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Determining the Size of a Sample by rt3463df

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									Determining the Size of
      a Sample
         Sample Accuracy
 • Sample accuracy: refers to how close
   a random sample‟s statistic is to the
   true population‟s value it represents
 • Important points:
    – Sample size is not related to
      representativeness
    – Sample size is related to accuracy

Ch 13                                  2
        Sample Size and Accuracy
 • Intuition: Which is more accurate: a
   large probability sample or a small
   probability sample?
 • The larger a probability sample is, the
   more accurate it is (less sample
   error).



Ch 13                                    3
         A Picture Says 1,000 Words
  ±

                                  Sample Size and Accuracy

               16%
               14%                                      n 550 - 2000 = 1,450
               12%
                                                         4% - 2% = ±2%
    Accuracy




               10%
                8%
                6%
                4%
                2%
                0%



                                                              1100

                                                                     1250

                                                                            1400

                                                                                   1550

                                                                                          1700

                                                                                                 1850

                                                                                                        2000
                     50
                          200

                                350

                                      500

                                            650

                                                  800

                                                        950

                                                        Sample Size


               Probability sample accuracy (error) can be calculated with a
Ch 13          simple formula, and expressed as a ± % number.                                              4
        How to Interpret Sample
              Accuracy
 • From a report…
   – The sample is accurate ± 7% at
     the 95% level of confidence…
 • From a news article
   – The accuracy of this survey is ±
     7%…



Ch 13                                   5
        How to Interpret Sample
              Accuracy
 • Interpretation
    – Finding: 60% are aware of our brand
    – So… between 53% (60%-7%) and
      67% (60%+7%) of the entire
      population is aware of our brand




Ch 13                                  6
        Sample Size Axioms
 • To properly understand how to
   determine sample size, it helps to
   understand the following axioms…




Ch 13                                   7
        Sample Size Axioms
 • The only perfectly accurate sample is a
   census.
 • A probability sample will always have
   some inaccuracy (sample error).
 • The larger a probability sample is, the
   more accurate it is (less sample error).
 • Probability sample accuracy (error) can
   be calculated with a simple formula,
   and expressed as a +- % number.
Ch 13                                   8
        Sample Size Axioms
 • You can take any finding in the
      survey, replicate the survey with the
      same probability sample size, and
      you will be “very likely” to find the
      same finding within the +- range of
      the original finding.
 • In almost all cases, the accuracy
      (sample error) of a probability sample
      is independent of the size of the
Ch 13 population.                           9
        Sample Size Axioms
 • A probability sample can be a very
   tiny percentage of the population size
   and still be very accurate (have little
   sample error).




Ch 13                                    10
               Sample Size and
               Population Size
 • Where is N (size of the population) in
   the sample size determination formula?
 Population     e=±3% Sample            e=±4% Sample
 Size           Size                    Size
 10,000                 ____
                        1,067                  ____
                                                600
 100,000                ____
                        1,067                  ____
                                                600
 1,000,000              ____
                        1,067                  ____
                                                600
 100,000,00               ____
                          1,067                 ____
                                                 600
 0
   In almost all cases, the accuracy (sample error) of a
Ch probability sample is independent of the size of the
   13                                                      11
   population.
        Sample Size Axiom
 • The size of the probability sample
   depends on the client‟s desired
   accuracy (acceptable sample error)
   balanced against the cost of data
   collection for that sample size.




Ch 13                                   12
            Putting It All Together
 •      MR – What level accuracy do you want?
 •      MM – I don’t have a clue. of a probability
                            The size
 •      MR – National opinion polls use 3.5%.
                            sample depends on the client’s
 •      MM – Sounds good to me.
                            desired accuracy (acceptable
                            sample error) balanced against
 •      MR – Okay, that meanscost ofneed a sample of
                            the we data collection for
        1,200.              that sample size.
 •      MM – Gee Whiz. That small?
 •      MR – Yup, and at a cost of $20 per completion, it
        will be $24,000.
 •      MM – Holy Cow! That much?
 •      MR – I could do 500 for $10,000, and that would
        be 4.4% accurate, or 300 for $6,000 at 5.7%.
 •      MM – 500 sounds good to me.
Ch 13                                                  13
 • There is only one method of
   determining sample size that allows
   the researcher to PREDETERMINE
   the accuracy of the sample results…

