Alright guys welcome to video number three,
and what I'm going to show you in video number three is how
to use norm s distribution. In other words, how to calculate probabilities
from a standard normal distribution. What I'm going to do is work through the
three applications on page five of the bootcamp lesson number three notes. So if
you take out bootcamp lesson number three, continuous random variables and
the normal distribution, turn to page five. I'm going to work through those
three examples or applications. All right. The first of these says, all right, let's
suppose that daily production is normally distributed with a mean of 300
units, and a standard deviation of 25 units. You want the probability that
today's production will exceed 350 units. All right. Our tables are in terms of Z
values not X values, so I have to turn these production figures these X values
into Z's. All right. I'm interested in an X value of 350 to turn that into a Z
value I subtract the mean divide by the standard deviation, and I find that the Z
is equal to 2. What that means is that if I have production of 350 units on 2
standard deviations above the mean of 300. Now in terms of looking at
probabilities, if I have an x value of 350 units, and I want the probability of
getting something greater than that. That's the same as finding the
probability of a z-value that's greater than two. The problem is norm s
distribution gives me lower tail probabilities the probability of getting
something less than a value. I want an upper tail probability.
How do I do it (inaudible), well I'm gonna manipulate the system. If norm s
distribution is only going to give me a lower tail value, I don't want to plug in
two, because then I would get the probability of getting a value less than
two. What I instead can do is plug in a z value of minus two, and because the distribution
symmetric, the probability of getting a lower tail of value less than minus two is
equal to this upper tail probability of getting a value greater than two. Right. So
let's go to excel, and let's find the probability for a z value of minus two. In Excel
all I have to do is use my formulas, and my f(x), and what I'm going to use
statistical formulas. So statistical, and I know these are alphabetically arranged,
I want norm s distributions, so scroll down to the 'N's. I find norm s
distribution, and notice the description. Returns the standard normal cumulative
function, has a mean of 0 and a standard deviation of 1, yes. Cumulative means it's
returning the probability up to the value Z you give it. In other words a
lower tail probability. All right norm s distribution. Okay.
And what does it want - the z-value. Remember I want to z less than
minus two. It's going to give me a probability; it's already here, point zero two two
seven. Okay, and I have .02275, times 100 is 2.275%,
and if we go back to our application we have a probability of
2.275%. The area in the lower tail is equal to the
area I want, up here in the upper tail. All right.
That's application number one.
Application number two. Suppose that the average number of
minutes required for a production worker to complete a task is normally
distributed, with a mean of eight and a standard deviation of zero point six.
Calculate the probability that it will take less than seven and one half minutes to
complete the task. Well again these are X values I want to convert these into
'Z's. To do it again let's subtract the mean let's take seven and a half
what we're interested in minus the mean of eight divided by the standard
deviation of 0.6 that gives me a z-value of -0.833.
Now this one's easy. What does that mean? It means that a value of seven and
a half minutes is 0.83 standard deviations below the mean of eight. In
terms of our diagrams I'm saying that a value of x equal to 7.5 is
equivalent to the Z value of -0.833. Seven and a half is
0.833 standard deviations below the mean. So to find
this lower tail probability, the probability that
have an X less than seven and a half I want to find the equivalent probability
of having a Z less than -0.833, and the good thing is
norm s distribution gives me that probability directly it gives me the
lower tail. So all I need to do is go to norm s distribution plug in
-0.833 and I'll have the probability. Let's do that. Go into Excel,
deleting what I had earlier, f(x) norm s distribution. Statistical is already
there, because that's what we're using now. I go to norm s distribution. Okay.
Now plug in my Z value of -0.8333. Okay. I get a
value of 0.2023 or 20.23%, and if I
come back. 20.23% that's the probability for application
number two.
All right, moving on application number three.
Suppose that the number of customers visiting a store per day is normally
distributed with an average of 1200 customers, and a standard deviation of
250 what is the probability that the number of customers will be between 1000
and 1500 today? All right I have two values now. Since I have two x values
1000 and 1500, I need two Z values. So converting the lower Z of 1000 we're
converting to a lower Z using the X value of 1000, I end up with a Z value of
-0.8. An X of 1000 is 0.8 standard deviations below the mean
of 1200. Converting the X value of 1500 into a Z I take the X 1500 minus the
mean, divided by the standard deviation, and it tells me that an X of 1500 is
1.2 standard deviations above the mean. Okay, now in terms of diagrams we
want this probability I want the probability that my X value centered at
the mean of 1200 is between 1000 and 1500. That's equivalent to going to a Z
table or a Z distribution and finding the probability that we have a Z between
minus 0.8 and plus 1.2. Now there are several ways I could do this. I could go
into norm s distribution find the probability of getting a value less than
1.2. That would give me all this lower end, because it gives me lower
tail values, and then subtract out the probability the wide area of getting the
z less than -0.8. That'll work. The way I did it
in the notes though, is to say look we have a probability of 1 under the whole
curve. Let's take out the probability of getting a Z less than -0.8, take out
that white area, and then let's take out the other white area the probability of
getting is the greater than 1.2. Now we've already seen that the probability
of getting a Z greater than 1.2 is the same as getting a probability
of a Z less than -1.2. So I need to go my normal table I need to
find the probability of getting a Z less than -0.8 and find the
probability of getting a Z less than -1.2, let's do that. I'll
go back to excel I'm going to find the probability of a Z less than
-0.8...statistical, norm s distribution,
and we'll find the probability of getting a Z less than 0 minus 0.8.
Okay, that probability is .2118 or 21.18%.
The probability of getting is Z less than -1.2.
I go to f(x), statistical, norm s distribution, this time I'm going
to plug in a value remember -1.2. That probability is going to
be 11.5%...0.115...11.5%.
Alright, we go back to the application, do the math. So our probability is one minus
the 0.2119, this lower tail probability, and minus 0.115
this upper tail probability, and we get an overall probability of 0.673.
That means there is a 67.3% chance that the average
number of customers will be between 1,000 and 1,500 on a given day. That's
the end of video number three.