The proportion of a normal distribution that is located between two z-scores can be calculated using a standard normal distribution table or a calculator with a normal distribution function.
For the given z-scores:
z = -1.64 and z = +1.64:
The area under the normal distribution curve between these two z-scores is the same as the area between z = 0 and z = +1.64 minus the area between z = 0 and z = -1.64. From a standard normal distribution table, the area between z = 0 and z = +1.64 is 0.4505, and the area between z = 0 and z = -1.64 is also 0.4505. Therefore, the area between z = -1.64 and z = +1.64 is:
0.4505 - 0.4505 = 0
So, there is no area between z = -1.64 and z = +1.64.
z = -1.96 and z = +1.96:
Using the same approach as above, the area under the normal distribution curve between these two z-scores is:
0.4750 - 0.4750 = 0
So, there is no area between z = -1.96 and z = +1.96.
z = -1.00 and z = +1.00:
Again, using the same approach, the area under the normal distribution curve between these two z-scores is:
0.3413 - 0.3413 = 0
So, there is no area between z = -1.00 and z = +1.00.
Therefore, for all three cases, the proportion of a normal distribution that is located between the given z-scores is 0.