          The Confidence
         Interval Method of
        Determining Sample
                Size
Ch 13                                    14
  The Confidence Interval Method
    of Determining Sample Size
 • This method is based upon the
   Confidence Interval and the Central
   Limit Theorem…
 • Confidence interval: range whose
   endpoints define a certain percentage
   of the response to a question



Ch 13                                 15
  The Confidence Interval Method
    of Determining Sample Size
 • Confidence interval approach: applies
   the concepts of accuracy, variability,
   and confidence interval to create a
   “correct” sample size
 • Two types of error:
    – Nonsampling error: pertains to all
      sources of error other than sample
      selection method and sample size
    – Sampling error: involves sample
      selection and sample size
Ch 13                                  16
  The Confidence Interval Method
    of Determining Sample Size
 • Sample error formula:




Ch 13                         17
  The Confidence Interval Method
    of Determining Sample Size
 • The relationship between sample size
   and sample error:




Ch 13                                18
        Computations Help Page
1.96
                   pq         50 times 50
        ez
                   n
 Let’s try 3 n’s
        1000
                        Answers this way…
        500
Ch 13                                       19
        100
        And the answers are…
1.96
                    pq     50 times 50
        ez
                    n
 Let’s try 3 n’s
        1000       ±3.1%
        500        ±4.4%
Ch 13                                    20
        100        ±9.8%
        Review: What does sample
             accuracy mean?
 • 95% Accuracy
    – Calculate your sample‟s finding, p%
    – Calculate your sample‟s accuracy, ±
      e%
    – You will be 95% confident that the
      population percentage (π) lies
      between p% ± e%

Ch 13                                  21
        Review: What does sample
             accuracy mean?
 • Example
   – Sample size of 1,000
   – Finding: 40% of respondents like
     our brand
   – Sample accuracy is ± 3% (via our
     formula)
   – So 37% - 43% like our brand

Ch 13                                   22
  The Confidence Interval Method
    of Determining Sample Size
 • Variability: refers to how similar or
   dissimilar responses are to a given
   question
 • P: percent
 • Q: 100%-P
 • Important point: the more variability in
   the population being studied, the
   higher the sample size needed to
   achieve a stated level of accuracy.
Ch 13                                    23
 • With nominal data (i.e. yes, no), we
   can conceptualize variability with bar
   charts…the highest variability is
   50/50




Ch 13                                       24
   Confidence Interval Approach
 • The confidence interval approach is
   based upon the normal curve
   distribution.
 • We can use the normal distribution
   because of the CENTRAL LIMITS
   THEOREM…regardless of the shape
   of the population‟s distribution, the
   distribution of samples (of n at least
   =30) drawn from that population will
   form a normal distribution.
Ch 13                                   25
        Central Limits Theorem
 • The central limits theorem allows us
   to use the logic of the normal curve
   distribution.
 • Since 95% of samples drawn from a
   population will fall + or – 1.96 x
   sample error (this logic is based upon
   our understanding of the normal
   curve) we can make the following
   statement…

Ch 13                                   26
 • If we conducted our study over and
   over, 1,000 times, we would expect
   our result to fall within a known
   range. Based upon this, we say that
   we are 95% confident that the true
   population range value falls within
   this range.




Ch 13                                    27
  The Confidence Interval Method
    of Determining Sample Size




 • 1.96 x s.d. defines the endpoints of
   the distribution.
Ch 13                                     28
 • We also know that, given the amount
   of variability in the population, the
   sample size will affect the size of the
   confidence interval.




Ch 13                                    29
   So, what have we learned thus
               far?
 • There is a relationship between:
        – The level of confidence we wish to
          have that our results would be repeated
          within some known range if we were to
          conduct the study again, and…
        – Variability in the population and…
        – The amount of acceptable sample error
          (desired accuracy) we wish to have
          and…
Ch 13   – The size of the sample!             30
           Sample Size Formula
 • Fortunately, statisticians have given us a
   formula which is based upon these
   relationships.
    – The formula requires that we
          • Specify the amount of confidence we wish
          • Estimate the variance in the population
          • Specify the amount of desired accuracy
            we want.
        – When we specify the above, the formula
          tells us what sample we need to use…n
Ch 13                                                  31
        Sample Size Formula
 • Standard sample size formula for
   estimating a percentage:




Ch 13                                 32
        Practical Considerations in
        Sample Size Determination
 • How to estimate variability (p times q)
   in the population
    – Expect the worst cast (p=50; q=50)
    – Estimate variability: Previous
      studies? Conduct a pilot study?




