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Lesson04_new Lesson04_new Presentation Transcript

  • Statistics for Management Lesson 4 Confidence Interval Estimation
  • Lesson Topics
    • Confidence Interval Estimation for the Mean
    • (  Known)
    • Confidence Interval Estimation for the Mean
    • (  Unknown)
    • Confidence Interval Estimation for the
    • Proportion
    • The Situation of Finite Populations
    • Sample Size Estimation
  • Mean,  , is unknown Population Random Sample I am 95% confident that  is between 40 & 60. Mean X = 50 Estimation Process Sample
  • Estimate Population Parameter... with Sample Statistic Mean  Proportion p p s Variance s 2 Population Parameters Estimated  2 Difference  -  1 2 x - x 1 2 X _ _ _
    • Provides Range of Values
      • Based on Observations from 1 Sample
    • Gives Information about Closeness to Unknown Population Parameter
    • Stated in terms of Probability
      • Never 100% Sure
    Confidence Interval Estimation
  • Confidence Interval Sample Statistic Confidence Limit (Lower) Confidence Limit (Upper) A Probability That the Population Parameter Falls Somewhere Within the Interval. Elements of Confidence Interval Estimation
  • Parameter = Statistic ± Its Error © 1984-1994 T/Maker Co. Confidence Limits for Population Mean Error = Error = Error Error
  • 90% Samples 95% Samples Confidence Intervals 99% Samples X _  x _
    • Probability that the unknown
    • population parameter falls within the
    • interval
    • Denoted (1 -  ) % = level of confidence e.g. 90%, 95%, 99%
      •  Is Probability That the Parameter Is Not Within the Interval
    Level of Confidence
  • Confidence Intervals Intervals Extend from (1 -  ) % of Intervals Contain  .   % Do Not. 1 -   /2  /2 X _  x _ Intervals & Level of Confidence Sampling Distribution of the Mean to
    • Data Variation
    • measured by 
    • Sample Size
    • Level of Confidence (1 -  )
    Intervals Extend from © 1984-1994 T/Maker Co. Factors Affecting Interval Width X - Z  to X + Z  x x
  • Mean  Unknown Confidence Intervals Proportion Finite Population  Known Confidence Interval Estimates
    • Assumptions
      • Population Standard Deviation Is Known
      • Population Is Normally Distributed
      • If Not Normal, use large samples
    • Confidence Interval Estimate
    Confidence Intervals (  Known)
  • Mean  Unknown Confidence Intervals Proportion Finite Population  Known Confidence Interval Estimates
    • Assumptions
      • Population Standard Deviation Is Unknown
      • Population Must Be Normally Distributed
    • Use Student’s t Distribution
    • Confidence Interval Estimate
    Confidence Intervals (  Unknown)
  • Z t 0 t ( df = 5) Standard Normal t ( df = 13) Bell-Shaped Symmetric ‘ Fatter’ Tails Student’s t Distribution
    • Number of Observations that Are Free
    • to Vary After Sample Mean Has Been
    • Calculated
    • Example
      • Mean of 3 Numbers Is 2 X 1 = 1 (or Any Number) X 2 = 2 (or Any Number) X 3 = 3 (Cannot Vary) Mean = 2
    degrees of freedom = n -1 = 3 -1 = 2 Degrees of Freedom ( df )
  • Upper Tail Area df .25 .10 .05 1 1.000 3.078 6.314 2 0.817 1.886 2.920 3 0.765 1.638 2.353 t 0 Assume: n = 3 df = n - 1 = 2   = .10  /2 =.05 2.920 t Values  / 2 .05 Student’s t Table
    • A random sample of n = 25 has = 50 and
    • s = 8. Set up a 95% confidence interval estimate for  .
       . . 46 69 53 30 Example: Interval Estimation  Unknown
  • Mean  Unknown Confidence Intervals Proportion Finite Population  Known Confidence Interval Estimates
    • Assumptions
      • Sample Is Large Relative to Population
        • n / N > .05
    • Use Finite Population Correction Factor
    • Confidence Interval (Mean,  X Unknown)
    X    Estimation for Finite Populations
  • Mean  Unknown Confidence Intervals Proportion Finite Population  Known Confidence Interval Estimates
    • Assumptions
      • Two Categorical Outcomes
      • Population Follows Binomial Distribution
      • Normal Approximation Can Be Used
      • n · p  5 & n· (1 - p )  5
    • Confidence Interval Estimate
    Confidence Interval Estimate Proportion
    • A random sample of 400 Voters showed 32 preferred Candidate A. Set up a 95% confidence interval estimate for p .
    p   .053 .107 Example: Estimating Proportion
  • Sample Size
    • Too Big:
    • Requires too
    • much resources
    • Too Small:
    • Won’t do
    • the job
    • What sample size is needed to be 90% confident of being correct within ± 5 ? A pilot study suggested that the standard deviation is 45.
    n Z Error     2 2 2 2 2 2 1 645 45 5 219 2 220  . . Example: Sample Size for Mean Round Up
    • What sample size is needed to be within ± 5 with 90% confidence? Out of a population of 1,000, we randomly selected 100 of which 30 were defective.
    Example: Sample Size for Proportion Round Up 228 
    • What sample size is needed to be 90% confident of being correct within ± 5 ? Suppose the population size N = 500.
    Example: Sample Size for Mean Using fpc Round Up where 153 
  • Lesson Summary
    • Discussed Confidence Interval Estimation for
    • the Mean (  Known)
    • Discussed Confidence Interval Estimation for
    • the Mean (  Unknown)
    • Addressed Confidence Interval Estimation for
    • the Proportion
    • Addressed the Situation of Finite Populations
    • Determined Sample Size