# Why is a 90% confidence interval narrower than a 95% confidence interval?

Nov 3, 2015

See the explanation below.

#### Explanation:

The answer is with particular reference to the Normal distribution.

A test procedure starts with fixing the level of significance at 5% or 10% or 1% and so on.
When we say that level of significance is 5%, we admit that our results are likely to be erroneous in 5% cases. That is why a test procedure with 5% level of significance reflects a 95% confidence interval.
From the area table of a standard normal curve, we find that Pr [$\mu$ - $\sigma$ < x < $\mu$ + $\sigma$} is about .30
Pr[$\mu$ - 2$\sigma$ < x < $\mu$ + 2$\sigma$ ] is about 0.68
Pr[$\mu$-3$\sigma$ < x < $\mu$ + 3$\sigma$] is about 0.95 and so on.
Here we see that as the probability on the right hand side increases, the interval widens and as it decreases, the interval narrows down.
. Hence the 90% confidence interval is narrower than 95% confidence interval.