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第 11 章 統計估計 - 區間估計

第 11 章 統計估計 - 區間估計. 學習目的. 本章結構. 區間估計. A point estimate is the statistic, computed from sample information, which is used to estimate the population parameter.

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第 11 章 統計估計 - 區間估計

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  1. 第 11章 統計估計-區間估計

  2. 學習目的

  3. 本章結構

  4. 區間估計

  5. A point estimate is the statistic, computed from sample information, which is used to estimate the population parameter. A confidence interval estimate is a range of values constructed from sample data so that the population parameter is likely to occur within that range at a specified probability. The specified probability is called the level of confidence. Point and Interval Estimates

  6. 信賴區間

  7. 信賴水準

  8. 圖11.1 母體平均數的信賴區間

  9. 區間估計

  10. 圖11.2 的抽樣分配

  11. 圖11.3 抽樣誤差小於等於1.96 的區間

  12. 圖11.4 母體平均數μ的信賴區間

  13. Interval Estimates - Interpretation For a 95% confidence interval about 95% of the similarly constructed intervals will contain the parameter being estimated. Also 95% of the sample means for a specified sample size will lie within 1.96 standard deviations of the hypothesized population

  14. 95%信賴水準

  15. 圖11.5 母體平均數μ 的信賴區間

  16. 圖11.6 （ ） 的機率區間

  17. 1-1 母體平均數的區間估計—大樣本

  18. 表11.1 不同信賴水準下母體平均數μ的信賴區間

  19. 圖11.7 信賴區間對話方塊

  20. 圖11.8 老年人看電視的時間

  21. 信賴區間長度的決定因素

  22. 母體平均數的區間估計—大樣本

  23. 表11.2 台北縣中永和地區房屋價格的估計

  24. 1-1 母體平均數的區間估計—小樣本

  25. 1-1 母體平均數的區間估計—小樣本

  26. t分配

  27. t分配的性質

  28. 圖11.9t分配為以0為中心的對稱分配

  29. Comparing the z and t Distributions when n is small

  30. 1. It is, like the z distribution, a continuous distribution. 2. It is, like the z distribution, bell-shaped and symmetrical. 3. There is not one t distribution, but rather a family of t distributions. All t distributions have a mean of 0, but their standard deviations differ according to the sample size, n. 4. The t distribution is more spread out and flatter at the center than the standard normal distribution As the sample size increases, however, the t distribution approaches the standard normal distribution, Characteristics of the t-distribution

  31. 圖11.10 標準常態分配與t 分配

  32. 表11.3 t 值表（部份）

  33. 圖11.11 t 分配的機率值

  34. 圖11.12t值

  35. 圖11.13 t 分配的對話方塊

  36. 圖11.14 t值的對話方塊

  37. 圖11.15 的機率區間

  38. 母體平均數的區間估計—小樣本

  39. Use Z-distribution If the population standard deviation is known or the sample is greater than 30. Use t-distribution If the population standard deviation is unknown and the sample is less than 30. Confidence Interval Estimates for the Mean

  40. When to Use the z or t Distribution for Confidence Interval Computation

  41. Confidence Interval for the Mean – Example using the t-distribution A tire manufacturer wishes to investigate the tread life of its tires. A sample of 10 tires driven 50,000 miles revealed a sample mean of 0.32 inch of tread remaining with a standard deviation of 0.09 inch. Construct a 95 percent confidence interval for the population mean. Would it be reasonable for the manufacturer to conclude that after 50,000 miles the population mean amount of tread remaining is 0.30 inches?

  42. Student’s t-distribution Table

  43. 表11.4 米酒容量的區間估計

  44. 圖11.16 的機率區間

  45. 母體比例的區間估計

  46. Confidence Interval for a Population Proportion- Example The union representing the Bottle Blowers of America (BBA) is considering a proposal to merge with the Teamsters Union. According to BBA union bylaws, at least three-fourths of the union membership must approve any merger. A random sample of 2,000 current BBA members reveals 1,600 plan to vote for the merger proposal. What is the estimate of the population proportion? Develop a 95 percent confidence interval for the population proportion. Basing your decision on this sample information, can you conclude that the necessary proportion of BBA members favor the merger? Why?

  47. Finite-Population Correction Factor • A population that has a fixed upper bound is said to be finite. • For a finite population, where the total number of objects is N and the size of the sample is n, the following adjustment is made to the standard errors of the sample means and the proportion: • However, ifn/N < .05, the finite-population correction factor may be ignored.

  48. Effects on FPC when n/N Changes Observe that FPC approaches 1 when n/N becomes smaller

  49. C.I. for the Proportion () C.I. for the Mean () C.I. for the Mean () Confidence Interval Formulas for Estimating Means and Proportions with Finite Population Correction

  50. There are 250 families in Scandia, Pennsylvania. A random sample of 40 of these families revealed the mean annual church contribution was $450 and the standard deviation of this was $75. Develop a 90 percent confidence interval for the population mean. Interpret the confidence interval. Given in Problem: N – 250 n – 40 s - $75 Since n/N = 40/250 = 0.16, the finite population correction factor must be used. The population standard deviation is not known therefore use the t-distribution (may use the z-dist since n>30) Use the formula below to compute the confidence interval: CI For Mean with FPC - Example

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