- [Instructor] A set of laptop prices
are normally distributed with a mean of $750
and a standard deviation of $60.
What proportion of laptop prices
are between $624
and $768?
So let's think about what they are asking.
So we have a normal distribution for the prices.
So it would look something like this.
This is just my hand-drawn sketch
of a normal distribution.
So it would look something like this.
It should be symmetric,
so I'm making it as symmetric as I can hand draw it.
And we have the mean
right in the center.
So the mean
would be right there.
And that is $750.
They also tell us that we have a standard deviation of $60.
So that means one standard deviation above the mean
would be roughly right over here,
and that would be 750 plus 60.
So that would be $810.
One standard deviation below the mean
would put us right about there,
and that would be 750 minus $60,
which would be $690.
And then they tell us,
what proportion of laptop prices
are between $624
and $768.
So the lower bound,
$624,
that's going to actually be more than
another standard deviation less,
so that's going to be right around here.
So that is
$624.
And 768 would put us
right at about
right at about there,
and once again this is just a hand-drawn sketch.
But that is 768.
And so what proportion are between those two values?
So we wanna find essentially the area
under this distribution between these two values.
The way we are going to approach it,
we're going to figure out the z-score
for 768, it's going to be positive
because it's above the mean,
and then we're going to use a z-table to figure out,
what proportion is below 768.
So essentially we're going to figure out
this entire area.
We're even going to figure out the stuff
that's below 624.
That's what that z-table will give us.
Then we'll figure out the z-score for 624.
That will be negative 2 point something,
and we will use the z-table again
to figure out the proportion that is less than that.
And so then we can subtract this red area
from the proportion that is less than 768
to get this area in between.
So let's do that.
Let's figure out first the z-score for 768.
And then we'll do it for 624.
The z-score for 768, I'll write it like that,
is going to be
768 minus 750
over the standard deviation, over 60.
So this is going to be equal to 18
over 60,
which is the same thing as six over,
let's see if we divide the numerator
and the denominator by three,
6/20, and this is the same thing
as 0.30.
So that is the z-score for this upper bound.
Let's figure out what proportion is less than that.
For that, we take out a z-table,
get our z-table.
And let's see we wanna get 0.30.
And so this is 0.3, this first column,
and we've done this in other videos,
this goes up until the tenths place
for our z-score,
and then if we wanna go to our hundredths place,
that's what these other columns give us.
But we're at 0.3,
so we're going to be in
this row,
and our hundredths place is right over here,
it's a zero, so this is the proportion
that is less than $768.
So .6179.
So 0.61, 0.6179.
So now let's do the same exercise
but do it for the proportion that's below $624.
The z-score
for 624 is going to be equal to
624 minus the mean of 750,
all of that over 60.
And so what is that going to be?
I'll get my calculator out for this one,
don't wanna make a careless error.
624 minus 750
is equal to,
and then divide by 60,
is equal to negative 2.1.
So that lower bound is 2.1 standard deviations
below the mean,
or you could say it has a z-score of negative 2.1.
Is equal to negative 2.1.
And so to figure out the proportion that is less than that,
this red area right over here,
we go back to our z-table.
And we actually go to the first part of the z-table.
So same idea but this starts at negative,
a z-score of negative 3.4,
3.4 standard deviations below the mean.
And just like we saw before,
this is our zero hundredths, one hundredth,
two hundredths, so on and so forth.
And we wanna go to negative 2.1.
We could say negative 2.10 just to be precise.
So this is going to get us,
let's see negative 2.1,
there we go.
And so we are
negative 2.1, and it's negative 2.10,
so we have zero hundredths,
so we're gonna be right here on our table.
So we see the proportion that is less than 624
is .0179 or 0.0179.
So 0.0179.
And so if we wanna figure out the proportion
that's in between the two,
we just subtract this red area
from this entire area,
the entire proportion that's less than 768
to get what's in between.
0.6179,
once again I know I keep repeating it,
that's this entire area right over here,
and we're gonna subtract out what we have in red,
minus 0.0179,
so we're gonna subtract this out,
to get 0.6.
So if we wanna give our answer to four decimal places,
it would be 0.6000,
or another way to think about it is
exactly 60% is
between 624 and 768.