Let 𝑋 be a normal random variable. Find the probability that 𝑋 is greater than 𝜇 plus 0.71𝜎.
Remember the graph of a curve representing the normal distribution with a mean of 𝜇 and a standard deviation of 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. In this case, we’re looking to find the probability that 𝑋 is greater than 𝜇 plus 0.71𝜎; that’s this shaded region. We know it must sit above the mean in our bell curve because the standard deviation can’t be negative.
Once we’ve established this, the next step with most normal distribution questions is to calculate the 𝑍-value. This is a way of scaling our data or standardising it in what becomes a standard normal distribution. Once we complete this step, we can work from a single standard normal table.
Now, it doesn’t really matter that we haven’t got a numerical value for the mean 𝜇 or the standard deviation 𝜎 of this data set. Let’s see what happens when we substitute everything we know into our formula for the 𝑍-value. Our value for 𝑋 is 𝜇 plus 0.71𝜎 and then we subtract 𝜇 and we divide through by 𝜎. 𝜇 minus 𝜇 is zero.
So our formula simplifies somewhat to 0.71𝜎 all divided by 𝜎. However, we can simplify a little further by dividing through by 𝜎. And we get 𝑍 is equal to 0.71. So we’re looking to find the probability that 𝑍 is greater than 0.71 since in the original question, it was asking us to find the probability that 𝑋 is greater than 𝜇 plus 0.71𝜎.
Our standard normal table though only gives probabilities between zero and 𝑍. In this case, that’s this side of the curve. So we find the probability that 𝑍 is greater than 0.71 by subtracting the probability that it’s less than 0.71 from one because we said that the area under the curve is 100 percent or one whole.
Looking up a 𝑍-value of 0.71 in our standard normal table and we can see that the probability that 𝑍 is less than 0.711 is 0.7611. That means the probability that 𝑍 is greater than 0.71 is one minus 0.7611; that’s 0.2389.
That means the probability that 𝑋 is greater than 𝜇 plus 0.71𝜎 is equal to 0.2389.