Which Method? Refer to Exercise 7 “Requirements” and assume that sampleof 12 voltage levels appears to be from a population with a distribution thatis substantially far from being normal. Should a 95% confidence intervalestimate of \(\sigma \)be constructed using the \({\chi ^2}\)distribution? If not, what othermethod could be used to find a 95% confidence interval estimate of\(\sigma \).

It is given that a sample of 12 voltage levels of smartphone batteries comes from a population that is normally distributed. A 95% confidence interval is to be constructed to estimate the standard deviation of voltage levels.

A strict requirement to compute the confidence interval estimate of\(\sigma \)using the\({\chi ^2}\)distribution is that the sample should be selected from a normally distributed population (even if the sample size is large).

As the sample of voltage levels does not come from a population that is normally distributed, the 95% confidence interval to estimate \(\sigma \) cannot be computed using the \({\chi ^2}\) distribution.

An alternate method that can be used to compute the confidence interval is discussed below:

• Obtain a set of 1000 or more bootstrap samples of size n=12 from the given sample.
• A bootstrap sample is a random sample obtained with the replacement of values from the given sample.
• Compute the sample standard deviation for each of the bootstrap samples.

\({P_{\frac{\alpha }{2} \times 100}} < \sigma < {P_{1 - \frac{\alpha }{2} \times 100}} = {P_{\frac{{0.05}}{2} \times 100}} < \sigma < {P_{1 - \frac{{0.05}}{2} \times 100}}\)

Thus, the limits are obtainedas follows:

\({P_{2.5}} < \sigma < {P_{97.5}}\)