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: 18

Answer

σ is between 5 and 8.5.

Work Step by Step

The mean can be counted by summing all the data and dividing it by the number of data: $\frac{57+...+47}{38}=54.76.$ Standard deviation=$\sqrt{\frac{\sum (x-\mu)^2}{n-1}}=\sqrt{\frac{(57-54.76)^2+...+(46-54.76)^2}{37}}=6.5613.$ $\alpha=1-0.98=0.02.$ By using the table we can find the critical chi-square values with with $df=sample \ size-1=38-1=37$. $X_{L}^2= X_{0.95}^2=63.691$ $ X_{R}^2= X_{0.05}^2=12.592$ Hence the confidence interval:$\sigma$ is between $\sqrt{\frac{(n-1)\cdot s^2}{ X_{R}^2}}=\sqrt{\frac{(37)\cdot 6.5613^2}{63.691}}=5$ and $\sqrt{\frac{(n-1)\cdot s^2}{ X_{L}^2}}=\sqrt{\frac{(37)\cdot 6.5613^2}{12.592}}=8.5.$
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