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 - Basic Skills and Concepts - Page 338: 18


a)0.82 b)p is between 0.82-0.0578=0.7622 and 0.82+0.0578=0.8778. c)The method is effective.

Work Step by Step

a) The best point estimate is equal to the proportion of the sample (x) divided by the sample size: $\hat{p}=\frac{x}{n}=\frac{239}{291}=0.82.$ b)$E=z_{\frac{\alpha}{2}}\cdot \sqrt{\frac{\hat{p}\cdot (1-\hat{p})}{n}}=2.575\cdot \sqrt{\frac{0.82\cdot (1-0.82)}{291}}=0.0578.$ Hence, the confidence interval: E is between $\hat{p}-E$ and $\hat{p}+E$, hence p is between 0.82-0.0578=0.7622 and 0.82+0.0578=0.8778. c) The method is effective, because without the method, the proportion would be approximately 50%, whilst here, according to b), 50% is much lower than the lower bound of the confidence interval.
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