/Franklin Statistics, 12 of 87 What is the Logic Behind Constructing a Confidence Interval? To construct a confidence interval for a population proportion, start with the sampling distribution of a sample proportion Agresti/Franklin Statistics, 13 of 87 The Sampling Distribution of the Sample Proportion Gives the possible values for the sample proportion and their probabilities Is approximately a normal distribution for large random samples Has a mean equal to the population proportion Has a standard deviation called the standard error Agresti/Franklin Statistics, 14 of 87 A 95% Confidence Interval for a Population Proportion Fact: Approximately 95% of a normal distribution falls within 1.96 standard deviations of the mean • That means: With probability 0.95, the sample proportion falls within about 1.96 standard errors of the population proportion Agresti/Franklin Statistics, 15 of 87 Margin of Error The margin of error measures how accurate the point estimate is likely to be in estimating a parameter The distance of 1.96 standard errors in the margin of error for a 95% confidence interval Agresti/Franklin Statistics, 16 of 87 Confidence Interval A confidence interval is constructed by adding and subtracting a margin of error from a given point estimate When the sampling distribution is approximately normal, a 95% confidence interval has margin of error equal to 1.96 standard errors Agresti/Franklin Statistics, 17 of 87 Section 7.2 How Can We Construct a Confidence Interval to Estimate a Population Proportion? Agresti/Franklin Statistics, 18 of 87 Finding the 95% Confidence Interval for a Population Proportion We symbolize a population proportion by p The point estimate of the population proportion is the sample proportion We symbolize the sample proportion by p̂ Agresti/Franklin Statistics, 19 of 87 Finding the 95% Confidence Interval for a Population Proportion A 95% confidence interval uses a margin of error = 1.96(standard errors) [point estimate ± margin of error] = p̂ 1.96(standard errors) Agresti/Franklin Statistics, 20 of 87 Finding the 95% Confidence Interval for a Population Proportion The exact standard error of a sample proportion equals: p (1 p ) n This formula depends on the unknown population proportion, p In practice, we don’t know p, and we need to estimate the standard error Agresti/Franklin Statistics, 21 of 87 Finding the 95% Confidence Interval for a Population Proportion In practice, we use an estimated standard error: se p ˆ (1 p ˆ) n Agresti/Franklin Statistics, 22 of 87 Finding the 95% Confidence Interval for a Population Proportion A 95% confidence interval for a population proportion p is: p̂ 1.96(se), with se p̂(1 - p̂) n Agresti/Franklin Statistics, 23 of 87 Example: Would You Pay Higher Prices to Protect the Environment? In 2000, the GSS asked: “Are you willing to pay much higher prices in order to protect the environment?” • Of n = 1154 respondents, 518 were willing to do so Agresti/Franklin Statistics, 24 of 87 Example: Would You Pay Higher Prices to Protect the Environment? Find and interpret a 95% confidence interval for the population proportion of adult Americans willing to do so at the time of the survey Agresti/Franklin Statistics, 25 of 87 Example: Would You Pay Higher Prices to Protect the Environment? 518 p̂ 0.45 1154 (0.45)(0.55) se 0.015 1154 p̂ 1.96(se) 1.96(0.015) 0.45 0.03 (0.42, 0.48) Agresti/Franklin Statistics, 26 of 87 Sample Size Needed for Large-Sample Confidence Interval for a Proportion For the 95% confidence interval for a proportion p to be valid, you should have at least 15 successes and 15 failures: np ˆ 15 and n(1- p̂) 15 Agresti/Franklin Statistics, 27 of 87 “95% Confidence” With probability 0.95, a sample proportion value occurs such that the confidence interval contains the population proportion, p With probability 0.05, the method produces a confidence interval that misses p Agresti/Franklin Statistics, 28 of 87 How Can We Use Confidence Levels Other than 95%? In practice, the confidence level 0.95 is the most common choice But, some applications require greater confidence To increase the chance of a correct inference, we use a larger confidence level, such as 0.99 Agresti/Franklin Statistics, 29 of 87 A 99% Confidence Interval for p pˆ 2.58(se) Agresti/Franklin Statistics, 30 of 87 Different Confidence Levels Agresti/Franklin Statistics, 31 of 87 Different Confidence Levels In using confidence intervals, we must compromise between the desired margin of error and the desired confidence of a correct inference • As the desired confidence level increases, the