Agresti/Franklin Statistics, 1 of 87  Section 7.2 How Can We Construct a Confidence Interval to Estimate a Population Proportion?

Agresti/Franklin Statistics, 2 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

Agresti/Franklin Statistics, 3 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] =

Agresti/Franklin Statistics, 4 of 87 Finding the 95% Confidence Interval for a Population Proportion The exact standard error of a sample proportion equals: 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, 5 of 87 Finding the 95% Confidence Interval for a Population Proportion In practice, we use an estimated standard error :

Agresti/Franklin Statistics, 6 of 87 Finding the 95% Confidence Interval for a Population Proportion A 95% confidence interval for a population proportion p is:

Agresti/Franklin Statistics, 7 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, 8 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, 9 of 87 Example: Would You Pay Higher Prices to Protect the Environment?

Agresti/Franklin Statistics, 10 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:

Agresti/Franklin Statistics, 11 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, 12 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, 13 of 87 A 99% Confidence Interval for p

Agresti/Franklin Statistics, 14 of 87 Different Confidence Levels

Agresti/Franklin Statistics, 15 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, 16 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

Agresti/Franklin Statistics, 17 of 87 What is the Error Probability for the Confidence Interval Method?

Agresti/Franklin Statistics, 18 of 87 Summary: Confidence Interval for a Population Proportion, p A confidence interval for a population proportion p is:

Agresti/Franklin Statistics, 19 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, 20 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, 21 of 87  Section 7.3 How Can We Construct a Confidence Interval To Estimate a Population Mean?

Agresti/Franklin Statistics, 22 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/ In practice, we estimate σ by the sample standard deviation, s

Agresti/Franklin Statistics, 23 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:

Agresti/Franklin Statistics, 24 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/ 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, 25 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/ 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, 26 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, 27 of 87 t-Distribution

Agresti/Franklin Statistics, 28 of 87 Summary: 95% Confidence Interval for a Population Mean A 95% confidence interval for the population mean µ is: To use this method, you need: Data obtained by randomization An approximately normal population distribution

Agresti/Franklin Statistics, 29 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, 30 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, 31 of 87 Example: eBay Auctions of Palm Handheld Computers “Buy-it-now” option: Bidding only:

Agresti/Franklin Statistics, 32 of 87 Example: eBay Auctions of Palm Handheld Computers Summary of selling prices for the two types of auctions: buy_now N Mean StDev Minimum Q1 Median Q3 no yes buy_now Maximum no yes

Agresti/Franklin Statistics, 33 of 87 Example: eBay Auctions of Palm Handheld Computers

Agresti/Franklin Statistics, 34 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, 35 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 = $ The sample standard deviation is: s = $14.64

Agresti/Franklin Statistics, 36 of 87 Example: eBay Auctions of Palm Handheld Computers The 95% confidence interval for the “buy-it- now” option is: which is ± or (220.03, )

Agresti/Franklin Statistics, 37 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, )

Agresti/Franklin Statistics, 38 of 87 Example: eBay Auctions of Palm Handheld Computers Notice that the two intervals overlap a great deal: “Buy-it-now”: (220.03, ) Bidding only: (220.70, ) There is not enough information for us to conclude that one probability distribution clearly has a higher mean than the other

Agresti/Franklin Statistics, 39 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 in each tail and the appropriate t-score is t.005

Agresti/Franklin Statistics, 40 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, 41 of 87 If the Population is Not Normal, is the Method “Robust”? How problematic is it if we use the t- confidence interval even if the population distribution is not normal?

Agresti/Franklin Statistics, 42 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, 43 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, 44 of 87 Cases Where the t- Confidence Interval Does Not Work With binary data With data that contain extreme outliers

Agresti/Franklin Statistics, 45 of 87 The Standard Normal Distribution is the t-Distribution with df = ∞

Agresti/Franklin Statistics, 46 of 87 The 2002 GSS asked: “What do you think is the ideal number of children in a family?” The 497 females who responded had a median of 2, mean of 3.02, and standard deviation of What is the point estimate of the population mean? a. 497 b. 2 c d. 1.81