Video Transcript
The heights of a group of students follow a normal distribution with a standard deviation of 20 centimeters. The probability that a student’s height is less than or equal to 180 centimeters is equal to the probability that a standard normal variable is less than or equal to 2.2. Find the mean height of the group of students.
We have been given the details about a normally distributed data set. And we’re given that the probability that a student’s height is less than or equal to 180 centimeters is equal to the probability that a standard normal variable is less than or equal to 2.2. It’s standard convention to call the height 𝑥 and the standard normal variable 𝑧. And we can therefore see that the probability that 𝑥 is less than or equal to 180 is equal to the probability that 𝑧 is less than or equal to 2.2.
So how does this help us find the mean? Well, when solving problems about normally distributed data, usually one of our first steps is to find the standard score or the associated 𝑧-value. And we use this formula 𝑧 is equal to 𝑥 minus 𝜇 all divided by 𝜎, where 𝜇 is the mean of the data set and 𝜎 is the standard deviation.
In fact, we already know the 𝑧-value that is associated with an 𝑥-value of 180. So we can substitute what we know about our data set into this formula and solve to find the mean. The standard deviation is 20. So substituting 2.2 for 𝑧 and 180 for 𝑥, we get 2.2 is equal to 180 minus 𝜇 all over 20. And to begin solving this equation, we multiplied both sides by 20. And we see that 44 is equal to 180 minus 𝜇. We could add 𝜇 to both sides. And we see that 44 plus 𝜇 is equal to 180. And finally, we subtract 44 from both sides. 180 minus 44 is 136.
And we can see that the mean height of the group of students is 136 centimeters.