margin of error gets larger Agresti/Franklin Statistics, 32 of 87 What is the Error Probability for the Confidence Interval Method? The general formula for the confidence interval for a population proportion is: Sample proportion ± (z-score)(std. error) which in symbols is pˆ z(se) Agresti/Franklin Statistics, 33 of 87 What is the Error Probability for the Confidence Interval Method? Agresti/Franklin Statistics, 34 of 87 Summary: Confidence Interval for a Population Proportion, p A confidence interval for a population proportion p is: p̂ z p̂(1 - p̂) n Agresti/Franklin Statistics, 35 of 87 Summary: Effects of Confidence Level and Sample Size on Margin of Error The margin of error for a confidence interval: • Increases as the confidence level increases • Decreases as the sample size increases Agresti/Franklin Statistics, 36 of 87 What Does It Mean to Say that We Have “95% Confidence”? If we used the 95% confidence interval method to estimate many population proportions, then in the long run about 95% of those intervals would give correct results, containing the population proportion Agresti/Franklin Statistics, 37 of 87 A recent survey asked: “During the last year, did anyone take something from you by force?” a. b. c. Of 987 subjects, 17 answered “yes” Find the point estimate of the proportion of the population who were victims .17 .017 .0017 Agresti/Franklin Statistics, 38 of 87 Section 7.3 How Can We Construct a Confidence Interval To Estimate a Population Mean? Agresti/Franklin Statistics, 39 of 87 How to Construct a Confidence Interval for a Population Mean Point estimate ± margin of error The sample mean is the point estimate of the population mean The exact standard error of the sample mean is σ/ n In practice, we estimate σ by the sample standard deviation, s Agresti/Franklin Statistics, 40 of 87 How to Construct a Confidence Interval for a Population Mean For large n… • and also For small n from an underlying population that is normal… The confidence interval for the population mean is: x z( n ) Agresti/Franklin Statistics, 41 of 87 How to Construct a Confidence Interval for a Population Mean In practice, we don’t know the population standard deviation Substituting the sample standard deviation s for σ to get se = s/ n introduces extra error To account for this increased error, we replace the z-score by a slightly larger score, the t-score Agresti/Franklin Statistics, 42 of 87 How to Construct a Confidence Interval for a Population Mean In practice, we estimate the standard error of the sample mean by se = s/ n Then, we multiply se by a t-score from the t-distribution to get the margin of error for a confidence interval for the population mean Agresti/Franklin Statistics, 43 of 87 Properties of the t-distribution The t-distribution is bell shaped and symmetric about 0 The probabilities depend on the degrees of freedom, df The t-distribution has thicker tails and is more spread out than the standard normal distribution Agresti/Franklin Statistics, 44 of 87 t-Distribution Agresti/Franklin Statistics, 45 of 87 Summary: 95% Confidence Interval for a Population Mean A 95% confidence interval for the population mean µ is: s x t ( ); df n - 1 n .025 To use this method, you need: • • Data obtained by randomization An approximately normal population distribution Agresti/Franklin Statistics, 46 of 87 Example: eBay Auctions of Palm Handheld Computers Do you tend to get a higher, or a lower, price if you give bidders the “buy-it-now” option? Agresti/Franklin Statistics, 47 of 87 Example: eBay Auctions of Palm Handheld Computers Consider some data from sales of the Palm M515 PDA (personal digital assistant) During the first week of May 2003, 25 of these handheld computers were auctioned off, 7 of which had the “buy-it-now” option Agresti/Franklin Statistics, 48 of 87 Example: eBay Auctions of Palm Handheld Computers “Buy-it-now” option: 235 225 225 240 250 250 210 Bidding only: 250 249 255 200 199 240 228 255 232 246 210 178 246 240 245 225 246 225 Agresti/Franklin Statistics, 49 of 87 Example: eBay Auctions of Palm Handheld Computers Summary of selling prices for the two types of auctions: buy_now N Mean StDev no 18 231.61 21.94 yes 7 233.57 14.64 buy_now Maximum no 255.00 yes 250.00 Minimum Q1 Median Q3 178.00 221.25 240.00 246.75 210.00 225.00 235.00 250.00 Agresti/Franklin Statistics, 50 of 87 Example: eBay Auctions of Palm Handheld Computers Agresti/Franklin Statistics, 51 of 87 Example: eBay Auctions of Palm Handheld Computers To construct a confidence interval using the t-distribution, we must assume a random sample