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Chapter 8

Chapter 8. Confidence Interval (Interval Estimate). Confidence Interval. Sample statistics are point estimates (“good guess”) for population parameters x-bar (sample mean) is a point estimator for μ (population mean)

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Chapter 8

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  1. Chapter 8 Confidence Interval (Interval Estimate)

  2. Confidence Interval • Sample statistics are point estimates (“good guess”) for population parameters • x-bar(sample mean) is a point estimator for μ (population mean) • p-bar (sample proportion) is a point estimator for p (population proportion)

  3. Confidence Interval • To compute the average age of Americans: • Collect the ages of the entire population (all Americans) and compute their average (more accurate), or • Collect the ages of a sample (1000 Americans) and compute their average (less time consuming, less expensive) • To compute the proportion of Americans who would vote against tax increase: • Collect the opinions from the entire population (all Americans), or • Collect the opinions from a sample (1000 Americans)

  4. Confidence Interval • As estimates, x-bar and p-bar have certain degree of inaccuracy /error (showing how much it varies from the actual population mean/proportion) • x-bar and p-bar do not provide the level of accuracy of the estimate • Confidence interval provides the level of accuracy of the estimate • Confidence level (c)

  5. Confidence Interval What is CONFIDENCE INTERVAL (INTERVAL ESTIMATE) for mean? • Interval • Around the sample mean (x-bar) • Where • We are c %confident (confidence level = c) • That this interval contains the actual population mean (μ)

  6. Confidence Interval • In this chapter, we will learn HOW TO COMPUTE THE CONFIDENCE INTERVAL for: (1) population mean (μ), and (2) population proportion (p) x-bar  margin of error = interval estimate (confidence interval) for population mean p-bar  margin of error = interval estimate (confidence interval) for population proportion

  7. Confidence Interval Examples of CONFIDENCE INTERVAL (INTERVAL ESTIMATE) for mean • For c = 90%, the confidence interval for sample mean (n = 500) is 6 ± 1 • For c = 90%, the confidence interval for sample mean (n = 100) is 6 ± 3 Which one is more accurate? Why?

  8. Margin of Error • Margin of error is the range of values above and below the sample statistic for certain confidence level x-bar  margin of error = interval estimate (confidence interval) for population mean p-bar  margin of error = interval estimate (confidence interval) for population proportion

  9. Margin of Error and Confidence Level (c) • How to compute the margin of error? • Margin of error is computed based on the confidence level (c) that we select beforehand

  10. Level of Confidence (c) • The level of confidence c: • The probability that the interval estimate contains the population parameter • Confidence level (c) is selected by the researcher

  11. Level of Confidence (c) • Most commonly used confidence level c: • 90% (least accurate) • “I am 90% confident that the confidence interval (interval estimate) contains the population mean” • 95% (more accurate) • “I am 95% confident that the confidence interval (interval estimate) contains the population mean” • 99% (most accurate) • “I am 99% confident that the confidence interval (interval estimate) contains the population mean”

  12. Computing interval estimate (confidence interval) of population mean for σ known • Population standard deviation (σ) is known • Use Formula 8.1 on page 313 • Where: • x̄= is the sample mean/average • zα/2is the z value for a given confidence level • For c = 90%, usez α/2 = 1.645 • For c = 95%, usez α/2 = 1.96 • For c = 99%, usez α/2 = 2.576 • σ is the known population standard deviation • nis the sample size Memorize these three values!

