Question

How large a sample must a pollster take in order to estimate with 95% confidence and to within 3 percentage points, the proportion of voters who are in favor of a certain measure

Answer 1

Answer:   1068

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Work Shown:

$n = \hat{p}*(1-\hat{p})\left(\frac{z}{E}\right)^2\\\\n \approx 0.5*(1-0.5)\left(\frac{1.96}{0.03}\right)^2\\\\n \approx 1067.111\\\\n \approx \boldsymbol{1068}$

Notes:

• At 95% confidence, the z critical value is roughly z = 1.96 which is determined using a Z table.
• E = 0.03 to represent the 3% error.
• We're not told the value of $\hat{p}$, so we assume the most conservative estimate of 0.5
• Always round up to the nearest integer. The value 1067.111 is closer to 1067, but we round up to 1068 to clear the hurdle needed.