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

See the explanation below.

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.