If the heights of 1500 students follow a normal distribution with mean 175 centimeters and standard deviation five centimeters, find the number of students whose heights are more than 180 centimeters.
We’ll use the random variable 𝑥 to represent the distribution of heights of the students. We’re told in the question that this follows a normal distribution with a mean of 175 centimeters and a standard deviation of five centimeters. To find the number of students whose heights are more than 180 centimeters, we must first find the probability of this random variable 𝑥 being greater than 180. Remember, the normal distribution is a bell-shaped curve symmetrical about its mean, which in this case is 175. The area underneath the full curve is one. And the area to the left of a particular value is the probability that the normally distributed random variable is less than or equal to that value.
The probability that 𝑥 is greater than 180 then is the area to the right of 180. That’s the area that I’ve shaded in pink. To work this out, we need to find the probability that 𝑥 is less than or equal to 180. That’s the area shaded in orange. And then, subtract this probability from one. To do so, we must first find the 𝑧-score associated with this value of 180. The 𝑧-score tells us how many standard deviations away from the mean a particular value is. The 𝑧-score for given value 𝑥 is defined as 𝑥 minus 𝜇 over 𝛿, where 𝜇 represents the mean of the normal distribution and 𝛿 represents its standard deviation.
In this question, 𝑥 is equal to 180. 𝜇, the mean, is 175. And 𝛿, the standard deviation, is five. So the 𝑧-score is equal to 180 minus 175 over five, which is equal to one. This means that this value of 180 is one standard deviation above the mean. Which makes sense when we remember that the mean of the distribution was 175 and the gap between 175 and 180 is five, which is the same as the standard deviation.
Next, we need to look at the probability associated with this 𝑧-score in our standard normal tables. These are the tables for the standard normal distribution. That is, the normal distribution with a mean of zero and a standard deviation of one. Remember, these tables give the cumulative probability of getting a 𝑧-score less than or equal to a particular value. If we look up the 𝑧-score of one, or in fact 1.00, in our table, we see that the probability of getting a 𝑧-score less than or equal to one is equal to 0.8413. This is in turn equal to the probability that our random variable 𝑥, which represented the heights of the students, was less than or equal to 180.
So to find the probability that 𝑥 is greater than 180, we subtract this probability from one, giving 0.1587. So now we know that the probability that a student from this distribution has a height that’s more than 180 centimeters is 0.1587. But the question asked us for the number of students out of the 1500 whose heights are more than 180 centimeters. So the final step is to multiply this probability by the number of students, which is 1500. And it gives 238.05. The number of students needs to be an integer. So rounding this to the nearest integer gives 238. And so, we can conclude that for this normal distribution, the number of students whose heights are more than 180 centimeters is 238.