Standard normal curve. Very confused

Each shaded area represents a probability. The first plot, for instance, represents the probability P (-z ≤ Z ≤ z) = 0.8064 for some number z that you have to find. To find these z, you'll need to be familiar with properties of the normal distribution as well as with lookup tables for z-scores or using inverse CDFs. (I'll provide a link in the comments to a calculator you can use.)

(a) The area under the curve between -z and z is 0.8064. This means the area outside this interval is 1 - 0.8064 = 0.1936. Because the distribution is symmetric about its mean (0 for the standard normal distribution), you know that the area under the curve to either side of the shaded region is half of this, or 0.0968. This area corresponds to the probability P (Z ≤ -z).

So you have

P (-z ≤ Z ≤ z) = P (Z ≤ z) - P (Z ≤ -z) … … … (a property of continuous distributions)

0.8064 = P (Z ≤ z) - 0.0968   ==>   P (Z ≤ z) = 0.9032

and looking up the z-score, you would find z = 1.3.

(b) Because this distribution is symmetric about the mean, the mean is the same as the median, which is the number z such that P (Z ≤ z) = 0.5. In other words, half of the entire distribution falls to either side of the mean.

We use this fact to write

P (0 ≤ Z ≤ z) = P (Z ≤ z) - P (Z ≤ 0) = P (Z ≤ z) - 0.5 = 0.4452

==>   P (Z ≤ z) = 0.9452

==>   z ≈ 1.6000

(c) Keep in mind that the CDF is defined as P (Z ≤ z), so in order to find a probability like P (Z ≥ z), you have to take the complement:

P (Z ≥ z) = 1 - P (Z ≤ z) = 0.1711   ==>   P (Z ≤ z) = 0.8289

==>   z ≈ 0.9498

(d) Same reasoning as in (c):

P (Z ≥ z) = 1 - P (Z ≤ z) = 0.9918   ==>   P (Z ≤ z) = 0.0082

==>   z ≈ -2.3999

(e) Much more straightforward:

P (Z ≤ z) = 0.9115   ==>   z ≈ 1.3501

10 down (bang1)

5 up

5down (2)

2.5 up

2.5 down (3)

1.25 up

1.25 down (4)

now add them up : )

hoped i lol