document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=251235>Question 251235</A>: <I><SPAN STYLE=\"word-break: break-all;\"> A marketing advisory service wants to estimate the proportion of the population that responds favorably to a new product. It wants the estimate to be correct within 4% at the 95% confidence level. How big of a sample should they take?\r<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #184324 by </B><A HREF=/tutors/aboutme.mpl?userid=Theo><B>Theo(13342)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=Theo><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=Theo><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=184324\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> you need three pieces of information to determine that, based on the following link.\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"http://www.custominsight.com/articles/random-sampling.asp\" target = \"_blank\">http://www.custominsight.com/articles/random-sampling.asp</a>\r<BR>\n" );
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document.write( "those pieces of information are:\r<BR>\n" );
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document.write( "population size<BR>\n" );
document.write( "desired accuracy<BR>\n" );
document.write( "confidence level.\r<BR>\n" );
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document.write( "you have 2 out of the 3.\r<BR>\n" );
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document.write( "you are missing the population size.\r<BR>\n" );
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document.write( "it matters.\r<BR>\n" );
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document.write( "if a small population, then you will need a proportionally larger sample size to get the same accuracy.\r<BR>\n" );
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document.write( "as the population size gets larger, the percentage of the population that needs to be samples is relatively smaller.\r<BR>\n" );
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document.write( "example:\r<BR>\n" );
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document.write( "if the population is 1,000 people, then you need to get 375 people to provide data in order to get 4% accuracy at the 95% confidence level./\r<BR>\n" );
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document.write( "if the population is 10,000 people, then you need to get 566.\r<BR>\n" );
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document.write( "if the population is 100,000 people, then you need to get 597.\r<BR>\n" );
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document.write( "The percentage of the population required for 4% accuracy at 95% confidence level is progressively smaller as the population gets larger.\r<BR>\n" );
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document.write( "The above mentioned article doesn't give you the formula to calculate the right percentages, but it does provide a calculator that provides you with the answer.\r<BR>\n" );
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document.write( "the following link uses a similar calculator that provides the same answer.\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"http://www.surveysystem.com/sscalc.htm\" target = \"_blank\">http://www.surveysystem.com/sscalc.htm</a>\r<BR>\n" );
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document.write( "the following link shows the formula they use.\r<BR>\n" );
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document.write( "<a  rel=nofollow HREF= \"http://www.surveysystem.com/sample-size-formula.htm\" target = \"_blank\">http://www.surveysystem.com/sample-size-formula.htm</a>\r<BR>\n" );
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document.write( "there is an additional factor called percent.   the default value for this is .5 (50%).   this would be a worst case scenario where only 50% of the respondents answer yes to a question and 50% of the respondents answer no.   if all answered yes, then the sample size could be smaller.  the safe bet is to assume the worst case which is why 50% is the default percent.\r<BR>\n" );
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document.write( "An example of how the formula works is as follows:\r<BR>\n" );
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document.write( "The formula is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pss+=+%28z%5E2+%2A+p+%2A+%281-p%29%29%2Fc%5E2 ALIGN=MIDDLE  ALT=\"pss+=+%28z%5E2+%2A+p+%2A+%281-p%29%29%2Fc%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "pss = preliminary sample size.   there is an adjustment to this value shown below which will result in fss (final sample size).\r<BR>\n" );
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document.write( "p = proportion of the sample respondents that picked the desired answer.   the worst case is .5 and that is the typical default value used.   a value of .5 requires the largest sample size for a desired accuracy level.\r<BR>\n" );
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document.write( "c = confidence interval which is the desired accuracy level you want to achieve.  if you want your answer to be +/- 4%, you would make c = .04\r<BR>\n" );
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document.write( "z is the z-score.  for 95% accuracy, a z-score of 1.96 would be required.  this is rounded to the nearest one hundredth.\r<BR>\n" );
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document.write( "example:\r<BR>\n" );
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document.write( "the following statement is extracted from above.\r<BR>\n" );
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document.write( "\"if the population is 100,000 people, then you need to get 597.\"\r<BR>\n" );
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document.write( "this means that a sample size of 597 would be required for a population of 100,000 at a 95% confidence level with a 4% confidence interval and a default worst case percent of 50%.\r<BR>\n" );
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document.write( "the calculator provided that answer.\r<BR>\n" );
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document.write( "the formula that the calcultor used is:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pss+=+%28z%5E2+%2A+p+%2A+%281-p%29%29%2Fc%5E2 ALIGN=MIDDLE  ALT=\"pss+=+%28z%5E2+%2A+p+%2A+%281-p%29%29%2Fc%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "pss is the preliminary value for what we want to find (sample size)<BR>\n" );
document.write( "z = 1.96 (95% confidence level)<BR>\n" );
document.write( "p = .5 (default value for worst case percent)<BR>\n" );
document.write( "c = .04 (confidence interval)\r<BR>\n" );
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document.write( "the formula becomes:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=pss+=+%28%281.96%29%5E2+%2A+.5+%2A+.5%29%2F%28.04%29%5E2 ALIGN=MIDDLE  ALT=\"pss+=+%28%281.96%29%5E2+%2A+.5+%2A+.5%29%2F%28.04%29%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "this becomes pss = 600.25\r<BR>\n" );
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document.write( "there is an adjustment to this answer for population size based on the following formula:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=fss+=+%28pss%2F%281%2B%28%28pss-1%29%2Fps%29%29%29 ALIGN=MIDDLE  ALT=\"fss+=+%28pss%2F%281%2B%28%28pss-1%29%2Fps%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "fss is the sample size we are looking for.<BR>\n" );
document.write( "pss is the preliminary sample size we derived above.<BR>\n" );
document.write( "ps is the population size.\r<BR>\n" );
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document.write( "in our example, we have the following values for this formula.\r<BR>\n" );
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document.write( "fss = what we want to find.<BR>\n" );
document.write( "pss = 600.25<BR>\n" );
document.write( "ps = 100,000\r<BR>\n" );
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document.write( "the formula becomes:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=fss+=+%28600.25%2F%281%2B%28%28600.25-1%29%2F100000%29%29%29 ALIGN=MIDDLE  ALT=\"fss+=+%28600.25%2F%281%2B%28%28600.25-1%29%2F100000%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "this becomes:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=fss+=+%28600.25%2F1.0059925%29 ALIGN=MIDDLE  ALT=\"fss+=+%28600.25%2F1.0059925%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "this becomes fss = 596.6744285 which rounds to fss = 597.\r<BR>\n" );
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document.write( "we set sample size equal to fss and we get:\r<BR>\n" );
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document.write( "sample size = 597 for a population of 100,000 with a desired confidence interval of 4% at a 95% confidence level with a default percent of 50%.\r<BR>\n" );
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document.write( "the question you asked can't be answered because you are missing a very important ingredient.  that's the sample size.\r<BR>\n" );
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document.write( "include that and you should be able to find the required sample size using the information and tools provided in the references.\r<BR>\n" );
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