Why are critical values nearly always 2.58, 1.96 or 1.645? Explain in layman's terms please! (Read details)

Question

I'm asking for layman's terms (Although you can include formulae as well if necessary - they justy need to be translated into words) because I'm a first-year university student and I'm only covering business statistics over one semester.

I understand that critical values are worked out using probability distributions, but in two tailed tests, the critical value - in my experience - is always 1.96 or 1.645.

Why is this so, and for the mean, what is the critical value there? Unless this is true, don't tell me it depends on question and statistics bering worked out - I'm talking about in terms of percentage here.

I know it's badly worded, but I've no idea how else to put it!! PLEASE help as best you can!

Answer 1

The critical value is related to the confidence level.
If there is a 95% confidence level ,the critical value is 1.96
90% confidence level the critical value is 1.645
99% confidence level the critical value is 2.575
These are the most common confidence levels used in stats, which is why your critical values are going to be the same.
The critical value is the z-value found for the shaded in region on your normal distribution graph.

For an explanation: http://www.isixsigma.com/library/content/c000709a.asp

A 95% degree confidence corresponds to = 0.05. Each of the shaded tails in the following figure has an area of = 0.025. The region to the left of and to the right of = 0 is 0.5 – 0.025, or 0.475. In the Table of the Standard Normal () Distribution, an area of 0.475 corresponds to a value of 1.96. The critical value is therefore = 1.96.

Answer 2

Ok, simple answer to your question. When you have a z-distribution (Normal distribution), the critical values 1.96 and 1.645 are the ones associated with 90% and 95% confidence intervals. 90% and 95% confidence intervals are the ones that get used most often as they strike a nice balance between confidence and accuracy. For example, a 99.999% confidence interval sounds like it should be the best, but in order to get a confidence that high, we need to have a very wide interval. To say that we are 99.999% confident that the mean is in the interval (-100, 100) is pretty well useless, but to say that we are 90% confident that it is in (- 5, 5) is much better. On the other end of things, we could have a very small interval, but the confidence would be small as well. A confidence interval of (-0.5, 0.5) sounds pretty good, but if we are just 65% confident that our mean really is in there, then it isn't so good. Again, those numbers come up in the Normal distribution because the 90% and 95% confidence intervals strike a nice balance.