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Confidence Interval Estimation for a Population Proportion

Understand confidence intervals & point estimates for population proportions with practical examples. Learn to compute 95% confidence intervals, interpret results, & choose optimal confidence levels.

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Confidence Interval Estimation for a Population Proportion

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  1. Confidence Interval Estimation for a Population Proportion Lecture 33 Section 9.4 Mon, Nov 6, 2006

  2. Point Estimates • Point estimate – A single value of the statistic used to estimate the parameter. • The problem with point estimates is that we have no idea how close we can expect them to be to the parameter. • That is, we have no idea of how large the error may be.

  3. Interval Estimates • Interval estimate – an interval of numbers that has a stated probability (often 95%) of containing the parameter. • An interval estimate is more informative than a point estimate.

  4. Interval Estimates • Confidence level – The probability that is associated with the interval. • If the confidence level is 95%, then the interval is called a 95% confidence interval.

  5. Approximate 95% Confidence Intervals • How do we find a 95% confidence interval for p? • Begin with the sample size n and the sampling distribution of p^. • We know that the sampling distribution is normal with mean p and standard deviation

  6. The Target Analogy • Suppose a shooter hits within 4 rings (4 inches) of the bull’s eye 95% of the time. • Then each individual shot has a 95% chance of hitting within 4 inches.

  7. The Target Analogy

  8. The Target Analogy

  9. The Target Analogy

  10. The Target Analogy

  11. The Target Analogy

  12. The Target Analogy

  13. The Target Analogy • Now suppose we are shown where the shot hit, but we are not shown where the bull’s eye is. • What is the probability that the bull’s eye is within 4 inches of that shot?

  14. The Target Analogy

  15. The Target Analogy

  16. The Target Analogy Where is the bull’s eye?

  17. The Target Analogy 4 inches

  18. The Target Analogy 4 inches 95% chance that the bull’s eye is within this circle.

  19. The Confidence Interval • In a similar way, 95% of the sample proportions p^ should lie within 1.96 standard deviations (p^) of the parameter p.

  20. The Confidence Interval p

  21. The Confidence Interval 1.96 p^ p

  22. The Confidence Interval 1.96 p^ p

  23. The Confidence Interval 1.96 p^ p

  24. The Confidence Interval 1.96 p^ p

  25. The Confidence Interval 1.96 p^ p

  26. The Confidence Interval 1.96 p^ p

  27. The Confidence Interval • Therefore, if we compute a single p^, then we expect that there is a 95% chance that it lies within a distance 1.96p^ of p.

  28. The Confidence Interval

  29. The Confidence Interval

  30. The Confidence Interval p^ Where is p?

  31. The Confidence Interval 1.96 p^ p^

  32. The Confidence Interval 1.96 p^ p^ 95% chance that p is within this interval

  33. Approximate 95% Confidence Intervals • Thus, the confidence interval is • The trouble is, to know p^, we must know p. (See the formula for p^.) • The best we can do is to use p^ in place of p to estimate p^.

  34. Approximate 95% Confidence Intervals • That is, • This is called the standard error of p^ and is denoted SE(p^). • Now the 95% confidence interval is

  35. Example • Example 9.6, p. 585 – Study: Chronic Fatigue Common. • Rework the problem supposing that 350 out of 3066 people reported that they suffer from chronic fatigue syndrome. • How should we interpret the confidence interval?

  36. Standard Confidence Levels • The standard confidence levels are 90%, 95%, 99%, and 99.9%. (See p. 588 and Table III, p. A-6.)

  37. The Confidence Interval • The confidence interval is given by the formula where z • Is given by the previous chart, or • Is found in the normal table, or • Is obtained using the invNorm function on the TI-83.

  38. Confidence Level • Rework Example 9.6, p. 585, by computing a • 90% confidence interval. • 99% confidence interval. • Which one is widest? • In which one do we have the most confidence?

  39. Probability of Error • We use the symbol  to represent the probability that the confidence interval is in error. • That is,  is the probability that p is not in the confidence interval. • In a 95% confidence interval,  = 0.05.

  40. Probability of Error • Thus, the area in each tail is /2.

  41. Which Confidence Interval is Best? • Which is better? • A large margin of error (wide interval), or • A small margin of error (narrow interval). • Which is better? • A low level of confidence, or • A high level of confidence.

  42. Which Confidence Interval is Best? • Why not get a confidence interval that has a small margin of error and has a high level of confidence associated with it? • Hey, why not a margin of error of 0 and a confidence level of 100%?

  43. Which Confidence Interval is Best? • Which is better? • A smaller sample size, or • A larger sample size.

  44. Which Confidence Interval is Best? • A larger sample size is better only up to the point where its cost is not worth its benefit. • That is why we settle for a certain margin of error and a confidence level of less than 100%.

  45. TI-83 – Confidence Intervals • The TI-83 will compute a confidence interval for a population proportion. • Press STAT. • Select TESTS. • Select 1-PropZInt.

  46. TI-83 – Confidence Intervals • A display appears requesting information. • Enter x, the numerator of the sample proportion. • Enter n, the sample size. • Enter the confidence level, as a decimal. • Select Calculate and press ENTER.

  47. TI-83 – Confidence Intervals • A display appears with several items. • The title “1-PropZInt.” • The confidence interval, in interval notation. • The sample proportion p^. • The sample size. • How would you find the margin of error?

  48. TI-83 – Confidence Intervals • Rework Example 9.6, p. 585, using the TI-83.

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