from an approximately normal population of selling prices Agresti/Franklin Statistics, 52 of 87 Example: eBay Auctions of Palm Handheld Computers Let µ denote the population mean for the “buy-it-now” option The estimate of µ is the sample mean: x = $233.57 The sample standard deviation is: s = $14.64 Agresti/Franklin Statistics, 53 of 87 Example: eBay Auctions of Palm Handheld Computers The 95% confidence interval for the “buy-itnow” option is: s 14.64 x t.025 ( ) 233.57 2.44( ) n 7 which is 233.57 ± 13.54 or (220.03, 247.11) Agresti/Franklin Statistics, 54 of 87 Example: eBay Auctions of Palm Handheld Computers The 95% confidence interval for the mean sales price for the bidding only option is: (220.70, 242.52) Agresti/Franklin Statistics, 55 of 87 Example: eBay Auctions of Palm Handheld Computers Notice that the two intervals overlap a great deal: • “Buy-it-now”: (220.03, 247.11) • Bidding only: (220.70, 242.52) There is not enough information for us to conclude that one probability distribution clearly has a higher mean than the other Agresti/Franklin Statistics, 56 of 87 How Do We Find a t- Confidence Interval for Other Confidence Levels? The 95% confidence interval uses t.025 since 95% of the probability falls between - t.025 and t.025 For 99% confidence, the error probability is 0.01 with 0.005 in each tail and the appropriate t-score is t.005 Agresti/Franklin Statistics, 57 of 87 If the Population is Not Normal, is the Method “Robust”? A basic assumption of the confidence interval using the t-distribution is that the population distribution is normal Many variables have distributions that are far from normal Agresti/Franklin Statistics, 58 of 87 If the Population is Not Normal, is the Method “Robust”? How problematic is it if we use the tconfidence interval even if the population distribution is not normal? Agresti/Franklin Statistics, 59 of 87 If the Population is Not Normal, is the Method “Robust”? For large random samples, it’s not problematic The Central Limit Theorem applies: for large n, the sampling distribution is bell-shaped even when the population is not Agresti/Franklin Statistics, 60 of 87 If the Population is Not Normal, is the Method “Robust”? What about a confidence interval using the t-distribution when n is small? Even if the population distribution is not normal, confidence intervals using t-scores usually work quite well We say the t-distribution is a robust method in terms of the normality assumption Agresti/Franklin Statistics, 61 of 87 Cases Where the t- Confidence Interval Does Not Work With binary data With data that contain extreme outliers Agresti/Franklin Statistics, 62 of 87 The Standard Normal Distribution is the t-Distribution with df = ∞ Agresti/Franklin Statistics, 63 of 87 The 2002 GSS asked: “What do you think is the ideal number of children in a family?” a. b. c. d. The 497 females who responded had a median of 2, mean of 3.02, and standard deviation of 1.81. What is the point estimate of the population mean? 497 2 3.02 1.81 Agresti/Franklin Statistics, 64 of 87 Section 7.4 How Do We Choose the Sample Size for a Study? Agresti/Franklin Statistics, 65 of 87 How are the Sample Sizes Determined in Polls? It depends on how much precision is needed as measured by the margin of error The smaller the margin of error, the larger the sample size must be Agresti/Franklin Statistics, 66 of 87 Choosing the Sample Size for Estimating a Population Proportion? First, we must decide on the desired margin of error Second, we must choose the confidence level for achieving that margin of error In practice, 95% confidence intervals are most common Agresti/Franklin Statistics, 67 of 87 Example: What Sample Size Do You Need For An Exit Poll? A television network plans to predict the outcome of an election between two candidates – Levin and Sanchez They will do this with an exit poll that randomly samples votes on election day Agresti/Franklin Statistics, 68 of 87 Example: What Sample Size Do You Need For An Exit Poll? The final poll a week before election day estimated Levin to be well ahead, 58% to 42% • So the outcome is not expected to be close The researchers decide to use a sample size for which the margin of error is 0.04 Agresti/Franklin Statistics, 69 of 87 Example: What Sample Size Do You Need For An Exit Poll? What is the sample size for which a 95% confidence interval for the population proportion has margin of error equal to 0.04? Agresti/Franklin Statistics, 70 of 87 Example: What Sample Size Do You Need For An Exit Poll? The 95% confidence interval for a population proportion p is: p ˆ 