Confidence Intervals about a Population Proportion - MATH 130

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					Confidence Intervals about a Population
             Proportion
      MATH 130, Elements of Statistics I


              J. Robert Buchanan

              Department of Mathematics


                       Fall 2009




          J. Robert Buchanan   Confidence Intervals about a Population Proportion
Motivation


  Example
      PRINCETON, NJ – Gallup Poll Daily tracking finds
      41% of Americans describing economic conditions as
      "poor," down slightly from the 2008 high of 44%, but
      still more than double the percentage who say the
      economy is "excellent" or "good" (17%). The vast
      majority, 85%, perceive the economy to getting worse.
      – Jeff Jones
  The results reported here are based on combined data from
  1,544 interviews conducted March 19-21, 2008. For results
  based on this sample, the maximum margin of sampling error is
  ±3 percentage points.



                    J. Robert Buchanan   Confidence Intervals about a Population Proportion
Background




  The point estimate for the population proportion p is the
                    ˆ
  sample proportion p.
  If 1544 are surveyed and 633 respond that the economy is
  getting worse then

                      ˆ
                      p = 633/1544 ≈ 0.410.




                     J. Robert Buchanan   Confidence Intervals about a Population Proportion
                         ˆ
Sampling Distribution of p



  Theorem
  For a simple random sample of size n such that n ≤ 0.05N (that
  is, the sample is less than or equal to 5% of the population),
                                                  ˆ
        the shape of the sampling distribution of p is approximately
        normal provided np(1 − p) ≥ 10,
                                                 ˆ
        the mean of the sampling distribution of p is µp = p,
                                                        ˆ
                                                             ˆ
      the standard deviation of the sampling distribution of p is

                                          p(1 − p)
                               σp =
                                ˆ                  .
                                             n




                     J. Robert Buchanan   Confidence Intervals about a Population Proportion
Confidence Interval for a Sample Proportion




  Suppose a simple random sample of size n is taken from a
  population. A (1 − α) · 100% confidence interval for p is given by

                                                                 ˆ     ˆ
                                                                 p(1 − p)
                                  ˆ
          Lower and Upper bounds: p ± zα/2 ·
                                                                    n

                                  ˆ     ˆ
  Note: it must be the case that np(1 − p ) ≥ 10 and n ≤ 0.05N to
  construct this interval.




                     J. Robert Buchanan   Confidence Intervals about a Population Proportion
Example




 Example
 Of 1500 people surveyed, 850 had eaten pizza within the last
 month. Construct the 95% confidence interval estimate of the
 population proportion of people who have eaten pizza within
 the last month.




                   J. Robert Buchanan   Confidence Intervals about a Population Proportion
Example




 Example
     In the city or area where you live, are you satisfied or
     dissatisfied with the quality of water?
 In the United States 1000 residents aged 15 or older were
 surveyed and 870 replied they were satisfied with the water
 quality. Construct the 99% confidence interval estimate of all
 US residents satisfied with their water quality.




                    J. Robert Buchanan   Confidence Intervals about a Population Proportion
Example



 Example
     Have recent price increases in gasoline caused any
     financial hardship for you or your household?

 In the United States 1025 residents aged 18 or older were
 surveyed and 646 replied “yes”. Construct the 90% confidence
 interval estimate of all US residents who report that the price
 increases in gasoline have caused some financial hardship for
 themselves or their household.




                    J. Robert Buchanan   Confidence Intervals about a Population Proportion
Estimating the Sample Size




                                 ˆ     ˆ
                                 p(1 − p)
                    E     = zα/2 ·
                                    n
                                   zα/2 2
                         ˆ     ˆ
                     n = p(1 − p)
                                    E

                                                          ˆ
  Remark: to use this formula we need a prior estimate of p or
  we must consider the worst case scenario.




                    J. Robert Buchanan   Confidence Intervals about a Population Proportion
Worst Case Scenario


       p1 p
      0.25


      0.20


      0.15


      0.10


      0.05


                                                                            p
              0.2           0.4        0.6             0.8            1.0




                J. Robert Buchanan   Confidence Intervals about a Population Proportion
Estimating the Sample Size


  The sample size required to obtain a (1 − α) · 100% confidence
  interval for p with a margin of error E is given by
                                                         2
                                              zα/2
                            ˆ     ˆ
                        n = p(1 − p)
                                               E

                                          ˆ
  (rounded up the next integer), where p is a prior estimate of p. If
  a prior estimate of p is unavailable, the sample size required is
                                                     2
                                           zα/2
                            n = 0.25
                                            E

  rounded up the next integer.



                      J. Robert Buchanan   Confidence Intervals about a Population Proportion
Example




 Example
 Determine the sample size necessary to estimate the true
 proportion of college students with blue eyes, if the estimate is
 to have a margin of error of 0.02 with 90% confidence.




                    J. Robert Buchanan   Confidence Intervals about a Population Proportion
Example




 Example
 An automobile manufacturer purchases bolts from a supplier
 who claim that the bolts are approximately 5% defective.
 Determine the sample size necessary to estimate the true
 proportion of defective bolts if the margin of error is to be 0.01
 with 95% confidence.




                     J. Robert Buchanan   Confidence Intervals about a Population Proportion
Homework




    Read Section 9.3.
    Pages 441-443: 5–25 odd




                 J. Robert Buchanan   Confidence Intervals about a Population Proportion