Elementary Statistics (12th Edition)

Published by Pearson
ISBN 10: 0321836960
ISBN 13: 978-0-32183-696-0

Chapter 7 - Estimates and Sample Sizes - 7-2 Estimating a Population Proportion - Beyond the Basics: 39

Answer

a)178 b)176

Work Step by Step

a) By using the given formula with $\hat{p}\cdot\hat{q}=0.25.$ $n=\frac{N\cdot z_{\alpha/2}^2\cdot 0.25}{ z_{\alpha/2}^2\cdot 0.25+(N-1)\cdot E^2}=\frac{200\cdot 1.96^2\cdot 0.25}{1.96^2\cdot 0.25+(200-1)\cdot0.025^2}\approx178.$ b)By using the given formula with $\hat{q}=1-\hat{p}.$ $n=\frac{N\cdot z_{\alpha/2}^2\cdot \hat{p}\cdot (1-\hat{p})}{ z_{\alpha/2}^2\cdot \hat{p}\cdot (1-\hat{p})+(N-1)\cdot E^2}=\frac{200\cdot 1.96^2\cdot 0.38\cdot(1-0.38)}{1.96^2\cdot 0.38\cdot(1-0.38)+(200-1)\cdot0.025^2}\approx176.$
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