Let 𝑥 be a random variable which is normally distributed with a mean of 68 and a standard deviation of three. Determine the probability that 𝑥 is greater than or equal to 61.7.
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. It can be really useful to sketch the curve out to help you decide the best way to calculate probabilities.
The mean of 𝑥 is 68, and its standard deviation is three. The question is asking us to calculate the probability that 𝑥 is greater than or equal to 61.7, which is represented by the area shaded. The first step with most normal distribution questions is to calculate the 𝑍 value. This is essentially 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. We’ve already specified our values for 𝜇 and 𝜎. So we need to substitute 61.7 into our formula for the 𝑍 value. This gives us negative 2.1. Now we’re actually interested in the probability that 𝑥 is greater than or equal to 61.7. That’s the shaded area. And our standard normal table only gives us cumulative probabilities. That’s the unshaded area.
We can therefore consider the symmetry of the bell curve and recognise that if we look up a 𝑍 value of positive 2.1. That will also tell us the probability that 𝑥 is greater than or equal to 61.7. Reading the value for 2.1 in our standard normal table gives us 0.98214. The probability, therefore, that 𝑥 is greater than or equal to 61.7 is 0.9821, correct to four significant figures.