Final answer:
The value of z when the area between -z and z under the standard normal curve is 0.6184 is approximately 0.88.
Step-by-step explanation:
To find z where the standard normal curve area between -z and z is 0.6184, we need to understand that this area represents the central portion of the normal distribution curve. The total area under the curve adds up to 1, so if we subtract the given area from 1 and then divide by 2, we will find the area in one tail. Subsequently, we can use a Z-table, statistical software, or a calculator to find the Z-score that corresponds to this tail area.
First, let's calculate the tail area:
1 - 0.6184 = 0.3816
0.3816 / 2 = 0.1908
The area to the left of z will then be 1 - 0.1908 = 0.8092.
Using the Z-table, we need to find a Z-score where the cumulative area to the left is 0.8092. This Z-score is the positive value of z we are looking for.
Searching the Z-table, we find that the Z-score that corresponds to an area of approximatively 0.8092 is around 0.88. Because we are dealing with a standard normal distribution that is symmetric, the Z-score will also be -0.88 for the negative value of z.
Therefore, z is approximately 0.88, meaning the area under the standard normal curve between -0.88 and 0.88 is 0.6184.