Introduction to Probability and Statistics                          10th Week (5/17)            Estimation Theory
Confidence IntervalsContent of this chapter Confidence Intervals for the Population  Mean, μ       when Population Stand...
Point and Interval Estimates     A point estimate is a single number,     a confidence interval provides additional     ...
Point Estimates  We can estimate a        with a SamplePopulation Parameter …        Statistic                         (a ...
Confidence Intervals   How much uncertainty is associated with a    point estimate of a population parameter?   An inter...
Confidence Interval Estimate   An interval gives a range of values:       Takes into consideration variation in sample  ...
Estimation Process                 Random Sample   I am 95%                                 confident that                ...
General Formula      The general formula for all       confidence intervals is:Point Estimate ± (Critical Value)(Standard...
Confidence Level   Confidence Level       Confidence for which the interval        will contain the unknown        popul...
Confidence Level, (1-α)                                                     (continued)   Suppose confidence level = 95%...
Estimates
Estimates
Estimates
Efficient estimator: If the sampling distributions of twostatistics have the same mean, the statistic with the smallervari...
Point/Interval Estimates•Point estimate of the parameter: An estimate of a population parametergiven by a single number•In...
Confidence Interval- A particular kind of interval estimate of a population  parameter and is used to indicate the reliabi...
Confidence Interval
Confidence Intervals                 Confidence                  Intervals    Population                Population      Me...
Confidence Interval for μ                  (σ Known)• Assumptions   – Population standard deviation σ is known   – Populat...
Finding the Critical Value, Z• Consider a 95% confidence interval: = ±1.96                                   Z            ...
Common Levels of           Confidence• Commonly used confidence levels are  90%, 95%, and 99%                 Confidence  ...
Intervals and Level of                   Confidence              Sampling Distribution of the Mean                α/2     ...
Example• A sample of 11 circuits from a large  normal population has a mean  resistance of 2.20 ohms. We know  from past t...
Example                                           (continued)• A sample of 11 circuits from a large  normal population has...
Interpretation• We are 95% confident that the true  mean resistance is between 1.9932  and 2.4068 ohms• Although the true ...
Confidence Intervals for Means
Confidence Intervals for Means
Confidence Intervals for Means
Confidence Intervals for Means
Confidence Intervals                 Confidence                  Intervals    Population                Population      Me...
Confidence Interval for μ               (σ Unknown)   If the population standard deviation σ is    unknown, we can substi...
Confidence Interval for μ                    (σ Unknown)                                                                  ...
Student’s t Distribution   The t is a family of distributions   The t value depends on degrees of    freedom (d.f.)    ...
Degrees of Freedom (df)Idea: Number of observations that are free to vary      after sample mean has been calculatedExampl...
Student’s t Distribution                   Note: t         Z as n increases                    Standard                   ...
Student’s t Table     Upper Tail Area                                Let: n = 3df    .25    .10      .05       df = n - 1 ...
t distribution values         With comparison to the Z valueConfidence    t            t           t         Z Level     (...
ExampleA random sample of n = 25 has X = 50 andS = 8. Form a 95% confidence interval for μ   d.f. = n – 1 = 24, so   tα/2...
Confidence Intervals for Means
Confidence Intervals for Means
Confidence Intervals for Means
Confidence Intervals for Means
Confidence Intervals                 Confidence                  Intervals    Population                Population      Me...
Confidence Intervals for the      Population Proportion, π   An interval estimate for the population    proportion ( π ) ...
Confidence Intervals for the     Population Proportion, π                                              (continued)   Reca...
Confidence Interval Endpoints   Upper and lower confidence limits for the    population proportion are calculated with th...
Example   A random sample of 100 people    shows that 25 are left-handed.   Form a 95% confidence interval for    the tr...
Example                                           (continued)   A random sample of 100 people shows    that 25 are left-h...
Interpretation   We are 95% confident that the true    percentage of left-handers in the population    is between        ...
Confidence Intervals for Proportions
Confidence Intervals for Proportions
Confidence Intervals for Proportions
Confidence Intervals for Proportions
Confidence Intervals for Differences and                 Sums
Confidence Intervals for Differences and                 Sums
Confidence Intervals for Differences and                 Sums
Confidence Intervals for Differences and                 Sums
Confidence Intervals for the Variance of a          Normal Distribution
Confidence Intervals for Variance Ratios
Maximum Likelihood Estimates
Determining Sample Size          Determining          Sample SizeFor the                  For the Mean                   P...
Sampling Error   The required sample size can be found to reach    a desired margin of error (e) with a specified    leve...
