When does it make sense to build a confidence interval for the mean with known standard deviation?

Question

While estimating with confidence interval the mean value for a population, there are two options:

1. If the standard deviation is known, and
2. If the standard deviation is unknown.

But in the first case where the standard deviation is known, the mean value should not also be known? Why do we need to estimate it, then? Because in order to compute the standard deviation, we already must have the mean... or is there any other way to know the standard deviation?

Answer 1

A couple of reasons:

1. We may be able to estimate the standard deviation without knowing the mean. For example, a random variable that takes only the values $0$ and $1$ has standard deviation at most $1/2$. This can be used to provide a conservative estimate for the confidence interval size.

2. In a textbook, this may be done just for the illustration of a mathematical technique.

Answer 2

Here is a real life example where $\mu$ is unknown and $\sigma$ known: We use a piece of lab equipment to test a small number portions of the same specimen.

In this case $\sigma$ is a property of the equipment, and likely known (printed in instruction manual, verified from extensive prior use). However, $\mu$ is a property of the specimen. The machine is known to produce normally distributed readings.

Say $n = 4,\,\sqrt{n} = 2.$ In this case there could be a big difference between the two 95% confidence intervals: $\bar X \pm 1.96\sigma/2$ and $\bar X \pm 3.18 S/2,$ where 3.18 is from t tables and $S$ is the (quite variable) sample standard deviation.

The figure below illustrates the considerable difference between the z-interval based on $\sigma = 10$ and the t-interval based on $S$. The 4-portion experiment was simulated 10,000 times. Each dot shows the $(\bar X, S)$ pair from one experiment. The z-interval covers $\mu = 100$ for dots between the green vertical lines. By contrast, the t-interval covers for dots in black (above the "V"). For each type of interval, about 95% of the 10,000 points correspond to CIs that cover $\mu,$ but there is a big difference in which experiments produce 'covering' intervals. In particular, the z-intervals cover when $\bar X$ is near $\mu$, while the t-intervals cover for a combination of small values of $|\bar X - \mu|$ and large values of $S$. All of the 95% z-CIs are of the same length; the t-CIs vary markedly in length (according to the random value of $S$).