Question Video: Calculating the Mean of Normally Distributed Random Variables Mathematics

Video Transcript

Suppose 𝑋 is normally distributed with mean 𝜇 and variance 196. Given that the probability 𝑋 is less than or equal to 40 is 0.0668, find the value of 𝜇.

We are told that 𝑋 is normally distributed with a mean 𝜇, which is what we’re looking to find, and a variance. That’s a 𝛿 squared of 196. We’re also told that the probability that 𝑋 is less than or equal to 40 is equal to 0.0668.

Now, our first step is usually to find associated 𝑍-scores. So in this example, if we were looking to find the probability that 𝑋 is less than or equal to 40, we would find the 𝑍-score associated with an 𝑋-value of 40. However, we don’t yet know the value of the mean. So we’re going to need to find the value of 𝑍 such that the probability that 𝑍 is less than or equal to lowercase 𝑧 is equal to 0.0668. Now, what we might do is look for this probability in the standard normal table.

However, if you try that, you’ll notice that the smallest possible value we get is 0.5. And that’s because the probability 0.0668 is such a small value it will lie all the way down here. So we need to find a 𝑍-value for which this is true.

And to do that, we recall that the standard normal curve is completely symmetrical about the mean. And the area under the curve is one whole. So this part up here will be 0.0688. And since the probabilities are cumulative, to find the 𝑍-value for which this is true, we’re going to subtract 0.0688 from one since that’s the total area under the curve. And that’s 0.9332.

Let’s find this value in the standard normal table. In fact, the 𝑍-value for which this is true is 1.5. And using the symmetry of our normal curve, we can see that the probability that 𝑍 is less than or equal to negative 1.5 is 0.0688. And that’s the value we’re looking for.

And now, we’re almost ready to substitute what we know into the formula for the 𝑍-value. Remember, this formula requires the standard deviation, not the variance. So we find the square root of 196. And that gives us 14. The standard deviation has to be positive. So we disregard any negative values for the square root of 196. And now, we can substitute what we know about 𝑋 into this formula. We have a 𝑍-value of negative 1.5. And that corresponds to an 𝑋-value of 40. And we’ve just shown that the standard deviation is 14.

We’re going to solve this equation for 𝜇. And we do that by multiplying both sides by 14. That gives us negative 21 equals 40 minus 𝜇. There are a number of different next steps we can take. One way is to multiply through by negative one. Or we can add 𝜇 to both sides. Adding 𝜇 gives us 𝜇 minus 21 is equal to 40. And finally, to solve for 𝜇, we add 21 to both sides. 40 plus 21 is 61.

And we can now see that the value of 𝜇, the mean, is 61.