Determining Sample Size             Determining             Sample Size   For the    Mean             Sampling error      ...
Determining Sample Size                                     (continued)               Determining               Sample Siz...
Determining Sample Size                                                    (continued)   To determine the required sample...
Required Sample Size ExampleIf σ = 45, what sample size is needed toestimate the mean within ± 5 with 90%confidence?      ...
If σ is unknown   If unknown, σ can be estimated when    using the required sample size formula       Use a value for σ ...
Determining Sample Size                                             (continued)                  Determining              ...
Determining Sample Size                                                          (continued)   To determine the required ...
Required Sample Size ExampleHow large a sample would be necessaryto estimate the true proportion defective ina large popul...
Required Sample Size Example                                               (continued)    Solution:    For 95% confidence,...
Ethical Issues   A confidence interval estimate (reflecting    sampling error) should always be included    when reportin...
10주차
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10주차

  1. 1. Introduction to Probability and Statistics 10th Week (5/17) Estimation Theory
  2. 2. Confidence IntervalsContent of this chapter Confidence Intervals for the Population Mean, μ  when Population Standard Deviation σ is Known  when Population Standard Deviation σ is Unknown Confidence Intervals for the Population Proportion, p Determining the Required Sample Size
  3. 3. Point and Interval Estimates  A point estimate is a single number,  a confidence interval provides additional information about variabilityLower UpperConfidence Confidence Point EstimateLimit Limit Width of confidence interval
  4. 4. Point Estimates We can estimate a with a SamplePopulation Parameter … Statistic (a Point Estimate) Mean μ XProportion π p
  5. 5. Confidence Intervals How much uncertainty is associated with a point estimate of a population parameter? An interval estimate provides more information about a population characteristic than does a point estimate Such interval estimates are called confidence intervals
  6. 6. Confidence Interval Estimate An interval gives a range of values:  Takes into consideration variation in sample statistics from sample to sample  Based on observations from 1 sample  Gives information about closeness to unknown population parameters  Stated in terms of level of confidence  Can never be 100% confident
  7. 7. Estimation Process Random Sample I am 95% confident that μ is betweenPopulation Mean 40 & 60.(mean, μ, is X = 50 unknown) Sample
  8. 8. General Formula  The general formula for all confidence intervals is:Point Estimate ± (Critical Value)(Standard Error)
  9. 9. Confidence Level Confidence Level  Confidence for which the interval will contain the unknown population parameter A percentage (less than 100%)
  10. 10. Confidence Level, (1-α) (continued) Suppose confidence level = 95% Also written (1 - α) = 0.95 A relative frequency interpretation:  In the long run, 95% of all the confidence intervals that can be constructed will contain the unknown true parameter A specific interval either will contain or will not contain the true parameter  No probability involved in a specific interval
  11. 11. Estimates
  12. 12. Estimates
  13. 13. Estimates
  14. 14. Efficient estimator: If the sampling distributions of twostatistics have the same mean, the statistic with the smallervariance is called a more efficient estimator of the mean.Clearly one would in practice prefer to have estimates that areboth efficient and unbiased, but this is not always possible.
  15. 15. Point/Interval Estimates•Point estimate of the parameter: An estimate of a population parametergiven by a single number•Interval estimate of the parameter: An estimate of a populationparameter given by two numbers between which the parameter•Reliability: A statement of the error or precision of an estimate
  16. 16. Confidence Interval- A particular kind of interval estimate of a population parameter and is used to indicate the reliability of an estimate.- It is an observed interval (i.e. it is calculated from the observations), in principle different from sample to sample, that frequently includes the parameter of interest, if the experiment is repeated.- How frequently the observed interval contains the parameter is determined by the confidence level or confidence coefficient.
