If 𝑍 is a standard normal random variable such that the probability that negative 𝑎 is less than or equal to 𝑍 which is less than or equal to 𝑎 is 0.874, find 𝑎.
If 𝑍 is a standard normal random variable, as we’re told in the question, then this means that it has a normal distribution with a mean of zero and a standard deviation of one. The standard normal distribution is a bell-shaped curve symmetrical about its mean. So, in this case, that’s zero. The area under the full curve is one. And the area to the left of a particular value gives the probability of a value from this distribution being less than or equal to that value.
In this question, we’re told that the probability that 𝑍 is between negative 𝑎 and 𝑎 is 0.874. So that’s this area that I’ve shaded in orange. We need to convert this to an area to the left of a particular value so that we can use our standard normal tables to find the value of 𝑎. Remember that this distribution is symmetrical, which means that the two orange areas, either side of the mean, are equal. So they’re each equal to half of 0.874 which is 0.437. The whole area to the left of the mean is 0.5. That’s half of the total area of one, again due to the symmetry of the distribution. This means that the area to the left of the value 𝑎 is 0.5, that’s for the pink region, plus 0.437, for the orange region, giving a total of 0.937.
Remember that the area to the left of a particular value represents the probability that 𝑍, our standard normal random variable, is less than or equal to this value. So we now know that the probability that 𝑍 is less than or equal to 𝑎 is 0.937. We can use our standard normal tables to look up this probability and find the 𝑍-score, which will be the value of 𝑎, associated with it.
Now, here is an extract from those standard normal tables. And we can see that a probability of 0.937, or 0.9370, is located here. Moving horizontally across from this probability, we can see that the 𝑍-score is 1.50. And then moving upwards from this probability, we can see that there’s a three in the second decimal place, the hundredths column. So we add 0.03 to 1.50. And this tells us that the value of 𝑎 is 1.53.