z-score
The values given by z-score tables represent the fraction of the area under a normal curve between -∞ and z. For example, for a given z, the value given by a table represents the following area:
However, in this exercise we must find the area under the curve between -z and z and not between -∞ and z. We are basically looking for an area like this one:
So the z in a z-score table that corresponds to 70% of the area is not the answer.
However, we still can find the value of z using a z-score table. Remember that the total area under this curve is equal to 1. We are told that the area between -z and z is the 70% so this area is equal to 0.7. Then the remaining area i.e. the sum of the areas at the left of -z and at the right of z is equal to 1-0.7=0.3. Another important property of the normal distribution curve is that it's symmetric so the area at the right of z is equal to that at the left of -z then the two green areas are equal and their sum is 0.3. This means that each green area is equal to 0.3/2=0.15. So basically we have the following:
- The area between -∞ and -z is equal to 0.15.
- The area between z and ∞ is equal to 0.15.
Remember that the z-scores tables give us the z-score associated with the area under the curve between -∞ and z. Then if we look at a z-score table and look for the value 0.15 the table will give us the value of -z and with it the value of z. So we must look for 0.15 in a z-score table:
0.14917 is the closest value to 0.15 in this table so it is useful. As you can see it's located at row -1 and column 0.04 which means that it corresponds to -1.04. Then -z=-1.04 and therefore z=1.04. Then the answer is:
[tex]-1.04,1.04[/tex]
