A prior study estimated hat p as 34 percent The analysts would like to conduct a second study on the same topic with a margin of error, cap e , of 0 point 0 2 7 and a confidence level of 90 percent ( z to the th power times -score 1 point 6 4 5 ). What is the minimum sample size that should be used so the estimate of hat p will be within the required margin of error of the population proportion? n is equal to hat p times open paren 1 minus hat p close paren times open paren the fraction with numerator z to the th power times and denominator cap e close paren squared
Use the sample-size formula n = p̂(1 − p̂)(z/e)². Substituting p̂ = 0.34, z = 1.645, and e = 0.027 gives n ≈ 0.34(0.66)(1.645/0.027)² ≈ 832.95. Since the sample size must be a whole number large enough to meet the margin of error, round up to 833; the correct choice is 833.
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