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Chapter 7 Statistical Inference: Confidence Intervals

Chapter 7 Statistical Inference: Confidence Intervals. Learn …. How to Estimate a Population Parameter Using Sample Data. Section 7.1. What Are Point and Interval Estimates of Population Parameters?. Point Estimate.

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Chapter 7 Statistical Inference: Confidence Intervals

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  1. Chapter 7Statistical Inference: Confidence Intervals • Learn …. How to Estimate a Population Parameter Using Sample Data

  2. Section 7.1 What Are Point and Interval Estimates of Population Parameters?

  3. Point Estimate • A point estimate is a single number that is our “best guess” for the parameter

  4. Interval Estimate • An interval estimate is an interval of numbers within which the parameter value is believed to fall.

  5. Point Estimate vs Interval Estimate

  6. 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

  7. 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

  8. 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

  9. 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

  10. 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

  11. 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

  12. 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

  13. 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

  14. 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

  15. 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

  16. 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

  17. 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

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

  19. 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

  20. 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] =

  21. 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

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

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

  24. 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

  25. 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

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

  27. 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:

  28. “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

  29. 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

  30. A 99% Confidence Interval for p

  31. Different Confidence Levels

  32. 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

  33. 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

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

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

  36. 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

  37. 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

  38. A recent survey asked: “During the last year, did anyone take something from you by force?” • Of 987 subjects, 17 answered “yes” • Find the point estimate of the proportion of the population who were victims • .17 • .017 • .0017

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