>> This is a quick tutorial on, I don't know, normal distribution probabilities. I guess
that's the best description of it. This first problem tells us that x is approximately normally
distributed with a mean of 100 and a standard deviation of 15. And then it asks to determine
the following, making sure to include the appropriate picture. So the first thing I'm
going to write out here is something we will just kind of get in the habit of writing.
Shape, center, and spread of our distribution. Seems a little overkill now, but later on
in the class we'll see why it's important. These won't always be so obvious. But for
now, the shape, it's just given to us. It says it's approximately normal. The center,
because it's approximately normal because we have the symmetric distribution, the best
measure of center is the mean, which in this case is given to us to be 100. And then similarly
because of the symmetry, the shape, the spread, the best measure of the spread to use is our
standard deviation, also given to us, equaled 15. Okay, so getting to the problem. The first
one says, "Find the" -- okay, the way to read this is, what's the probability that x is
approximately normally distributed variable is greater than 145? So we can draw our little
distribution here. Try to make it as symmetric as I can. We know the middle is 100 and the
standard deviation is 15, so I can count up by one standard deviation, two and three.
And then similarly count down 100 minus 15 is 85, minus another 15 is 70 and another
gets us 55. So this question is asking us, what's the probability that x is greater than
145? 145 is way up here. We want to know this probability out here. It goes on forever,
but it gets so small we can kind of stop our picture there. And the way we can find this
probability -- turns out our calculator figures out exactly this. Use the normal CDF function,
which stands for normal cumulative distribution function. And what you put in as your input
into the normal CDF function is the lower bound, so in this case 145, which is our lower
bound. Then the upper bound, so you're starting at 145, how far do you want to go? Well, we
want to go forever to the right, so we just have to put in some really big number here.
There's no way to tell it infinity. You can use scientific notation to give it a huge
number, but it'll turn out that any big number will suffice. I usually just go down a bunch
of nines and call it good enough. So this is our upper bound. Then you need the mean
and the standard deviation. So those were both given to us. The mean is 100 and the
standard deviation is 15. So if I tell my calculator all this information, what it'll
give me is this area that's shaded in here. So how do you do this on your calculator?
Well, first of all we need to find this normal CDF function, and that's hidden under this
distribution menu, which is above variables. So to get there I hit second, and the variables,
and it brings me into my distribution list and I want the second one on the list here,
normal CDF. So hit normal CDF. And then just copy all your inputs. You want to know from
145 all the way up to some big number with a mean of 100 and the standard deviation of
15. If I hit enter it'll give me that probability. In this case it's point .0013, maybe 135,
or 0.135 percent if you move this decimal place over twice. In other words, not a whole
lot of area. If your mean is 100 and your standard deviation is 15, it's pretty unlikely
that you're going to be greater than 145. All right, moving on to the second part. It's
a similar question. Same picture, it turns out. Let me make a little bit more room. We
still have this normal distribution. We still have a mean of 100. But now what we want,
we're kind of given the answer. We're told that the probability that x is less than a
given point is 60 percent. So I have some point. I'll put it over here on the right
side, because I know that the area to the left of it, this shaded region out here -- I'll
try to stay inside the lines if I can -- ll this green shaded region which goes on forever
out here, it tells me that's equal to 60 percent. This right here is .60. So what my question
is is what is this value x not that has this much area to the left of it? And notice to
the right of the mean, because right on the mean has 50 percent to the left of it, half,
so since 60 percent is more than 50 percent, I want it over here. So your calculator can
figure this out, too. It's a different function, though. You use the inverse norm function,
and using the inverse norm function, the arguments you give it are, first you have to tell it
the area below. It's important that it is the area below. In this problem we were given
the area below is 60 percent, which is nice. That won't always be the case, and I think
we'll see later on this example when we solve for this. So you've got the area below the
mean and the standard deviation. This is the mean and this is the standard deviation. And
what it'll return to you is the value of x, what x not is here, that has 60 percent below
it. To find the inverse norm, you go into the same menu as before, the distribution
menu. So I'll hit second and then variables to get under distribution, and then the inputs
that I copied down there, .6, 100, and 15, and if I hit Enter it'll give me the value:
103.8. So what that's saying is that if the mean is 100 and the standard deviation is
15, exactly 60 percent of the data will be below 103.8. So that's the answer to this
question. All right. These next two questions have to do with the standard normal distribution.
