Jitter in a water power system. Jitter is a term used to describe the variation in conduction time of a water power system. Low throughput jitter is critical to successful waterline technology. An investigation of throughput jitter in the opening switch of a prototype system (Journal of Applied Physics) yielded the following descriptive statistics on conduction time for n = 18 trials:$\overline{x}=334.8$ nanoseconds, s = 6.3 nanoseconds. (Conduction time is defined as the length of time required for the downstream current to equal 10% of the upstream current.)

a. Construct a 95% confidence interval for the true standard deviation of conduction times of the prototype system.

b. Practically interpret the confidence interval, part a.

c. A system is considered to have low throughput jitter if the true conduction time standard deviation is less than 7 nanoseconds. Does the prototype system satisfy this requirement? Explain.

It is given that$n=18,\overline{x}=334.8,s=6.3$

a.

The 95% confidence interval can be calculated using the formula,

$\sqrt{\frac{\left(n-1\right)s^2}{{\chi }_{\frac{\alpha }{2}}^2}}⩽\sigma ⩽\sqrt{\frac{\left(n-1\right)s^2}{{\chi }_{\left(1-\frac{\alpha }{2}\right)}^2}}$

The degrees of freedom is 17, at 0.05 level of significance, from the table value we have

${\chi }_{0.025}^2=30.191$and${\chi }_{0.975}^2=7.564$

Substitute the values to get the required confidence interval.

$$\begin{gathered} \sqrt{\frac{\left(18-1\right){6.3}^2}{30.191}}⩽\sigma ⩽\sqrt{\frac{\left(18-1\right){6.3}^2}{7.564}} \\ 4.7274⩽\sigma ⩽9.4447 \end{gathered}$$

Therefore, the 95% confidence interval for$\sigma$ is (4.7274,9.4447).

b.

From the 95% confidence interval we can say that, we are 95% confidence that the population standard deviation of conduction times of the prototype system will lies between 4.7274 nanoseconds and 9.4447 nanoseconds.

c.

It is given that, a system is considered to have low throughout jitter if the true conduction time standard deviation is less than 7 thousands. This condition is satisfies by the given prototype system because then confidence interval is obtained for the population standard deviation is contains the given true population standard deviation.