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!

Ackh... ALL the answers are equally good in different ways! I think I need help from you voters!

Answer 1

It will turn out this is not always so. These critical value do come up a lot because they come from a standard normal distribution. For two tailed tests, the critical values are 1.645 for alpha = 0.1, 1.96 for alpha = 0.05, and 2.58 for alpha = 0.01.

However, if your test statistic has, say, a t distribution instead of a standard normal distribution, then these would not be the critical values. Then the critical value would depend on the degrees of freedom for that particular distribution. That is because the t has wider tails than the standard normal.

Critical values can also come from the chi-square or the F distributions, and there you can get other possibilities as well.

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.

Answer 3

Confidence Interval 1.96

Answer 4

1.96 Confidence Interval

Answer 5

Based on your critical values, that is how confident you are in the results.
1.645 is 90%
1.96 is 95%
2.58 is 99%
Level of Confidence c Critical Value
0.75, or 75% 1.15
0.80, or 80% 1.28
0.85, or 85% 1.44
0.90, or 90% 1.645
0.95, or 95% 1.96
0.98, or 98% 2.33
0.99, or 99% 2.58

Answer 6

The speed of sound is approx 700mph - the speed of light is way faster (light takes approx 8 mins to get from the sun to hear and thats millions and millions of equivalent miles - you could get your calculator out and work out how long sound would take but of course, sound doesn't travel in space due to the vacuum). My understanding of the theory of relativity (and it may not be entirely accurate) is that for a body to achieve the speed of light (or "infinite" speed) you would require "infinite" energy (i.e. all the energy in the universe) and therefore would require infinite mass (i.e. all the mass or matter in the universe). As you can imagine, for a layman such as myself, that is why it is easy to agree with Einstein, that speed of light travel is not possible. Of course, noone told also those light photons that go around! The theory also goes on about: what if you were on a train at the back of the carriage, travelling at the speed of light and then moved to the front, you would in fact be travelling at faster than light speeds (which Mr E stated is impossible). Lastly, Mr E bangs on about if you were on the train doing the Speed of Light (S.O.L.) your perception would be different to someone standing by the side of the "train track" - imagine you are looking at a tree; what you are actually seeing is the light (that took 8 mins to arrive from the sun) bounce of the tree to your eyes, which sends the electric signal to your brain which tells you you're seeing a tree. Apply that example to the bystander watching the S.O.L. train - what can you see when the train is moving at the same speed as the light that needs to bounce off the train to reach your eyes? I think this is why Mr E called it the theory of relativity because of this taking into account of the relative perceptions of the train passenger and the track bystander. As I said, this is my understanding and I could be way wrong - I would recommend you read "A Brief History of Time" by Stephen Hawkings, although I've read it 3 times (last time was about 5 years ago) and I still feel like a caveman grappling with a Nintendo DS! Good luck in your journey towards enlightenment - may the force be with you!