Suppose Z has a standard normal distribution with a mean of 0 and standard deviation of 1. So 96% of the possible Z values are between ________ and ________ (symmetrically distributed about the mean).
2007-06-27 06:38:55 · 4 answers · asked by LG 1 in Science & Mathematics ➔ Mathematics
-2 and +2, approximately.
2 std devs either way of the mean.
2007-06-27 06:43:35 · answer #1 · answered by Dr D 7 · 1⤊ 0⤋
If you look up a standard normal distribution table (with mean 0, standard deviation = 1), or use the NORMSINV table in Excel, it will give you the cumulative distribution from -infinity to that value. Since the normal distribution is symmetrical, you want 2% area on each side. Looking up 2% (and 1-2%=98%) in the table gives Z-values -2.054 and +2.054.
2007-06-27 13:47:44 · answer #2 · answered by Vince 2 · 1⤊ 0⤋
A standard normal is one where P(-3z <= x <= 3z) ~ 1.000; the probability of a random value x(z) falls between or on plus and minus three standard deviations (z) is almost 1.000. You should keep in mind what the probability of x falling between 2z and 1z is also...it'll be helpful for checking your work.
P(-3z <= x <= 3z) ~ .999 (almost one)
P(-2z <= x <= 2z) ~ .95
P(-1z <= x <= 1z) ~ .68
These are really good relationships to memorize. For example, in your problem you are asking P(-Nz <= x <= Nz) = .96 and what's Nz equal to? Well, from above we can see that Nz ~ 2z because .96 ~ .95; so as a sanity check, if you come up with something afar from 2z, you know your answer is wrong.
NB: That 96% you cite could very well be a confidence interval...contrary to what someone wrote below. But, and this is a BIG BUT, it's not very likely. CI's are usually given as 95% or 90% confident when doing hypothesis tests. That's because z tables, t tables, and such provide the .90 and .95 data and not data for something like .96. Even so, 96% represents an interval even though it's not usually cited as a confindence interval. And through interpolation, one can find the z values for 96%.
2007-06-27 14:05:00 · answer #3 · answered by oldprof 7 · 0⤊ 0⤋
Others have already given you the two values you need to fill in your blanks, but I want to point out something to you. Your question asked about confidence intervals and the normal distribution. The interval you are seeking is NOT a confidence interval. It is called a probability interval, which is a special case of a tolerance interval.
Math (and Stats) Rule!
2007-06-27 14:19:27 · answer #4 · answered by Math Chick 4 · 0⤊ 0⤋