## Question 474:

1

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We need to work backwards from a Margin of Error to arrive at the sample size.

1. We're given a Margin of Error of +/- .02 or 2%.
2. The Margin of Error is made up of the Standard Error of the Mean times a critical value for the confidence level. In this case we're given a 95% Confidence Level. When we lookup .95 using the percentile to z-score calculator we get the z-critical value of 1.96
3. The standard error of the mean is made up of the standard deviation divided by the square root of the sample size. We don't have either here, but we can fill in some values. For proportional data, the largest sample size comes when the proportion is at 50%. We can plan the sample size for this worst case scenario.
4. If we assume the proportion to be .5, then the standard deviation is the square root of p * 1-p or SQRT(.5*.5), which turns out to be .5.
5. We setup an equation and solve for the unknown sample size.
1. ( .5/SQRT(n) ) *1.96 = .02
2.  .5/SQRT(n) = .0102
3. .5 = SQRT(n)*.0102
4. 48.98 = SQRT(n)
5. 2400 = n

So we'd need to plan on a sample size of 2400, assuming about an equal split between the preference for the grading system to achieve a margin of error of .02 with 95% confidence.

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