The heights of a sample of flowers are normally distributed with mean 𝜇 and standard deviation 12. Given that 10.56 percent of the flowers are shorter than 47 centimeters, determine 𝜇.
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. A sketch of this curve can be a really useful way to decide how to answer a problem about normally distributed data.
In our case, we have a mean of 𝜇 and a standard deviation 𝜎 of 12. Now, 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 probability.
We are told that 10.56 percent of the flowers have a height of less than 47 centimeters. On our curve, that’s roughly this area. Now, when we standardize our values using the formula for the 𝑍-value, the values on the left-hand side of the mean that’s down here are negative.
Since we don’t have negative 𝑍-values in our standard normal table, we’ll need to use the symmetry of the curve to help us find the relevant value to use. Since the normal curve is symmetrical about the mean, we have this value up here, which is also equal to 10.56 percent. This is the 𝑍-value we’re interested in finding.
Remember though the standard normal table tells us the probability between zero and 𝑍. Since the area under the curve sums to 100, we can subtract 10.56 percent from 100 to find the value that we need to look up. 100 minus 10.56 is 89.44. In decimal form, that’s 0.8944.
We need to find the associated value for 𝑍 that corresponds to a probability of 0.8944 in our standard normal table. In fact, a 𝑍-value of 1.25 has a probability of 0.8944 as required. This means then that a 𝑍-value of negative 1.25 must have that probability that we were trying to find earlier of 10.56 percent.
We’ll use this 𝑍-value in our formula in addition to the value for the standard deviation given in the question to calculate the value of the mean. We said the 𝑍-value that had an associated probability of 10.56 percent was negative 1.25. 𝑥 is 47 and standard deviation 𝜎 is equal to 12.
Let’s solve this equation by multiplying both sides by 12. That’s negative 15 is equal to 47 minus 𝜇. Next, we’ll subtract 47 from both sides and we get negative 62 is equal to negative 𝜇. Finally, we’ll multiply everything by negative one. And we get 62 is equal to 𝜇.
We have shown that the mean for this population is 62 centimeters.