Confidence Intervals for a Population Proportion by hkksew3563rd

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									Math 221: Confidence Intervals for a Population Proportion
S. K. Hyde
Chapter 19, (Moore, 5th Ed.)
The tobacco industry closely monitors all surveys that involve smoking. One survey showed that among
785 randomly selected subjects who completed four years of college, 144 smoke (based on data from the
American Medical Association).

  1. Large-Sample Confidence Interval for a Population Proportion
     An approximate level C confidence interval for p is

                                        x                                            p(1 − p)
                                                                                     ˆ     ˆ
            p±m
            ˆ          where       p=
                                   ˆ      ,     m = z ∗ SEp ,
                                                          ˆ       and       SEp =
                                                                              ˆ
                                        n                                               n
      Use this interval only when the number of successes and failures in the sample are both at
      least 15.
     Find a 99% confidence interval for the proportion of smokers among the population of people who
     have completed four years of college.




  2. The “Plus-Four” Confidence Interval for a Population Proportion
     A more accurate level C confidence interval for p is

                                       x+2                                            p(1 − p)
                                                                                      ˜     ˜
           p±m
           ˜          where       p=
                                  ˜        ,      m = z ∗ SEp ,
                                                            ˜      and      SEp =
                                                                              ˜
                                       n+4                                              n+4
      This is the same as the above method, with the addition of adding two imaginary successes
      and two imaginary failures (four overall) to your sample. Hence, the x + 2 and n + 4 in the
                    ˜
      definition of p. Use this interval when the confidence level is at least 90% and the sample size
      n is at least 10.
     Find a 99% “plus four” confidence interval for the proportion of smokers among the population of
     people who have completed four years of college.
                                             Confidence Interval for a Population Proportion, page 2


3. Sample Size for a Desired Margin of Error
   The level C confidence interval for a population proportion p will have margin of error approx-
   imately equal to a specified value m when the sample size is
                                             z∗   2
                                       n=             p∗ (1 − p∗ ),
                                             m
   where p∗ is a guessed value for the sample proportion. The margin of error will be less than
   or equal to m if you take the guess p∗ to be 0.5.
  Suppose the American Medical Association wants to determine the sample size needed to have a
  margin of error of ±1%?




4. Suppose the American Medical Association wants to determine the sample size needed to have a
   margin of error of ±1%? What would be the sample size required if you assume that an estimate
   for p is not known?
                                                        Confidence Interval for a Population Proportion, page 3


Solution
 1. Find a 99% confidence interval for the proportion of smokers among the population of people who
    have completed four years of college.
                               144
                         ˆ
   The estimate for p is p =   785   = 0.1834395. Thus, the standard error is

                                                              144
                                            p(1 − p)
                                            ˆ     ˆ           785   1 − 144
                                                                        785
                           SEp =
                             ˆ                       =                      = 0.01381357.
                                               n                    785
   Thus, the margin of error is m = z ∗ SEp = 2.576(0.01381357) = 0.03558374.
                                          ˆ

   So a 99% confidence interval for the proportion of smokers among the population of people who
   have completed four years of college is

                          p ± m =⇒ 0.1834395 ± 0.03558374 =⇒ (0.148, 0.219)
                          ˆ

   Note: When using a TI-83 or TI-84
   calculator, select STAT −→ TESTS −→ 1-PropZInt , and enter the appropriate data or statistics.


 2. Find a 99% “plus four” confidence interval for the proportion of smokers among the population of
    people who have completed four years of college.

   Here we repeat the above procedure, but use x + 2 for x and n + 4 for n.
                               144+2          146
                         ˜
   The estimate for p is p =   785+4      =   789   = 0.1850444. Thus, the standard error is

                                                              146
                                            p(1 − p)
                                            ˜     ˜           789   1 − 146
                                                                        789
                           SEp =
                             ˜                       =                      = 0.01382504.
                                              n+4                   789
   Thus, the margin of error is m = z ∗ SEp = 2.576(0.01382504) = 0.03561330.
                                          ˆ

   So a 99% confidence interval for the proportion of smokers among the population of people who
   have completed four years of college is

                          p ± m =⇒ 0.1850444 ± 0.03561330 =⇒ (0.149, 0.221)
                          ˆ

   To use the TI-83 or TI-84 calculator for the “plus four” intervals, select STAT −→ TESTS −→
    1-PropZInt , and you should enter x + 2 for “x” and n + 4 for “n”.


 3. Suppose the American Medical Association wants to determine the sample size needed to have a
    margin of error of ±1%? Here, we will choose the estimate for p to be p∗ = p = 144 = 0.1834395.
                                                                               ˆ 785
                                                                    2
                           z∗    2                         2.576        144          144
                     n=              p∗ (1 − p∗ ) =                             1−           = 9939.692
                           m                                .01         785          785

   The American Medical Association needs to have a sample size of 9,940 people to reduce the
   margin of error to ±1%.


 4. Suppose the American Medical Association wants to determine the sample size needed to have a
    margin of error of ±1%? Assume that an estimate for p is not known. Here, we will choose the
    estimate for p to be p∗ = p = 2 .
                              ˆ 1
                                                                        2
                                z∗    2                      2.576          1        1
                       n=                 p∗ (1 − p∗ ) =                        1−         = 16589.44
                                m                             .01           2        2

   If no estimate for p is known, then the American Medical Association needs to have a sample size
   of 16,590 people to reduce the margin of error to ±1%.

								
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