document.write( "Question 1206969: Students at a certain school were​ surveyed, and it was estimated that 15​% of college students abstain from drinking alcohol. To estimate this proportion in your​ school, how large a random sample would you need to estimate it to within 0.06 with probability 0.99​, if before conducting the study​ (a) you are unwilling to predict the proportion value at your school and​ (b) you use the results from the surveyed school as a guideline. \n" ); document.write( "
Algebra.Com's Answer #844724 by Theo(13342)\"\" \"About 
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p = .15 = proportion that satisfy the requirement.
\n" ); document.write( "q = .85 = 1 - p = proportion that don't satisfy the requirement.
\n" ); document.write( "moe = .06 = margin of error.
\n" ); document.write( "ci = .99 = confidence interval = 99%.
\n" ); document.write( "s = sqrt(p * q / n) = standard error.
\n" ); document.write( "n = sample size.\r
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\n" ); document.write( "\n" ); document.write( "z = critical z-score at 99% confidence interval.
\n" ); document.write( "this leaves a tail outside the confidence interval called alpha.
\n" ); document.write( "if it's a two-tail confidence interval, than half of the tail is to the left of the confidence interval and half is to the right.
\n" ); document.write( "those tails are equal to half the alpha = .01 / 2 = .005.\r
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\n" ); document.write( "\n" ); document.write( "s = sqrt(p * q / n) = sqrt(.15 * .85 / n) = standard error.
\n" ); document.write( "you will need to solve for n.\r
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\n" ); document.write( "\n" ); document.write( "z = (x - m) / s = z-score formula.
\n" ); document.write( "at 99% two tail confidence interval, z = plus or minus 2.5758 rounded to 4 decimal places.
\n" ); document.write( "since the normal distribution is symmetric about the mean, you only have to solve for the high side margin of error.
\n" ); document.write( "the low side margin of error will be the same.\r
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\n" ); document.write( "\n" ); document.write( "margin of error = (x - m) when you use the critical z-score.
\n" ); document.write( "z = (x - m) / s becomes 2.5758 = .06 / sqrt(.15 * .85 / n)
\n" ); document.write( "solve for sqrt(.15 * .85 / n) to get:
\n" ); document.write( "sqrt(.15 * .85 / n) = .06 / 2.5758.
\n" ); document.write( "square both sides of this equation to get.
\n" ); document.write( ".15 * .85 / n = (.06 / 2.2758)^2.
\n" ); document.write( "solve for n to get:
\n" ); document.write( "n = (.15 * .85) / (.06 / 2.2758)^2 = 234.9805748.\r
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\n" ); document.write( "\n" ); document.write( "when n = 234.9805748, s becomes sqrt(.15 * .85 / 234.9805748) = .023293734 = standard of error.\r
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\n" ); document.write( "\n" ); document.write( "z-score formula of z = (x - m) / s becomes 2.5758 = (x - .15) / .023293734.
\n" ); document.write( ".15 is the population mean proportion that was given.
\n" ); document.write( "solve for x to get x = .21.
\n" ); document.write( "the margin of error = (x - m) = .21 - .15) = .06, as desired.\r
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\n" ); document.write( "\n" ); document.write( "the formula says that the sample size is 234.9805748.
\n" ); document.write( "this needs to be rounded to the next highest integer to get n = 235.
\n" ); document.write( "that's the smallest sample size that will get a margin of error less than .06.
\n" ); document.write( "that should be your solution.\r
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