The monthly salaries of workers at a factory are normally distributed with mean 53 pounds and standard deviation 15 pounds. Find the percentage of workers whose wages are less than 98 pounds.
Remember, the graph of the curve representing the normal distribution with a mean of 𝜇 and a standard deviation 𝜎 is bell shaped and symmetric about the mean. And the total area under the curve is 100 percent or one. A sketch of the curve can be a really useful way to help decide how to answer a problem about normally distributed data.
This term, we’ve got a mean, that’s 𝜇, of 53 pounds and a standard deviation, that’s 𝜎, of 15 pounds. In this case, we’re looking to find the probability that 𝑋 is less than 98. That’s this area shaded. The next step with most normal distribution questions is to calculate the 𝑍 value. This is a way of scaling our data or standardizing it in what becomes a standard normal distribution.
Once we complete this step, we can work from a single standard normal table. Remember, 𝑍 is equal to 𝑋 minus the mean all divided by the standard deviation. In this case, our value for 𝑋 is 98, the mean is 53, and the standard deviation is 15. 98 minus 53 all divided by 15 is three.
So we can find the probability that 𝑍 is less than three by looking up this said value in the standard normal table. That’s 0.9987. That also means the probability of randomly choosing a worker whose wages are less than 98 pounds, 𝑋 is less than 98, is also 0.9987.
Since we’re being asked to find the percentage of workers whose wages are less than 98 pounds, we need to multiply this number by 100. 0.9987 multiplied by 100 is 99.87. So the percentage of workers whose wages are less than 98 pounds is 99.87 percent.