If 𝑧 is a standard normal variable such that the probability that 𝑧 is greater than or equal to negative 𝑎 and less than or equal to 𝑎 is 0.733, find the value of 𝑎.
Let’s recall what is meant by a standard normal variable to begin with. If 𝑧 is a standard normal variable, then it means that 𝑧 has a normal distribution with a mean 𝜇 of zero and the standard deviation 𝛿 of one. We write that 𝑧 follows a normal distribution zero, one squared because in general we write a normally distributed random variable in terms of its parameters 𝜇 and 𝛿 squared.
The standard normal distribution is a bell-shaped curve symmetrical about its mean. So, in this case, it’s symmetrical about zero. 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 greater than or equal to negative 𝑎 but less than or equal to 𝑎 is 0.733, which means that the probability that 𝑧 takes a value between these values is 0.733.
This means that the area between negative 𝑎 and 𝑎 on our standard normal distribution curve is 0.733. Now we want to convert this to an area which is to the left of a particular value so that we can use our standard normal tables to find the value of 𝑎.
Now the two shaded areas either side of the mean are each half of that value of 0.733 because our normal distribution curve is symmetrical about its mean. Half of 0.733 is 0.3665. We also know that the full area to the left of the mean is 0.5 because that’s half of the total area of one, again due to the symmetry of the distribution. This means the total area to the left of this value 𝑎 is 0.5 plus 0.3665, which is equal to 0.8665. And so, we know that the probability that 𝑧 is less than or equal to 𝑎 is equal to 0.8665.
We can now look this probability up in our standard normal tables to find the 𝑧-score associated with this value. The table of 𝑧-scores gives us the values from the standard normal distribution for which the probability that 𝑧 is less than or equal to each of these 𝑧-scores is equal to a particular probability. We first need to locate our probability of 0.8665 in the table. And it is here. Looking across from this value, we see that it corresponds to a 𝑧-score of 1.10. And if you look up from this value, we see that we need to add 0.01. So, we need to add a one in the second decimal place
We see then that this probability of 0.8665 is associated with the 𝑧-score of 1.11. This means that the probability that a value from the standard normal distribution is less than or equal to 1.11 is 0.8665. And by reversing our earlier logic, it means that the probability that this value is between negative 1.11 and 1.11 is 0.733. And hence, we found the value of 𝑎. The value of 𝑎 is 1.11.