Final answer:
The Z value is found by using the Z-table to look up the given areas under the normal distribution curve. For each part of the question, we use the area provided to search for the corresponding Z value that represents where that area falls on the standard normal distribution.
Step-by-step explanation:
To find the Z value, we need to look up the area given in the question on the standard normal distribution table (also known as the Z-table) and find the corresponding Z value.
a) Area to the left of the Z value is 0.0089:
For this, we find the Z value such that the area to its left under the normal curve is 0.0089. This is typically a negative Z value since it is in the lower tail of the normal distribution. You would look for the closest area in the Z-table and find the corresponding Z score.
b) Area to the right and to the left of Z = 2.59 is 0.9877:
This means the total area to the left of Z = 2.59 is 0.9877. Since the areas to the left and right of a specific Z value in a standard normal distribution (Z-table) add up to 1, the area to the right of Z = 2.59 would be 1 - 0.9877, which equals 0.0123.
c) Area to the right of the z value is 0.5793:
To find the Z value for the area to the right being 0.5793, we subtract this from 1 to get the area to the left, which is 1 - 0.5793 = 0.4207. Then we look up this area in the Z-table to find the corresponding Z value.