sample standard deviation \(= 70\)

confidence interval \(= 99\%\)

sample size \(= 20\)

degree of freedom \(= 19\)

a)For \(99\%\) confidence interval,

The upper critical value has right tail area \(=\frac{1\ -\ 0.99}{2}=0.005\) and for 19 degree of freedom the critical value is 38.582

The lower critical value has right tail area \(=\frac{1\ +\ 0.99}{2}=0.995\) and for 19 degree of freedom the critical value is 6.844

The confidence interval for variance is calculated as shown below

Lower limit \(=\frac{(n\ -\ 1)s^{2}}{X_{0.005}^{2}}=\frac{(20\ -\ 1)70^{2}}{38.582}=2413.04235\)

Upper limit \(=\frac{(n\ -\ 1)s^{2}}{X_{0.095}^{2}}=\frac{(20\ -\ 1)70^{2}}{6.844}=13603.156\)

The \(99\%\) confidence interval for variance is

Lower limit \(= 2413.04\) (rounded to 2 decimals)

Upper limit \(= 13603.16\) (rounded to 2 decimals)

b)The \(99\%\) confidence interval of standard deviation is calculated by applying square root to

\(99\%\) confidence interval of variance.

The \(99\%\) confidence interval for population standard deviation is given below

Lower limit \(=\sqrt{\frac{(n\ -\ 1)s^{2}}{X_{0.005}^{2}}}=\sqrt{\frac{(20\ -\ 1)70^{2}}{38.582}}=\sqrt{2413.04235}=49.1227\)

Upper limit \(=\sqrt{\frac{(n\ -\ 1)s^{2}}{X_{0.095}^{2}}}=\sqrt{\frac{(20\ -\ 1)70^{2}}{6.844}}=\sqrt{13603.156}=116.6326\)

The \(99\%\) confidence interval for standard deviation is

Lower limit \(= 49.12\) (rounded to 2 decimals)

Upper limit \(= 116.63\) (rounded to 2 decimals)