The monthly salaries of workers at a factory are normally distributed with mean 𝜇 and standard deviation 200 pounds. Given that 82.12 percent of the workers earn more than 1851 Egyptian pounds, find 𝜇.
Sketching this out can be a really useful way to help decide how to answer a problem about normally distributed data. Remember, the graph of the curve representing the normal distribution is bell-shaped and symmetric about the mean. And the total area under the curve is 100 percent or one.
If we knew the value of the mean, our next step will be to scale our data by calculating the 𝑧-value. In this case, since we don’t know the mean, we’ll need to work backwards from the information provided about the percentage of workers who earn more than 1851 Egyptian pounds.
We are given that 82.12 percent of the workers earn more than 1851 Egyptian pounds. On our curve, that’s this area. However, when we scale our data and then look up 𝑧-values in our standard normal table, we’re given percentages between zero and 𝑧. Instead, we’ll need to consider the symmetry of the curve to help us find the associated value in the standard normal table.
82.12 percent is the same as 0.8212. We need to find the value for 𝑧 in our standard normal table that has an associated probability of 0.8212. It’s 0.92. So the probability that 𝑧 is less than 0.92 is equal to 0.8212.
Since the curve is symmetrical about the mean, that must mean that the probability that 𝑧 is greater than negative 0.92 must also be 0.8212. We’re, therefore, going to substitute a value of 𝑧 is equal to negative 0.92 into our formula for the 𝑧-value, alongside 𝑥 as 1851 and 𝜎 as 200.
That gives us negative 0.92 is equal to 1851 minus 𝜇 all over 200. We can solve this equation for 𝜇 by multiplying both sides by 200, which gives us negative 184 equals 1851 minus 𝜇. Next, we’ll subtract 1851 from both sides and we get negative 2035 is equal to negative 𝜇. And if we multiplied both sides by negative one, that tells us that 𝜇 is equal to 2035.
𝜇, that’s the mean monthly salary of workers at this factory, is 2035 Egyptian pounds.