If 𝑥 is a normal random variable whose mean is 75 and standard deviation is four, find the probability that 𝑥 is less than 85.
So, we know that 𝑥 is a normal random variable, the mean is equal to 75, and standard deviation is four. The normal distribution can be described completely by these two parameters, the mean and standard deviation. The mean is the center of the distribution, and the standard deviation is the measure of the variation around that mean.
A normal distribution follows a bell-shaped curve, where the mean is in the very middle. And then, we have our standard deviation of four. So, we would have 79, 83, 87, and 91 because they’re four apart, and the same for the other side. So, how do we know where these are placed? Well, 68 percent of the values are going to be within one standard deviation of the mean. 95 percent of the values are within two standard deviations of the mean. 99.7 percent of the values are within three standard deviations of the mean.
And this is what the normal distribution would look like. 50 percent of the values are on one-half of this bell curve. And the other 50 percent are on the other half. So, what is our question asking? It’s saying find the probability that 𝑥 is less than 85. Well, here is around where 85 would be. So, what is the probability that 𝑥 would be less than 85, so in this range?
It seems pretty high. It almost takes up the entire curve, the area under it. We can find this by using the z-score. So, instead of saying the probability that 𝑥 is less than 85, we can go by the z-value. But we don’t know what it is yet. The way that we find the z-value, so we take our value of 85 subtract the mean of 75 and divide by the standard deviation of four.
So, we have 10 divided by four which is 2.5. But what is this value of 2.5? What is the z-value? The value of z gives the number of standard deviations a particular value of 𝑥 lies above or below the mean. In our example, 𝑥 is equal to 75 lies 2.5 standard deviations away from the mean. And looking at our graph, here’s our mean 75. Here’s one standard deviation, two standard deviations. And then, 85 is directly in between that second standard deviation and the third standard deviation. So, 2.5 standard deviations away from the mean.
And this is helpful because there will be a z-chart or a z-table that tells us this exact probability. And we find it to be 0.9938. Again, this is from the z-chart. So, we can either leave this probability as the decimal 0.9938 or change it into a percent. So, we would keep our decimal point, then shift all of the digits to the left two, then add the percentage symbol. So, as we’ve said, the probability that 𝑥 is less than 85 should be pretty high. And again, 99.38 percent is pretty close to 100 percent. Or the decimal is very close to one.