All right, welcome to video number four
And what I'm going to do now is stay with bootcamp lesson number three
Continuous random variables in the normal distribution, but now I'm going to show you how to use
Normsinv the inverse standard normal function in excel so Normsinv
If you'll go to page six
The bootcamp lesson three notes. I'm going to work through those three
examples or
Applications and the one on page seven. All right. So, Normsinv
Alright first application and all of these are going to use these same numbers
suppose that the number of customers visiting a store each day is normally distributed with a mean of
278 and a standard deviation of
46
number one
there's a
25% probability one chance in four that there will be fewer than what number of customers
Okay
again
We're working with Z distribution so I'm gonna find a Z value first
Then I'm gonna convert that back into an x value a number of customers. These are axes
This has a meaning of 278 not 0 it has a standard deviation of 46 not a standard deviation of 1
so this is X
I have to find Z's from excel though. All right
and I want a
25% chance of having a value less than some Z. So I'm gonna put
25% in the lower tail of this distribution
Find what the Z is. Let's go to excel
In Excel I'll use f of X
And this time statistical Normsinv, let's go down to the ends
Normsssssinv
make sure the S is in there
I want a lower tail probability remember again, this is going to give you a description
Returns the inverses of standard normal cumulative distribution what's telling you it's giving it a number back instead of the probability
So you give it the probability?
.25 do not put 25 in there. You'll get an error message
point two five
Okay
That gives me a z-value of -0.67449
That tells me that 25% of Z's
Are less than
0.675. I have to convert that back into an X though if you give your manager a Z value
He's going to look at you funny if your manager tells you I want the probability, but I want a 25% probability
There'll be fewer what number of customers you come back with a Z value?
What's the Z. You have to convert it back into an X so
I want the X that corresponds to this is Z of
0.67449
negative
All right converting from a Z value back into an x value number of customers your x value is equal to the mean
278
plus Z value of -0.67449
Times the standard deviation of 46 so on .674 standard deviations
Below the mean that gives me an x value of 246 in other words
this x-value
246 so go back to your manager and say there's a 25% probability
that there will be less than
246 customers today
Application number 2
If we can find it application number 2
There is a 75% probability
That there will be fewer than what number of customers with 75% is going to be on the upper end of the distribution, isn't it?
Cuz there's 50% in each end
75% will be fewer than what number of customers well again
I'm gonna find a Z value from Normsinv
Then I want to convert that back into
an
x value
with a mean of 278
Alright, so I'm going to find the Z from Normsinv then convert that back into
realistic numbers interpretable numbers X
All right, we're gonna plug then 75% into Normsinv
Now you can go through all the steps again or you can notice up here in your formula bar
Normsinv is already there because that's what we just used
Why don't you just plug in the
0.75 in place at the 0.25
Hit enter
And you'll get a probability of .67449
In other words
75% of Z values are less than .67449
Let's go back to our problem. I
Know then that this Z is .67449
remember
Your manager doesn't want to Z value. You don't wanna Z value. You want a number of customers you want an x value?
So I need to convert that Z back into an X. How do I do that?
Same as before
use the formula and we'll take my mean of 278 and
Then add the Z value of this time positive
0.67449 times the standard deviation of
46 and I'll come up with
309 so your conclusion
there's a
75% probability that there will be less than
309 customers on any given day
That's application number two
Moving on let's go to the application number three
Application number three says using the same setup a mean of 278 standard deviation of 46
There is a 20% probability that there will be more than what number of customers
Well, we've played this game before we have converted into a Z
So I want to Z value and I want this up for tail
So there's 20% chance of getting a Z greater than this value
well
problem is
Excel gives us lower tail values, doesn't it?
so
How do I do it an
upper tail value of 20%
There's a couple of ways I could do it. One of the easiest is to say look
There's 20% in this upper end. There must be 80%
Down here in the lower end. That's one way to do it
Let's find
Normsinv with a
probability of
80%
80% less than Z then will be equivalent to 20% greater than the C. Let's go to Normsinv
Again since I already have pulled this up in Excel. All I need is a Normsinv for 0.8
80 percent or less than what value?
Let's find out
.8416
so
80% of Z's are less than in other words
20% of Z's are greater than
0.8416
let's go back to our
math
How do we convert Z value of 0.8416 into an X?
to 78 plus the .8416 times the standard deviation and we get
317 your conclusion then is
There is a 20% chance that your daily comer of customers will be greater than
317
That's application number 3
Moving on
The last application
Is number four and it asks you a little tougher problem it says look there's a 90% probability
That the number of customers will allow within what two values symmetric around the mean
What does that look like well in terms of a Z
It means we're gonna put 10% in the two tails.
I want five percent in each tail since it's symmetric
So I'm going to put five percent in the upper tail find that Z
Put five percent in the lower tail find the negative Z
The two together then add up to the 10 percent in the tails
90 percent remaining between those two values
So, 90 percent between the two
Symmetric Z's plus and minus means I'm putting five percent probability in each tail
Well, that's easy with Normsinv. I just look for the negative Z with five percent in the lower tail
Let's go to Normsinv
And again since I already have it pulled up
I'm only gonna put five percent don't go 0.5
That's 50 percent guys go .05, five percent in the lower tail
Hit enter
And you'll get a minus Z of -1.64485
Since this is symmetric then the positive Z will be +1.66485
Going back to my math
That means this is -1.644. This is +1.644
I don't want Z values though. I need to convert those into
X's
customers centered at 278
to do that
Let's do a little calculation
The lower value is going to be the 278 minus 1.64485 times 46
202 customers the upper value will be 278 plus
the 1.64485 times 46 or
354
So what you will conclude
looking at our diagram down here
Is that 90% of the time your daily customers will be between 202 and
354. We will restate it: On 90% of days, your number of customers will lie between
202 and
354