Given a normal random variable 𝑋 such that the probability that 𝑋 is greater than or equal to 𝜇 minus 𝑘𝜎 and less than or equal to 𝜇 plus 𝑘𝜎 is equal to 0.8558. Find the value of 𝑘.
Remember, the graph of s curve representing a normal distribution with a mean 𝜇 and a standard deviation 𝜎 is symmetrical about the mean, and the total area under the curve is 100 percent or one. Let’s add in what we know about our random variable. Now our next step would usually be to find the 𝑍 value for 𝜇 plus 𝑘𝜎 and 𝜇 minus 𝑘𝜎. This is a way of scaling our data or standardizing it in what becomes a standard normal distribution.
Once we complete this step, we can then work from a single standard normal table. Now we shouldn’t worry too much that we don’t currently know the values of 𝜇 and 𝜎. Let’s substitute each other of our values of 𝑋 into the formula and see what happens. For an 𝑋 value of 𝜇 plus 𝑘𝜎, the 𝑍 becomes 𝜇 plus 𝑘𝜎 minus 𝜇 all over 𝜎. 𝜇 minus 𝜇 is zero. So these cancel out. And we’re left with 𝑘𝜎 all over 𝜎. The 𝜎s then cancel out and we’re left with the 𝑍 value of 𝑘.
For the next value of 𝜇 minus 𝑘𝜎, it looks like this. Once again, the 𝜇s and the 𝜎s cancel out and we’re left with a 𝑍 value of negative 𝑘. So we now know that the probability that 𝑍 is greater than or equal to negative 𝑘 and less than or equal to 𝑘 is 0.8558. We do need to be a bit careful here since we can’t currently look this value up in a standard normal table. When we look up a value of 𝑍 in the table, it tells us the probability that 𝑍 is less than that value. In this case though, we need to find the probability between two values of 𝑍.
At this point then, we instead need to remember that the curve is symmetrical about the mean. And since 𝑘 is a constant, we can see that we have complete symmetry here. What we will instead do is subtract the probability that we’ve been given from one whole. This will tell us the probability that 𝑍 is greater than 𝑘 or less than negative 𝑘. So the area represented by the two pink shaded regions on our curve is 0.1442. Halving this, and it tells us the probability that 𝑍 is greater than 𝑘 is 0.0721.
We still can’t look up this value in the table. But if we subtract 0.0721 from one, it tells us the probability that 𝑍 is less than or equal to 𝑘. That’s 0.9279. This is the probability we’re going to look for in our standard normal table. In fact, a value of 1.46 gives us that probability. This therefore means that the probability that 𝑍 is less than or equal to 1.46 is 0.9279. And we have shown that 𝑘 is equal to 1.46.
Remember, when we usually work with a normal random variable, we would say the probability that 𝑍 is less than 𝑘 for example, rather than less than or equal to. However, since normal is a continuous distribution, the difference between these two is minimal. And we can therefore use less than or less than or equal to interchangeably.