The confidence interval for the standard deviation has bounds that can be obtained by taking the square root of the corresponding bounds of the confidence interval for the variance.
The given values are
\(\begin{array}{l}{\rm{s = 2}}{\rm{.81}}\\{{\rm{s}}^{\rm{2}}}{\rm{ = 7}}{\rm{.9}}\\{\rm{n - 1 = 9 - 1 = 8}}\end{array}\)
In order to obtain \({\rm{95\% Cl}}\)for \({\rm{\sigma and }}{{\rm{\sigma }}^{\rm{2}}}\), value \({\rm{\alpha }}\)can be found from
\({\rm{100(1 - \alpha ) = 95\alpha = 0}}{\rm{.05}}\)
Therefore,
\({\rm{1 - \alpha /2 = 1 - 0}}{\rm{.05/2 = 0}}{\rm{.975\chi }}_{{\rm{1 - \alpha /2,n - 1}}}^{\rm{2}}{\rm{ = \chi }}_{{\rm{0}}{\rm{.975,8}}}^{\rm{2}}\mathop {\rm{ = }}\limits^{{\rm{(1)}}} {\rm{2}}{\rm{.18}}\)
and also
\(\begin{array}{l}{\rm{\alpha /2 = 0}}{\rm{.05/2 = 0}}{\rm{.025}}\\{\rm{\chi }}_{{\rm{\alpha /2,n - 1}}}^{\rm{2}}{\rm{ = \chi }}_{{\rm{0}}{\rm{.025,8}}}^{\rm{2}}\mathop {\rm{ = }}\limits^{{\rm{(1)}}} {\rm{17}}{\rm{.543}}\end{array}\)
(1) : the values can be found at the end of the book at the appendix's table or can be computed by software.
Finally, the confidence interval for the variance can be computed using formula from the beginning as
\(\left( {\frac{{{\rm{(n - 1)}}{{\rm{s}}^{\rm{2}}}}}{{{\rm{\chi }}_{{\rm{\alpha /2,n - 1}}}^{\rm{2}}}}{\rm{,}}\frac{{{\rm{(n - 1)}}{{\rm{s}}^{\rm{2}}}}}{{{\rm{\chi }}_{{\rm{1 - \alpha /2,n - 1}}}^{\rm{2}}}}} \right){\rm{ = }}\left( {\frac{{{\rm{8 \times 7}}{\rm{.9}}}}{{{\rm{17}}{\rm{.543}}}}{\rm{,}}\frac{{{\rm{8 \times 7}}{\rm{.9}}}}{{{\rm{17}}{\rm{.543}}}}} \right){\rm{ = (3}}{\rm{.6,28}}{\rm{.98)}}{\rm{.}}\)
The confidence interval for the standard deviation is
\((\sqrt {3.6} ,\sqrt {28.98} ) = (1.9,5.38)\)
Explanations 
Textbooks 
Flashcards 
91影视 AI 
Notes 
Study Plans 
Study Sets 
Find a degree 
Magazine 
