Elementary Statistics (12th Edition)

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

Chapter 7 - Estimates and Sample Sizes - 7-4 Estimating a Population Standard Deviation or Variance - Basic Skills and Concepts: 12

Answer

a)σ is between 7.87 and 14.14. b)σ is between 8.87 and 15.92. c)There seems to be no difference in the variation.

Work Step by Step

a)$\alpha=1-0.99=0.01.$ By using the table we can find the critical chi-square values with with $df=sample \ size-1=40-1=39$. $X_{L}^2= X_{0.975}^2=20.707$ $ X_{R}^2= X_{0.025}^2=66.766$ Hence the confidence interval:$\sigma$ is between $\sqrt{\frac{(n-1)\cdot s^2}{ X_{R}^2}}=\sqrt{\frac{(39)\cdot 10.3^2}{66.766}}=7.87$ and $\sqrt{\frac{(n-1)\cdot s^2}{ X_{L}^2}}=\sqrt{\frac{(39)\cdot 10.3^2}{66.766}}=14.14.$ b) $\alpha=1-0.99=0.01.$ By using the table we can find the critical chi-square values with with $df=sample \ size-1=40-1=39$. $X_{L}^2= X_{0.975}^2=20.707$ $ X_{R}^2= X_{0.025}^2=66.766$ Hence the confidence interval:$\sigma$ is between $\sqrt{\frac{(n-1)\cdot s^2}{ X_{R}^2}}=\sqrt{\frac{(39)\cdot 11.6^2}{66.766}}=8.87$ and $\sqrt{\frac{(n-1)\cdot s^2}{ X_{L}^2}}=\sqrt{\frac{(39)\cdot 11.6^2}{66.766}}=15.92.$ c)There seems to be no difference in the variation because the intervals in a) and b) overlap.
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