In a school with 1000 students, the heights of students are normally distributed with mean 113 centimetres and standard deviation five centimetres. How many students are shorter than 121 centimetres?
To answer this question, we’ll begin by considering the normal distribution to help us find the probability that a randomly selected student is shorter than 121 centimetres. Once we know this, we can calculate the number of students out of 1000 that we expect to be below this height.
So how do we calculate this probability? Well, we already said that our data set is normally distributed. It has a mean, 𝜇, of 113 centimetres and a standard deviation, 𝜎, of five centimetres. It can be useful to represent this information on that bell shaped curve that we associate with the normal distribution. And since we’re looking to find the probability that a student is less than 121 centimetres tall, we see that we’re looking to find this shaded area. And to do that, we need to find the associated 𝑧-score.
We use this formula 𝑧 is equal to 𝑥 minus 𝜇 over 𝜎. This is essentially a way of standardizing our data. And once we’ve done that, we can look up our values on the standard normal table. We’re looking to find the probability that a student is shorter than 121 centimetres. So our 𝑥-value is 121. And we see that 𝑧 is equal to 121 minus the mean, 113, all over standard deviation, which is five. And that gives us a 𝑧-value of 1.6.
Then, to find the probability that 𝑧 is less than 1.6, we look up a 𝑧-value of 1.6 in our standard normal table. That gives us 0.9452. And in turn, it means that the probability that 𝑥 is less than 121 is also 0.9452. And since there are 1000 students, we can say that we expect 1000 multiplied by 0.9452 students to be shorter than 121 centimetres. 1000 multiplied by 0.9452 is 945.2. We’ll round this to 945 as it makes no sense to have 0.2 of a person.
And we can see that we expect 945 students to be shorter than 121 centimetres.