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: 40

Answer

p is between 0.375-0.3355=0.0395 and 0.375+0.3355=0.7105. The results are not that close.

Work Step by Step

The best point estimate is equal to the proportion of the sample (x) divided by the sample size: $\hat{p}=\frac{x}{n}=\frac{3}{8}=0.375.$ $E=z_{\frac{\alpha}{2}}\cdot \sqrt{\frac{\hat{p}\cdot (1-\hat{p})}{n}}=1.96\cdot \sqrt{\frac{0.375\cdot (1-0.375)}{8}}=0.3355.$ Hence, the confidence interval: E is between $\hat{p}-E$ and $\hat{p}+E$, hence p is between 0.375-0.3355=0.0395 and 0.375+0.3355=0.7105. The results are not that close.
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