  17. 17. Confidence Interval
  18. 18. Confidence Intervals Confidence Intervals Population Population Mean Proportionσ Known σ Unknown
  19. 19. Confidence Interval for μ (σ Known)• Assumptions – Population standard deviation σ is known – Population is normally distributed – If population is not normal, use large sample• Confidence interval estimate: σ X±Z n X σ/ nwhere is the point estimate Z is the normal distribution critical value for a probability of α/2 in
  20. 20. Finding the Critical Value, Z• Consider a 95% confidence interval: = ±1.96 Z 1− α = 0.95 α α = 0.025 = 0.025 2 2Z units: Z= -1.96 0 Z= 1.96 Lower UpperX units: Confidence Point Estimate Confidence Limit Limit
  21. 21. Common Levels of Confidence• Commonly used confidence levels are 90%, 95%, and 99% Confidence Confidence Coefficient, Z value Level 1− α 80% 0.80 1.28 90% 0.90 1.645 95% 0.95 1.96 98% 0.98 2.33 99% 0.99 2.58 99.8% 0.998 3.08 99.9% 0.999 3.27
  22. 22. Intervals and Level of Confidence Sampling Distribution of the Mean α/2 1− α α/2 x Intervals μx = μextend from x1 σ x2 (1-α)x100% X+Z of intervals n to constructed σ contain μ; X−Z n (α)x100% do not. Confidence Intervals
  23. 23. Example• A sample of 11 circuits from a large normal population has a mean resistance of 2.20 ohms. We know from past testing that the population standard deviation is 0.35 ohms.• Determine a 95% confidence interval for the true mean resistance of the population.
  24. 24. Example (continued)• A sample of 11 circuits from a large normal population has a mean resistance of 2.20 ohms. We know from past testing that the population standard deviation±is 0.35 ohms. X Z σ n• Solution: = 2.20 ± 1.96 (0.35/ 11) = 2.20 ± 0.2068 1.9932 ≤ µ ≤ 2.4068
  25. 25. Interpretation• We are 95% confident that the true mean resistance is between 1.9932 and 2.4068 ohms• Although the true mean may or may not be in this interval, 95% of intervals formed in this manner will contain the true mean
  26. 26. Confidence Intervals for Means
  27. 27. Confidence Intervals for Means
  28. 28. Confidence Intervals for Means
  29. 29. Confidence Intervals for Means
  30. 30. Confidence Intervals Confidence Intervals Population Population Mean Proportionσ Known σ Unknown
  31. 31. Confidence Interval for μ (σ Unknown) If the population standard deviation σ is unknown, we can substitute the sample standard deviation, S This introduces extra uncertainty, since S is variable from sample to sample So we use the t distribution instead of the normal distribution
  32. 32. Confidence Interval for μ (σ Unknown) (continued) Assumptions  Population standard deviation is unknown  Population is normally distributed  If population is not normal, use large sample Use Student’s t Distribution Confidence Interval Estimate: S X ± t n-1 n (where t is the critical value of the t distribution with n -1 degrees of freedom and an area of α/2 in each tail)
  33. 33. Student’s t Distribution The t is a family of distributions The t value depends on degrees of freedom (d.f.)  Number of observations that are free to vary after sample mean has been calculated d.f. = n - 1
  34. 34. Degrees of Freedom (df)Idea: Number of observations that are free to vary after sample mean has been calculatedExample: Suppose the mean of 3 numbers is 8.0 Let X1 = 7 If the mean of these three Let X2 = 8 values is 8.0, What is X 3 ? then X3 must be 9 (i.e., X3 is not free to vary)Here, n = 3, so degrees of freedom = n – 1 = 3 – 1 = 2(2 values can be any numbers, but the third is not free to varyfor a given mean)
  35. 35. Student’s t Distribution Note: t Z as n increases Standard Normal (t with df = ∞) t (df = 13)t-distributions are bell-shaped and symmetric, buthave ‘fatter’ tails than the t (df = 5)normal 0 t
  36. 36. Student’s t Table Upper Tail Area Let: n = 3df .25 .10 .05 df = n - 1 = 2 α = 0.101 1.000 3.078 6.314 α/2 = 0.052 0.817 1.886 2.9203 0.765 1.638 2.353 α/2 = 0.05 The body of the table contains t values, not 0 2.920 t probabilities
  37. 37. t distribution values With comparison to the Z valueConfidence t t t Z Level (10 d.f.) (20 d.f.) (30 d.f.) ____ 0.80 1.372 1.325 1.310 1.28 0.90 1.812 1.725 1.697 1.645 0.95 2.228 2.086 2.042 1.96 0.99 3.169 2.845 2.750 2.58 Note: t Z as n increases
  38. 38. ExampleA random sample of n = 25 has X = 50 andS = 8. Form a 95% confidence interval for μ d.f. = n – 1 = 24, so tα/2 , n−1 = t 0.025,24 = 2.0639The confidence interval is S 8 X ± t α/2, n-1 = 50 ± (2.0639) n 25 46.698 ≤ μ ≤ 53.302
  39. 39. Confidence Intervals for Means
  40. 40. Confidence Intervals for Means
  41. 41. Confidence Intervals for Means
  42. 42. Confidence Intervals for Means
  43. 43. Confidence Intervals Confidence Intervals Population Population Mean Proportionσ Known σ Unknown