1.96( se) If the sample size is such that 1.96(se) = 0.04, then the margin of error will be 0.04 Agresti/Franklin Statistics, 71 of 87 Example: What Sample Size Do You Need For An Exit Poll? Find the value of the sample size n for which 0.04 = 1.96(se): 0.04 1.96 pˆ (1 pˆ n solve algebraically for n : n (1.96) pˆ (1 pˆ ) /(0.04) 2 Agresti/Franklin Statistics, 72 of 87 2 Example: What Sample Size Do You Need For An Exit Poll? A random sample of size n = 585 should give a margin of error of about 0.04 for a 95% confidence interval for the population proportion Agresti/Franklin Statistics, 73 of 87 How Can We Select a Sample Size Without Guessing a Value for the Sample Proportion In the formula for determining n, setting p̂ = 0.50 gives the largest value for n out of all the possible values to substitute for p̂ Doing this is the “safe” approach that guarantees we’ll have enough data Agresti/Franklin Statistics, 74 of 87 Sample Size for Estimating a Population Parameter The random sample size n for which a confidence interval for a population proportion p has margin of error m (such as m = 0.04) is pˆ (1 pˆ ) z n m 2 2 Agresti/Franklin Statistics, 75 of 87 Sample Size for Estimating a Population Parameter The z-score is based on the confidence level, such as z = 1.96 for 95% confidence You either guess the value you’d get for the sample proportion based on other information or take the safe approach of setting p̂ = 0.50 Agresti/Franklin Statistics, 76 of 87 Sample Size for Estimating a Population Mean The random sample size n for which a 95% confidence interval for a population mean has margin of error approximately equal to m is 4s n m 2 2 To use this formula, you guess the value you’ll get for the sample standard deviation, s Agresti/Franklin Statistics, 77 of 87 Sample Size for Estimating a Population Mean In practice, since you don’t yet have the data, you don’t know the value of the sample standard deviation, s You must substitute an educated guess for s • You can use the sample standard deviation from a similar study Agresti/Franklin Statistics, 78 of 87 Example: Finding n to Estimate Mean Education in South Africa A social scientist plans a study of adult South Africans to investigate educational attainment in the black community How large a sample size is needed so that a 95% confidence interval for the mean number of years of education has margin of error equal to 1 year? Agresti/Franklin Statistics, 79 of 87 Example: Finding n to Estimate Mean Education in South Africa No prior information about the standard deviation of educational attainment is available We might guess that the sample education values fall within a range of about 18 years Agresti/Franklin Statistics, 80 of 87 Example: Finding n to Estimate Mean Education in South Africa If the data distribution is bell-shaped, the range from x – 3s to x + 3 s will contain nearly all the distribution The distance x – 3 s to x + 3s equals 6s Solving 18 = 6s for s yields s = 3 So ‘3’ is a crude estimate of s Agresti/Franklin Statistics, 81 of 87 Example: Finding n to Estimate Mean Education in South Africa The desired margin of error is m = 1 year The required sample size is: 2 2 4s 4(3) n 36 m 1 2 2 Agresti/Franklin Statistics, 82 of 87 What Factors Affect the Choice of the Sample Size? The first is the desired precision, as measured by the margin of error, m The second is the confidence level Agresti/Franklin Statistics, 83 of 87 What Other Factors Affect the Choice of the Sample Size? A third factor is the variability in the data • If subjects have little variation (that is, s is small), we need fewer data than if they have substantial variation A fourth factor is financial • Cost is often a major constraint Agresti/Franklin Statistics, 84 of 87 What if You Have to Use a Small n? The t- methods for a mean are valid for any n • However, you need to be extra cautious to look for extreme outliers or great departures from the normal population assumption Agresti/Franklin Statistics, 85 of 87 What if You Have to Use a Small n? In the case of the confidence interval for a population proportion, the method works poorly for small samples Agresti/Franklin Statistics, 86 of 87 Constructing a Small-Sample Confidence Interval for a Proportion Suppose a random sample does not have at least 15 successes and 15 failures The confidence interval formula: p ˆz p ˆ (1 p ˆ) n ● Is still valid if we use it after adding ‘2’ to the original number of successes and ‘2’ to the original number of failures This results in adding ‘4’ to the sample size n Agresti/Franklin Statistics, 87 of 87