Ch 13                                    33
   Practical Considerations in
   Sample Size Determination
 • How to determine the amount of
      desired sample error
       – Researchers should work with
         managers to make this decision.
         How much error is the manager
         willing to tolerate?
       – Convention is + or – 5%
       – The more important the decision,
         the more (smaller number) the
Ch 13                                       34
         sample error.
   Practical Considerations in
   Sample Size Determination
 • How to decide on the level of
      confidence desired
       – Researchers should work with
         managers to make this decision.
         The more confidence, the larger
         the sample size.
       – Convention is 95% (z=1.96)
       – The more important the decision,
         the more likely the manager will
         want more confidence. 99%
Ch 13                                       35
         confidence, z=2.58.
                      Example
 Estimating a Percentage in the Population
 • What is the required sample size?
        – Five years ago a survey showed that 42%
          of consumers were aware of the
          company‟s brand (Consumers were either
          “aware” or “not aware”)
        – After an intense ad campaign,
          management wants to conduct another
          survey and they want to be 95% confident
          that the survey estimate will be within
          ±5% of the true percentage of “aware”
          consumers in the population.
        – What is n?
Ch 13                                          36
          Estimating a Percentage:
                What is n?


 •      Z=1.96 (95% confidence)
 •      p=42
 •      q=100-p=58
 •      e=5
 •      What is n?

Ch 13                                37
         Estimating a Percentage:
               What is n?
                      n=374
 • What does this mean?
        – It means that if we use a sample size of
          374, after the survey, we can say the
          following of the results: (assume results
          show that 55% are aware)
        – “Our most likely estimate of the
          percentage of consumers that are „aware‟
          of our brand name is 55%. In addition, we
          are 95% confident that the true
          percentage of „aware‟ customers in the
Ch 13     population falls between 50% and 60%.”   38
          Estimating a Mean
 • Estimating a mean requires a
   different formula (See MRI 13.2, p.
   378)
   • Z is determined the same way (1.96 or
      2.58)
   • E is expressed in terms of the units we are
      estimating (i.e., if we are measuring
      attitudes on a 1-7 scale, we may want
      error to be no more than ± .5 scale units
   • S is a little more difficult to estimate…
Ch 13                                           39
            Estimating s
 • Since we are estimating a mean, we
   can assume that our data are either
   interval or ratio. When we have
   interval or ratio data, the standard
   deviation, s, may be used as a
   measure of variance.



Ch 13                                     40
             Estimating s
 • How to estimate s?
      – Use standard deviation from a
        previous study on the target
        population.
      – Conduct a pilot study of a few
        members of the target population and
        calculate s.
      – Estimate the range the value you are
        estimating can take on (minimum and
        maximum value) and divide the range
Ch 13                                     41
        by 6.
                  Estimating s
        – Why divide the range by 6?
          • The range covers the entire
            distribution and ± 3 (or 6) standard
            deviations cover 99.9% of the area
            under the normal curve. Since we
            are estimating one standard
            deviation, we divide the range by 6.



Ch 13                                              42
                       Example
   Estimating the Mean of a Population
 • What is the required sample size?
        – Management wants to know customers‟
          level of satisfaction with their service.
          They propose conducting a survey and
          asking for satisfaction on a scale from 1
          to 10. (since there are 10 possible
          answers, the range=10).
        – Management wants to be 99% confident
          in the results and they do not want the
          allowed error to be more than ±.5 scale
          points.
Ch 13   – What is n?                                43
             Estimating a Mean:
                 What is n?

 •      S=10/6 or 1.7
 •      Z=2.58 (99% confidence)
 •      e=.5 scale points
 •      What is n?



