document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1197972>Question 1197972</A>: <I><SPAN STYLE=\"word-break: break-all;\"> In preparing a report on the​ economy, we need to estimate the percentage of businesses that plan to hire additional employees in the next 60 days.<BR>\n" );
document.write( "​a) How many randomly selected employers must we contact in order to create an estimate in which we are 99​% confident with a margin of error of ​6%?<BR>\n" );
document.write( "​b) Suppose we want to reduce the margin of error to 4​%. What sample size will​ suffice?<BR>\n" );
document.write( "​c) Why might it not be worth the effort to try to get an interval with a margin of error of 1​%?<BR>\n" );
document.write( "a) A sample size of ? is needed.<BR>\n" );
document.write( "Part 2<BR>\n" );
document.write( "​b) A sample size of ? is needed.\r<BR>\n" );
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document.write( "Part (a)\r<BR>\n" );
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document.write( "At 99% confidence, the z critical value is roughly z = 2.576<BR>\n" );
document.write( "Use a table like this<BR>\n" );
document.write( "<a  rel=nofollow HREF= \"https://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf\">https://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf</a><BR>\n" );
document.write( "to get that value. Look at the bottom row labeled \"Z\" and above the 99% confidence level.\r<BR>\n" );
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document.write( "You can also use a stats calculator or spreadsheet to determine this z critical value.\r<BR>\n" );
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document.write( "The desired margin of error is 6%, which means we want E = 0.06 or smaller.\r<BR>\n" );
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document.write( "We're not told the value of phat, which is the sample proportion of businesses that plan to hire additional employees in the next 60 days. \r<BR>\n" );
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document.write( "Use phat = 0.5 as a conservative estimate. This is the default value of phat if none is stated.\r<BR>\n" );
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document.write( "n = min sample size<BR>\n" );
document.write( "n = phat*(1-phat)*(z/E)^2<BR>\n" );
document.write( "n = 0.5*(1-0.5)*(2.576/0.06)^2<BR>\n" );
document.write( "n = 460.817778  approximately<BR>\n" );
document.write( "n = 461  always round UP to the nearest whole number\r<BR>\n" );
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document.write( "Here's why we round up to the nearest whole number.<BR>\n" );
document.write( "Let's try n = 460 in the margin of error formula<BR>\n" );
document.write( "E = z*sqrt(phat*(1-phat)/n)<BR>\n" );
document.write( "E = 2.576*sqrt(0.5*(1-0.5)/460)<BR>\n" );
document.write( "E = 0.060053<BR>\n" );
document.write( "We're slightly over the 6% target.<BR>\n" );
document.write( "Now try n = 461<BR>\n" );
document.write( "E = z*sqrt(phat*(1-phat)/n)<BR>\n" );
document.write( "E = 2.576*sqrt(0.5*(1-0.5)/461)<BR>\n" );
document.write( "E = 0.059988<BR>\n" );
document.write( "Now the error is either 6% or less, which meets the goal we're after.<BR>\n" );
document.write( "This is why we round up to the nearest whole number for min sample size problems. This is to clear the hurdle needed.\r<BR>\n" );
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document.write( "Answer: <font color=red>461 employers</font>\r<BR>\n" );
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document.write( "Part (b)\r<BR>\n" );
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document.write( "Repeat the same set of steps as part (a), but this time we use E = 0.04<BR>\n" );
document.write( "Keep everything else the same.\r<BR>\n" );
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document.write( "n = phat*(1-phat)*(z/E)^2<BR>\n" );
document.write( "n = 0.5*(1-0.5)*(2.576/0.04)^2<BR>\n" );
document.write( "n = 1036.84<BR>\n" );
document.write( "n = 1037 \r<BR>\n" );
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document.write( "Answer: <font color=red>1037 employers</font>\r<BR>\n" );
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document.write( "Part (c)\r<BR>\n" );
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document.write( "Now use E = 0.01\r<BR>\n" );
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document.write( "n = phat*(1-phat)*(z/E)^2<BR>\n" );
document.write( "n = 0.5*(1-0.5)*(2.576/0.01)^2<BR>\n" );
document.write( "n = 16589.44 <BR>\n" );
document.write( "n = 16590\r<BR>\n" );
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document.write( "We need to sample a lot more employers at this point (more than ten times as much compared to the result of part (b)), so it's more practical to go with the 6% or 4% margin of error instead. \r<BR>\n" );
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document.write( "Answer: <font color=red>Minimum sample size gets way too large</font><BR>\n" );
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