Probabilities for Standard Normal Random Variables?

look the value 2.33 up in your book and that is the probability that x is less than or equal to that value. I'd imagine it's something like .995 something because that's more than 2 standard deviations from the mean.

the probability for the second one is P(z<2.58) - [1 - p(z<2.58)]

to calculate a negative z just use the symmetries of the table -> find the probability of the positive z and subtract that from 1, and you have the value for a negative z.

a z table just shows the probability that a value z is less than a equal to a certain value x

the way that you would find z is as follows: z = (x - mu)/variance [x - average/standard deviation squared]

then you look that value up on the z table and that is the prob that x is less than or equal your specified value. ex. find P(x < 4) mu = 3, standard deviation = 2, that is:

find z: (4-3)/2^2 = 1/4, look up the value of .25 on the z table and that is the probability

alternatively is the question were P(x > 4) you would do the same thing, but subtract from 1 since that probability reflects the area less than or equal to the specified x. ie. if the probability that something is less than something is .4 the probability that it is greater than the same thing has to be .6, because probabilities add to 1. (1-0.4 = 0.6)

I've said it several times thus far but it's because it's important: that table only shows less than or equal to, but since a normal distribution is continuous less than or equal to is the same as less than. I'm sure you've dealt with discrete random variables already which you must then distinguish between less than and less than or equal to.

you need to look up the z score in a chart in your book somewhere

P(z<2.33)=0.990096924

for the second problem

look up the two values

subtract the smaller from the larger

P(-2.58<z<2.58)=0.990119968

use a calculator or Standard Normal Distribution table