The monthly salaries of workers at a factory are normally distributed with mean 210 pounds and standard deviation 10 pounds. Determine the probability of choosing at random a worker with a salary between 184 and 233 pounds.
We are told that the data set is normally distributed. It has a mean 𝜇 of 210 pounds and a standard deviation 𝛿 of 10 pounds. We’re looking to find the probability of choosing a worker with a salary of between 184 and 233 pounds. If we choose 𝑥 to represent the salary of the worker. We’re trying to find the probability that 𝑥 is greater than 184 and less than 233. And to do this, we’re going to need to find the 𝑧-scores that correspond to 184 and 233.
We use the formula 𝑥 minus 𝜇 over 𝛿. And what this does is it standardizes the data. And it allows us to read probability from a standard normal table, with a mean of zero and a standard deviation of one. If we substitute 184 as 𝑥 in this formula, we get that the 𝑧-value is 184 minus the mean, that’s 210, all over the standard deviation 10. That’s negative 2.6. And for the 𝑥-value of 233, it becomes 233 minus 210, again, over 10, which is 2.3.
So, we can say the probability we’re interested in can be found by finding the probability that 𝑧 is greater than negative 2.6 and less than 2.3. And at this point, let’s consider the shape of the curve made by normally distributed data. It’s this bell shape. It’s completely symmetrical about the mean and the area underneath the curve is one.
We need to find the shaded area. And since the probabilities are cumulative, we can say that we can find this by subtracting the probability that 𝑧 is less than negative 2.6, that’s the lower one, from the probability that 𝑧 is less than 2.3. Now the probability that 𝑧 is less than 2.3 can be found by looking up 2.3 in the standard normal table. It’s 0.9893. But the probability that 𝑧 is less than negative 2.6 is a little trickier, since we don’t have negative values in our table.
And this is where the symmetry of the curve’s important. We can see that since the curve is symmetrical about the mean, the probability that 𝑧 is less than negative 2.6 is the same as the probability that 𝑧 is greater than 2.6. But once again we can’t find that probability by looking up 2.6 in the table. The table will tell us the probability that 𝑧 is less than 2.6.
So, instead we use the fact that the area under the curve is one. And we can see that this shaded area can be found by subtracting the probability that 𝑧 is less than 2.6 from one. If we look up a 𝑧-value of 2.6 in this table, that gives us 0.9953. And one minus 0.9953 is 0.0047.
So, the probability that 𝑧 is greater than 2.6 and, therefore, the probability that 𝑧 is less than negative 2.6 is 0.0047. 0.9893 minus 0.0047 is 0.9846. And therefore, the probability of choosing at random a worker with a salary between 184 and 233 pounds is 0.9846.