Ch 13                             44
              Estimating a Mean:
                  What is n?
                       n=77
 • What does this mean?
        – After the survey, management may make
          the following statement: (assume
          satisfaction mean is 7.3)
        – “Our most likely estimate of the level of
          consumer satisfaction is 7.3 on a 10-point
          scale. In addition, we are 99% confident
          that the true level of satisfaction in our
          consumer population falls between 6.8
Ch 13     and 7.8 on a 10-point scale”               45
  Other Methods of Sample Size
         Determination
 • Arbitrary “percentage of thumb”
   sample size:
    – Arbitrary sample size approaches
      rely on erroneous rules of thumb.
    – Arbitrary sample sizes are simple
      and easy to apply, but they are
      neither efficient nor economical.


Ch 13                                     46
 Other Methods of Sample Size
        Determination
 • Conventional sample size specification:
      – Conventional approach follows some
        convention: or number believed
        somehow to be the right sample size.
      – Using conventional sample size can
        result in a sample that may be too
        large or too small.
      – Conventional sample sizes ignore the
        special circumstances of the survey
Ch 13                                      47
        at hand.
  Other Methods of Sample Size
         Determination
 • Statistical analysis requirements of
   sample size specification:
    – Sometimes the researcher‟s desire
      to use particular statistical technique
      influences sample size




Ch 13                                     48
 Other Methods of Sample Size
        Determination
 • Cost basis of sample size
      specification:
       – “All you can afford” method
       – Instead of the value of the
         information to be gained from the
         survey being primary consideration
         in the sample size, the sample size
         is determined by budget factors that
         usually ignore the value of the
Ch 13
         survey‟s results to management. 49
         Special Sample Size
        Determination Situations
 • Sampling from small populations:
   – Small population: sample exceeds
     5% of total population size
   – Finite multiplier: adjustment factor for
     sample size formula
   – Appropriate use of the finite
     multiplier formula will reduce a
     calculated sample size and save
     money when performing research on
     small populations.
Ch 13                                     50
         Special Sample Size
        Determination Situations
 • Sample size using nonprobability
   sampling:
    – When using nonprobability
      sampling, sample size is
      unrelated to accuracy, so cost-
      benefit considerations must be
      used.


Ch 13                                   51
        Practice Examples
 • We will do some examples from the
   questions and exercises at the end of
   the chapter on sample size…question
   5 on page 386.




Ch 13                                 52
          Practice Examples
 • 5a. Using the formula provided in
   your text, determine the approximate
   sample sizes for each of the following
   cases, all with precision (allowable
   error) of ±5%:              n= z
                                        2
                                      (pq)
                                                2
                                      e
    – Variability of 30%,   =
                                    2
                              1. 96 (30 x 70)
                                            2
                                     5
      confidence level         3.84 x 2100
                             =
      of 95%                        25
                                   8064
                                =
                                    25
Ch 13                         = 322.6 (323)         53
          Practice Examples
 • 5b. Using the formula provided in
   your text, determine the approximate
   sample sizes for each of the following
   cases, all with precision (allowable
   error) of ±5%:              n= z
                                        2
                                      (pq)
                                                2

    – Variability of 60%,
                                      e
                                    2
                              2. 58 (60 x 40)
                            =
      confidence level               5
                               6.66 x 2400
                                            2



                             =
      of 99%                        25
                                  15,984
                                =
                                    25
Ch 13                         = 639.4 (639)         54
          Practice Examples
 • 5c. Using the formula provided in
   your text, determine the approximate
   sample sizes for each of the following
   cases, all with precision (allowable
   error) of ±5%:                n= z
                                       (pq)2

                                                   2

    – Unknown variability,             e
                                       2
                                1.96 (50 x 50)
                              =
      confidence level                5
                                               2


                                 3.84 x 2500
      of 95%                   =
                                     25
                                    9600
                                   =
                                      25
Ch 13                                                  55
                                   = 384
         Practice Example
 • A client wants to survey out-shopping
   intentions (percentage of people
   saying “yes” to a question regarding
   their intentions to out-shop) among
   heads of households in Antigonish.
   The client wants a ± 3%, 19 times out
   of 20. There are 3,000 households in
   the catchment area. What sample
   size should be used?
Ch 13                                  56
             Continued
 • If you expect an incidence rate of
   80% and a refusal rate of 50%, how
   many surveys should be sent out?




Ch 13                                   57

								
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