So what you'll note down here is that we're not told anything about the shape, the center
and the spread. Actually you don't even need to be told it's the standard normal distribution.
Just by giving this question and using the letter z here, in our class, we will assume
that the shape is approximately normal... ...and that the center, which is given by
the mean, is equal to zero. And the spread, given by the standard deviation, is equal
to 1. So just because there's a z here, we're supposed to assume all this stuff. But once
you get that, it turns out to be a lot like the first two problems, we just have a different
mean and a different standard deviation to deal with. Easy enough. We can go through
and calculate these. I'll try my best to draw the picture. First we've got the mean, zero,
so we can count up one, two, three standard deviations, and then down one, two, three
standard deviations. And then let's see, in this one they want to know the probability
that z is between negative 1.4 and 2.5. So negative 1.4 is somewhere over here, and 2.5
is somewhere out here. And what we want to know is the probability that were between
those two; in other words, the area of the region below the curve bounded by those two
points. In other words, this shaded green region right here. Now, we can figure that
out. We've done this before on our calculator. We're going to use that normal CDF function
again. And the inputs we're going to give it is our left bound, negative 1.4; our right
bound, which is 1.5; our mean, which is zero; and our standard deviation, which is 1. And
so if you type that all into a calculator, I can jump here. You could go back into normal
CDF, which is second and then distribution. It'd be nice if I could pull this back up,
this old normal CDF. I know it's not the last thing I typed. It turns out you can. If you
hit second and Enter, it brings up the last thing you typed. Second Enter again, it'll
bring up the thing before that. So normal CDF, I want to start at negative 1.4. Make
sure you use this key for the negative, not this key. This is minus. It's an operation.
This is negative to write a negative number. I think you'll get an error if you use the
wrong one. Negative 1.4, I want to go all the way up to 2.5. And then I want my mean
to be zero and my standard deviation to be 1. Negative 1.4 to 2.5, mean is zero, standard
deviation is 1. Whoa. Make that a little bit neater maybe. I'll just delete a couple of
these extra decimal spots. All right. And if I hit Enter, it's going to give me the
answer that I need. It tells me that it's .913, in other words 91.3 percent. So the
answer to this question, the probability that a z, our standard normal distribution, standard
normal distribution random variable, is between negative 1.4 and 2.5. It's 91.3 percent. All
right. Finally, the probability that z is greater than z not is equal to 5 percent.
What that's saying is that, if you drew your picture here, and you got your zeroes in the
middle, there's some value way out here to the right -- maybe I'll put it out here -- so
that only 5 percent of the data is above it. In other words, this shaded region in here
is only 5 percent. This right here is .05. Okay, and I can figure that out. The way you
figure that out is we're going to use our inverse normal function again; however, you
might remember from up here that our inverse normal, the first input, the thing we needed
was the area below the point in question. In this case, we're given the area above the
point in question, so what we have to do is figure out the area below. Not too bad. If
5 percent is above, 95 percent is below. So then I can use inverse norm with the .95.
Now, .95 gives us the area below. One extra step in here. It's easy to make a mistake.
Your standard is zero, your spread is 1. If I type this in, inverse norm, I'll just find
it out of distributions, third thing on the list. Inverse norm, I want .95... comma zero,
...comma 1. If I hit Enter, it will give me that value, 1.645. It will turn out this is
a value that we'll use so frequently later in the class, you'll probably just end up
memorizing it, but you don't need to memorize it. So that's the answer here. That's what
z not is equal to. The point that only has 6 percent above it is 1.645, if your mean
is zero and your standard deviation is 1. One little note. This value... of z... is
also called z sub .05. So if you ever see this notation, a z and then in the subscript
.05, you're supposed to assume that we're looking for the standard normal variable that
only has 5 percent of the area above it. A little bit tricky. You usually talk about
below; in this case we're talking about above. So this answer that I found here can also
be referred to just in this notation. Just wanted to get you comfortable with both the
notations you'll see. Didn't want to confuse you there at the end. If that last part confused
you, ignore it for now. We won't see it for a few chapters. That is the end of this tutorial.