OK, now we want to know what the probability is that our x is

__less than__ 123. In applications, that x of course will stand for something. So I've made this IQ scores. Supposedly the mean is 100, by definition. I sort of made up the 15 - I had a problem with IQ scores once and I think it was around that. So the problem wants to know the probability that some randomly selected person has an IQ of less than 123.

As a side note, it wouldn't matter if it included "equal" or not. That is, "less than" and "less than or equal to" are the same thing.

The green area is the area we are looking at, cause that's the "less than" part. We already know that's the equivalent of 1.53 in z's. So in a sense, it's like asking what's the probability that z is less than 1.53?

The pictures work nicely. I always draw them because I'm very visual. It helps a lot when you're learning them. So less than is to the left and includes all that green area. Something important to note is the dark purple line at the mean, the halfway point. What do all probabilities have to add up to? One. That means everything under that curve equals one. And what's half that? 5. So everything to the left of the purple line is .5 and everything to the right of that purple line is .5. (Note that I'm talking about the

**area** now, not the line itself. Line is x's or z's. Probabilities is the area.)

So notice that our area includes the entire left side, PLUS a portion of the right side. So we know our probability is greater than .5. Now the tricky part is explaining how to get the probability off the z chart, because there charts can be different. So here's where you have to learn how to interpret the chart. (Aren't you the one I gave the link to of different ones so you could pick which one looks like yours?)

If your chart looks like my example, that is, with over half of it shaded in, then you've got it made, cause all you have to do is look up 1.53 and you've got your answer. If, however, you've got the kind that shades from the center to some point to the side, that's only showing from the halfway mark. That means you'd have to add .5 to it so that it also includes the entire left side.

The probability is .937. Check out your chart and see if you can figure out how to come up with that.

I'd show another example for the probability of x being greater than the 123, but I'm getting kind of sleepy now.

But the concept works the same for all of them. Draw the chart. Put the mean in the middle. You can put the numbers at each standard deviation if you like - it can help give an idea about where your x is located. (For instance, I knew my z was 1.something cause of where it's located.) The z line underneath is just for the visual. Use the z-score equation to get the z for any given x the problem gives you. Then shade the area you want. Less than is to the left, more than is to the right. Remember that "no more than" means to the left. Then go to the z chart to find the probability.

This is for a population. There's a slight twist on it when doing sample means, but it's not a huge deal. But that would take some more explanation... the concept is exactly the same. So if you can figure this out, you'll be able to figure that out.