Probability and Z-Score?

Question

Find the value of the standard normal random variable z, called z(sub)0 [z_0] such that:

P(z (is less than or equal to z_0 ) = 0.6279
I got z_0: .325 by looking up value on a chart.

P(-z_0 less than or equal to] z [less than or equal to] z_0 ) = ?

Not sure how to go about this, tried about a million different ways.

Also,

P(-z_0 [less than or equal to] z [less than or equal to] 0 ) = 0.2671
z_0 =

That should say,

P(-z_0 less than or equal to] z [less than or equal to] z_0 ) = .3272

Thanks for the help; I'm beginning to the think the programming of the website has some sort of error on this problem.

Answer 1

P(z <= z0) = 0.6279 implies z0 = 0.3263 so if you used a chart, your answer of 0.325 is pretty close.

P(-z0 <= z <= z0). If you draw a picture of a normal curve you will see that this is equal to the area to the left of z0 minus the area to the left of -z0. This is also equal to twice the area under the curve from 0 to z0 (do you see why?) Since you are told that this probability is 0.3272 then we know that P(0 <= z <= z0) = 0.3272/2 = 0.1636. Therefore, since the area to the left of 0 is 0.5, we have P( z <= z0) = 0.5 + 0.1636 = 0.6636. You have to look up this probability in your table and read off what the closest z- value is. That will be your z0. I get z0 = 0.4223


P(-z0 <= z <= 0) = 0.2671. Therefore, using symmetry, P(0 <= z <= z0) = 0.2671. Therefore because the area under the curve to the left of 0 is 0.5, P(z <= z0) = 0.5 + 0.2671 = 0.7671. Therefore z0 = 0.7293 (or about 0.73).

Math (and Stats) Rule!

Answer 2

I hope I can help. Your nomenclature is the pits.

In your example, you found the z for which the cumulative probability of the normal distribution (0,1) is 0.6279. z is positive because p > 0.5

In the second example, you are looking for the
z for the same distribution when the probability is LESS THAN 0.5. In fact, this is your second number problem. So you go to the chart, read the z corresponding to the 26.71 % of the cumulative probability, and you should get a value of about
-0.6 roughly.

You might want to remember some points on this chart. For a p=0.16, z=-1. For a p=0.84, z = +1.
For a p=0.025, z=-2, and for a p=0.975, z=+2.

Answer 3

Find the value z0.16