Video: Using Normal Distribution Probabilities to Calculate an Unknown

Given a normal random variable π such that π(π β ππ β€ π β€ π + ππ) = 0.8558, Find the value of π.

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Video Transcript

Given a normal random variable π such that the probability that π is greater than or equal to π minus ππ and less than or equal to π plus ππ is equal to 0.8558. Find the value of π.

Remember, the graph of s curve representing a normal distribution with a mean π and a standard deviation π is symmetrical about the mean, and the total area under the curve is 100 percent or one. Letβs add in what we know about our random variable. Now our next step would usually be to find the π value for π plus ππ and π minus ππ. 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 shouldnβt worry too much that we donβt currently know the values of π and π. Letβs substitute each other of our values of π into the formula and see what happens. For an π value of π plus ππ, the π becomes π plus ππ minus π all over π. π minus π is zero. So these cancel out. And weβre left with ππ all over π. The πs then cancel out and weβre left with the π value of π.

For the next value of π minus ππ, it looks like this. Once again, the πs and the πs cancel out and weβre left with a π value of negative π. So we now know that the probability that π is greater than or equal to negative π and less than or equal to π is 0.8558. We do need to be a bit careful here since we canβt currently look this value up in a standard normal table. When we look up a value of π in the table, it tells us the probability that π is less than that value. In this case though, we need to find the probability between two values of π.

At this point then, we instead need to remember that the curve is symmetrical about the mean. And since π is a constant, we can see that we have complete symmetry here. What we will instead do is subtract the probability that weβve been given from one whole. This will tell us the probability that π is greater than π or less than negative π. 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 π is greater than π is 0.0721.

We still canβt look up this value in the table. But if we subtract 0.0721 from one, it tells us the probability that π is less than or equal to π. Thatβs 0.9279. This is the probability weβ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 π is less than or equal to 1.46 is 0.9279. And we have shown that π is equal to 1.46.

Remember, when we usually work with a normal random variable, we would say the probability that π is less than π 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.