If 𝑍 is a standard normal variable such that the probability that negative 𝑎 is less than or equal to 𝑍, which is less than or equal to 𝑎, equals 0.733, find the value of 𝑎.
First, let’s think about what we know about standard normal variables and their probabilities. If 𝑍 is a standard normal variable, then its probability can be represented as the area under the normal curve. This probability, 0.733, represents the area underneath this curve. We also know that zero is halfway between negative 𝑎 and 𝑎. If we draw a line at zero, it divides this area into two equal pieces. The piece on the left of zero and the piece on the right of zero would have the same area. And that area can be found by dividing the original area by two.
0.733 divided by two equals 0.379. The area between negative 𝑎 and zero is 0.379. And the area between zero and positive 𝑎 is 0.379, which means that the probability that 𝑍 falls between zero and 𝑎 equals 0.379. And this format allows us to use a table of areas that will tell us what 𝑎 value gives a probability of 0.379.
On the table, you look for 0.379 and see what the correlated value is. In this case, it would be 1.17. The probability that 𝑍 will fall between zero and 1.17 is 0.379. We could also rewrite our original statement to say that the probability that 𝑍 falls between negative 1.17 and positive 1.17 equals 0.733. But in this question, our main job was to find the value of 𝑎, which we’ve done.
𝑎 equals 1.17.