  13. Computing interval estimate (confidence interval) of population mean for σ known • Example: Exercise 2 on page 315 • n = 50 ; sigma = 6 ; x-bar = 32 • Problem a. For c = 90%, usez α/2 = 1.645 • Problem b. For c = 95%, usez α/2 = 1.96 • Problem c. For c = 99%, usez α/2 = 2.576

  14. Computing interval estimate (confidence interval) of population mean for s known • Sample standard deviation (s) is known • Use Formula 8.2 on page 319 • Where: • x-bar is the sample mean • tα/2is the t value given the confidence level with (n – 1) degree of freedom (d.f) • 90% confidence level = t for 0.05 and d.f = (n-1) • 95% confidence level = t for 0.025 and d.f = (n-1) • 99% confidence level = t for 0.005 and d.f = (n-1) • s is the sample standard deviation • n is the sample size

  15. How to find the t-value? • tα/2 is the t value given the confidence level with (n – 1) degree of freedom (d.f) • 90% confidence level = t α/2 for 0.05 and d.f = (n-1) • 95% confidence level = t α/2 for 0.025 and d.f = (n-1) • 99% confidence level = t α/2 for 0.005 and d.f = (n-1) • The t-value must be found in the “t distribution” table (Table 2 in Appendix B) • Look at the t α/2-value in the table based on: • d.f. (degrees of freedom in the first column) = (n – 1) • Area in the upper tail in the first row • 90% confidence level →Area in the upper tail = 0.05 • 95% confidence level →Area in the upper tail = 0.025 • 99% confidence level →Area in the upper tail = 0.005 Memorize these three values!

  16. Computing interval estimate (confidence interval) of population mean for s known • Example: Exercise 14 on page 323 • n = 54 ; x-bar = 22.5 ; s = 4.4 • Degree of freedom is (n – 1) = (54 – 1) = 53 • Problem a. 90% confidence level means t α/2 value for 0.05 and d.f = 53 • Problem b. 95% confidence level means t α/2 value for 0.025 and d.f = 53 • Problem c. 99% confidence level means t α/2 value for 0.005 and d.f = 53

  17. Computing interval estimate (confidence interval) of population mean for s known • Example: Exercise 22 • n = 25 • x-bar = 3.35 (use formula 3.1 on page 83 to compute x-bar) • s = 2.29 (use formula 3.5 on page 93 to compute variance and s is the square root of the variance) • Degree of freedom is (n – 1) = (25 – 1) = 24 • 95% confidence level means t value for 0.025 and d.f = 24

  18. REMEMBER:To compute the confidence interval (interval estimate) for population mean (μ)STEP 1. Identify what kind of standard deviation is givenσ (POPULATION standard deviation) OR s (SAMPLE standard deviation) STEP 2.Compute the confidence interval by using the right formula based on STEP 1

  19. REMEMBERFor population mean:IF the problem gives you σ (POPULATION standard deviation) use the first formula (Formula 8.1)IF the problem gives you s (SAMPLE standard deviation) use the second formula (Formula 8.2)

  20. Tips for solving the problems • Chapter 8 • Keywords: confidence interval, margin of error • The problem asks you to find the margin of error or the confidence interval (interval estimate) • Remember: • confidence interval = x-bar margin of error

  21. Tips for solving the problems • If the problem description gives you the population standard deviation σ, use the z α/2–value • If the problem says that “the standard deviation” is such and such without specifying the type of the standard deviation, it is assumed that the standard deviation is a population standard deviation • If the problem description gives you the sample standard deviation s, use the t α/2-value (found from t-distribution table based on the confidence level and d.f.)

  22. Population Proportion • Interval estimate for a population proportion: • See Formula 8.6 on page 329 • p-bar is the sample proportion • zα/2is the z value given the confidence level • For c = 90%, usez α/2 = 1.645 • For c = 95%, usez α/2 = 1.96 • For c = 99%, usez α/2 = 2.576 • n is the sample size

  23. Population Proportion • Examples: • Exercise 31 • n = 400 ; p-bar = 100/400 • For c = 95%, z = 1.96

  24. REMEMBER: • To compute the confidence interval (interval estimate) for population proportion • STEP 1. Compute the p-bar (most of the problems do not give you the p-bar). Use the formula (p-bar = x/n) on page 280 to compute the p-bar. • STEP 2.Compute the confidence interval or the interval estimate by using the formula 8.6 Remember: You only have one formula for population proportion, and this formula only uses z α/2, NEVERt α/2

  25. REMEMBER: Population proportion (p) • ONE FORMULA • ALWAYS zα/2 • NEVERtα/2

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