I have a question about confidence interval width.
I have a normal distribution and depending on whether the standard deviation is known or unknown, the confidence interval will be different. I would like to know which interval will be shorter when a number of samples are drawn from this normal distribution.
I think is the one with unknown standard deviation - is that correct? can somebody explain why?
Population SD known. If $X_1, X_2, \dots, X_n$ is a random sample from a normal distribution with unknown mean $\mu$ and known standard deviation $\sigma,$ then a 95% confidence interval (CI) for $\mu$ is $$\bar X \pm 1.96 \sigma\,/\sqrt{n},$$ where $\bar X = \frac{1}{n}\sum_i X_i$ is the sample mean. The width of this CI is a fixed value $2(1.96 \sigma\,/\sqrt{n}).$
Population SD unknown. If $X_1, X_2, \dots, X_n$ is a random sample from a normal distribution with unknown mean $\mu$ and unknown standard deviation $\sigma,$ then a 95% confidence interval (CI) for $\mu$ is $$\bar X \pm t^* S\,/\sqrt{n},$$ where $\bar X = \frac{1}{n}\sum_i X_i$ is the sample mean, $S = \sqrt{\frac{1}{n-1}\sum_i(X_i - \bar X)^2}$ is the sample standard deviation, and $t^*$ cuts 2.5% of the probability from the upper tail of Student's t distribution with $n - 1$ degrees of freedom. The width of this CI is a random variable because $S$ is a random variable.
On average, the CI is longer in the latter case when $\sigma$ is unknown and estimated by $S$ because $t^* > 1.96.$
As $n$ becomes large $t^*$ approaches 1.96 and $S$ becomes an increasing accurate estimate of $\sigma.$ So for large $n$ (say roughly, $n > 100$) there is very little difference in the lengths of the two kinds of intervals. Intuitively, this makes sense because for large $n$ the population standard deviation $\sigma$ is "almost" known.
Figure for 70,000 normal samples. In the figure below, $S$ is plotted against $\bar X$ for each of 70,000 random samples of size $n = 5$ from $\mathsf{Norm}(\mu=100,\,\sigma=15).$ There are 70,000 dots.
When $\sigma = 15$ is known, the horizontal distance between the vertical red lines is the width of the first kind of CI discussed above. About 5% of the dots (samples) are outside the red lines (about 3500 of them). These represent samples for which the CI does not cover the true value of $\mu.$
When $\sigma$ is unknown, the width of the CI for a particular value of $S$ is the width of the green region at $S.$ Thus, when $S = 15$ (horizontal blue line), one can see that the second kind of CI is a little wider than for the first. Here it is the black points (once again, about 3500 of them) that represent samples for which the CI does not cover the true value of $\mu.$