  44. 44. Confidence Intervals for the Population Proportion, π An interval estimate for the population proportion ( π ) can be calculated by adding an allowance for uncertainty to the sample proportion ( p )
  45. 45. Confidence Intervals for the Population Proportion, π (continued) Recall that the distribution of the sample proportion is approximately normal if the sample size is large, with standard deviation π (1− π ) σp = n We will estimate this with sample data: p(1− p) n
  46. 46. Confidence Interval Endpoints Upper and lower confidence limits for the population proportion are calculated with the formula p(1− p) p±Z n where  Z is the standard normal value for the level of confidence desired  p is the sample proportion  n is the sample size
  47. 47. Example A random sample of 100 people shows that 25 are left-handed. Form a 95% confidence interval for the true proportion of left-handers
  48. 48. Example (continued) A random sample of 100 people shows that 25 are left-handed. Form a 95% confidence interval for the true proportion of left-handers. p ± Z p(1− p)/n = 25/100 ± 1.96 0.25(0.75)/100 = 0.25 ± 1.96 (0.0433) 0.1651 ≤ π ≤ 0.3349
  49. 49. Interpretation We are 95% confident that the true percentage of left-handers in the population is between 16.51% and 33.49%. Although the interval from 0.1651 to 0.3349 may or may not contain the true proportion, 95% of intervals formed from samples of size 100 in this manner will contain the true proportion.
  50. 50. Confidence Intervals for Proportions
  51. 51. Confidence Intervals for Proportions
  52. 52. Confidence Intervals for Proportions
  53. 53. Confidence Intervals for Proportions
  54. 54. Confidence Intervals for Differences and Sums
  55. 55. Confidence Intervals for Differences and Sums
  56. 56. Confidence Intervals for Differences and Sums
  57. 57. Confidence Intervals for Differences and Sums
  58. 58. Confidence Intervals for the Variance of a Normal Distribution
  59. 59. Confidence Intervals for Variance Ratios
  60. 60. Maximum Likelihood Estimates
  61. 61. Determining Sample Size Determining Sample SizeFor the For the Mean Proportion
  62. 62. Sampling Error The required sample size can be found to reach a desired margin of error (e) with a specified level of confidence (1 - α) The margin of error is also called sampling error  the amount of imprecision in the estimate of the population parameter  the amount added and subtracted to the point estimate to form the confidence interval
  63. 63. Determining Sample Size Determining Sample Size For the Mean Sampling error (margin of error) σ σX±Z e=Z n n
  64. 64. Determining Sample Size (continued) Determining Sample Size For the Mean σ 2 Z σ 2e=Z Now solve for n to get n= n e 2
  65. 65. Determining Sample Size (continued) To determine the required sample size for the mean, you must know:  The desired level of confidence (1 - α), which determines the critical Z value  The acceptable sampling error, e  The standard deviation, σ
  66. 66. Required Sample Size ExampleIf σ = 45, what sample size is needed toestimate the mean within ± 5 with 90%confidence? 2 2 2 2 Z σ (1.645) (45) n= 2 = 2 = 219.19 e 5 So the required sample size is n = 220 (Always round up)
  67. 67. If σ is unknown If unknown, σ can be estimated when using the required sample size formula  Use a value for σ that is expected to be at least as large as the true σ  Select a pilot sample and estimate σ with the sample standard deviation, S
  68. 68. Determining Sample Size (continued) Determining Sample Size For the Proportion π (1− π ) Now solve Z 2 π (1− π )e=Z for n to get n= 2 n e
  69. 69. Determining Sample Size (continued) To determine the required sample size for the proportion, you must know:  The desired level of confidence (1 - α), which determines the critical Z value  The acceptable sampling error, e  The true proportion of “successes”, π  π can be estimated with a pilot sample, if necessary (or conservatively use π = 0.5)
  70. 70. Required Sample Size ExampleHow large a sample would be necessaryto estimate the true proportion defective ina large population within ±3%, with 95%confidence?(Assume a pilot sample yields p = 0.12)
  71. 71. Required Sample Size Example (continued) Solution: For 95% confidence, use Z = 1.96 e = 0.03 p = 0.12, so use this to estimate π Z π (1− π ) (1.96) (0.12)(1− 0.12) 2 2n= 2 = 2 = 450.74 e (0.03) So use n = 451
  72. 72. Ethical Issues A confidence interval estimate (reflecting sampling error) should always be included when reporting a point estimate The level of confidence should always be reported The sample size should be reported An interpretation of the confidence interval estimate should also be provided

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