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Section 7.2

Section 7.2. How Can We Construct a Confidence Interval to Estimate a Population Proportion?. 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

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Section 7.2

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  1. Section 7.2 How Can We Construct a Confidence Interval to Estimate a Population Proportion?

  2. 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

  3. 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] =

  4. 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

  5. Finding the 95% Confidence Interval for a Population Proportion • In practice, we use an estimated standard error:

  6. Finding the 95% Confidence Interval for a Population Proportion • A 95% confidence interval for a population proportion p is:

  7. 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

  8. 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

  9. Example: Would You Pay Higher Prices to Protect the Environment?

  10. 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:

  11. “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

  12. 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

  13. A 99% Confidence Interval for p

  14. Different Confidence Levels

  15. 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

  16. 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

  17. What is the Error Probability for the Confidence Interval Method?

  18. Summary: Confidence Interval for a Population Proportion, p • A confidence interval for a population proportion p is:

  19. 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

  20. 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

  21. Section 7.3 How Can We Construct a Confidence Interval To Estimate a Population Mean?

  22. 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

  23. 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:

  24. 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

  25. 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

  26. 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

  27. t-Distribution

  28. 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

  29. 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?

  30. 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

  31. 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

  32. 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 18 231.61 21.94 178.00 221.25 240.00 246.75 yes 7 233.57 14.64 210.00 225.00 235.00 250.00 buy_now Maximum no 255.00 yes 250.00

  33. Example: eBay Auctions of Palm Handheld Computers

  34. 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

  35. 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

  36. Example: eBay Auctions of Palm Handheld Computers • The 95% confidence interval for the “buy-it-now” option is: • which is 233.57 ± 13.54 or (220.03, 247.11)

  37. 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)

  38. 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

  39. 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

  40. 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

  41. 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?

  42. 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

  43. 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

  44. Cases Where the t- Confidence Interval Does Not Work • With binary data • With data that contain extreme outliers

  45. The Standard Normal Distribution is the t-Distribution with df = ∞

  46. 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 1.81. What is the point estimate of the population mean? • 497 • 2 • 3.02 • 1.81

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