2.14. Other types of confidence intervals

There are several other confidence intervals that you might come across in your career. We merely mention them here and don’t cover their derivation. What is important is that you understand how to interpret a confidence interval. Hopefully the previous discussion achieved that.

2.14.1. Confidence interval for the variance

This confidence interval finds a region in which the normal distribution’s variance parameter, \(\sigma\), lies. The range is obviously positive, since variance is a positive quantity. For reference, this range is:

\[\left[\frac{(n-1)S^2}{\chi^2_{n-1, \alpha/2}} \quad\text{to}\quad \frac{(n-1)S^2}{\chi^2_{n-1, 1-\alpha/2}} \right]\]
  • \(n\) is the number of samples

  • \(S^2\) is the sample variance

  • \(\chi^2_{n-1, \alpha/2}\) are values from the \(\chi^2\) distribution with \(n-1\) and \(\alpha/2\) degrees of freedom

  • \(1-\alpha\): is the level of confidence, usually 95%, so \(\alpha = 0.05\) in that case.

2.14.2. Confidence interval for the ratio of two variances

One way to test whether we can pool (combine) two variances, taken from two different normal distributions, is to construct the ratio: \(\dfrac{s^2_1}{s^2_2}\). We can construct a confidence interval, and if this interval contains the value of 1.0, then we have no evidence to presume they are different (i.e. we can assume the two population variances are similar).

\begin{alignat*}{4} F_{\alpha/2, \nu_1, \nu_2}\dfrac{s_2^2}{s_1^2} &\qquad<\qquad& \dfrac{\sigma_2^2}{\sigma_1^2} &\qquad<\qquad& F_{1-\alpha/2, \nu_1, \nu_2}\dfrac{s_2^2}{s_1^2} \end{alignat*}

where we use \(F_{\alpha/2, \nu_1, \nu_2}\) to mean the point along the cumulative \(F\)-distribution which has area of \(\alpha/2\) using \(\nu_1\) degrees of freedom for estimating \(s_1\) and \(\nu_2\) degrees of freedom for estimating \(s_2\). For example, in R, the value of \(F_{0.05/2, 10, 20}\) can be found from qf(0.025, 10, 20) as 0.2925. The point along the cumulative \(F\)-distribution which has area of \(1-\alpha/2\) is denoted as \(F_{1-\alpha/2, \nu_1, \nu_2}\), and \(\alpha\) is the level of confidence, usually \(\alpha = 0.05\) to denote a 95% confidence level.

2.14.3. Confidence interval for proportions: the binomial proportion confidence interval

Sometimes we measure the proportion of successes (passes). For example, if we take a sample of \(n\) independent items from our production line, and with an inspection system we can judge pass or failure. The proportion of passes is what is important, and we wish to construct a confidence region for the population proportion. This allows one to say the population proportion of passes lies between the given range. As in the proportion of packaged pizzas with 20 or more pepperoni slices is between 86 and 92%.

Incidentally, it is this confidence interval that is used in polls to judge the proportion of people that prefer a political party. One can run this confidence interval backwards and ask: how many independent people do I need to poll to achieve a population proportion that lies within a range of \(\pm 2\%\), 19 times out of 20? The answer actually is function of the poll result! But the worst case scenario is a split-poll, and that requires 2400 respondents.