In a school, the weights of students are normally distributed with a mean of 61 kilograms and a standard deviation of eight. What percentage of students weigh between 50.6 kilograms and 61.64 kilograms?
Remember, the graph of a curve representing the normal distribution is bell shaped and symmetric about the mean, and the total area under the curve is 100 percent. It can be really useful to sketch the curve out to help you decide the best way to calculate probabilities. The mean weight of the students is 61 kilograms. And the standard deviation, which represents a measure of spread, is eight. The question is asking us to calculate the percentage of students with weights between 50.6 and 61.64 kilograms, 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 have already specified our value for 𝜇, the mean, and 𝜎, the standard deviation. Substituting in our topmost weight of 61.64 gives 61.64 minus 61 all over eight, which equals 0.08. We can therefore find the percentage of students who weigh less than 61.64 kilograms by looking up a 𝑍-value of 0.08 in the table.
The probability that 𝑍 is less than 0.08 is 0.53188 or 53.188 percent. The probability that the weight is less than 61.64 is therefore 53.188 percent. Now let’s substitute 50.6 into our formula for the 𝑍-value. Doing so, we get 50.6 minus 61 all over eight, which is equal to negative 1.3. Looking up a value of negative 1.3 in the standard normal table gives us 0.09680 or 9.68 percent. The probability that the weight is less than 50.6 is 9.68 percent.
Now, we wanted to know the percentage of students who weigh between 50.6 and 61.64 kilograms. In this case, we subtract the percentage who weigh less than 50.6 from the percentage who weigh less than 61.4. That’s 53.188 minus 9.68. That will give us the percentage represented by the shaded region. That gives us a value of 43.508 percent. The percentage of students who weigh between 50.6 kilograms and 61.64 kilograms is therefore 43.51 percent, correct to two decimal places.