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## Normal distribution calculations

# Standard normal table for proportion below

AP.STATS:

VAR‑2 (EU)

, VAR‑2.B (LO)

, VAR‑2.B.3 (EK)

CCSS.Math: ## Video transcript

- [Instructor] A set of middle
school students' heights are normally distributed with
a mean of 150 centimesters and a standard deviation
of 20 centimeters. Darnell is a middle school school student with a height of 161.4 centimeters. What proportion of student heights are lower than Darnell's height? So, let's think about
what they are asking. So, they're saying that heights
are normally distributed. So, it would have a shape that
looks something like that. That's my hand-drawn version of it. There's a mean of 150
centimeters, so right over here. That would be 150 centimeters. They tell us that there's a standard deviation of 20 centimeters and Darnell has a height
of 161.4 centimeters. So, Darnell is above the mean, so let's say he is right over here, and I'm not drawing it
exactly, but you get the idea. That is 161.4 centimeters. And we wanna figure out what
proportion of students' heights are lower than Darnell's height. So, we wanna figure out what is the area under the normal curve right over here. That will give us the proportion that are lower than Darnell's height. And I'll give you a
hint on how to do this. We need to think about how
many standard deviations above the mean is Darnell,
and we can do that because they tell us what
the standard deviation is and we know the difference between Darnell's height
and the mean height, and then once we know how
many standard deviations he is above the mean, that's our z score, we can look at a z table
to tell us what proportion is less than that amount
in a normal distribution. So let's do that. So, I have my TI-84 Emulator
right over here and let's see. Darnell is 161.4 centimeters, 161.4. Now, the mean is 150,
minus 150 is equal to, we coulda done that in our
head, 11.4 centimeters. Now, how many standard deviations
is that above the mean? Well, they tell us that
the standard deviation in this case for this
distribution is 20 centimeters, so we'll take 11.4 divided by 20. So, we will just take our previous answer, so this just means our previous answer divided by 20 centimeters
and that gets us 0.57. So we can say that this is 0.57 standard deviations,
deviations above the mean. Now, why is that useful? Well, you could take this
z score right over here and look at a z table to
figure out what proportion is less than 0.57 standard
deviations above the mean. So, let's get a z table over here. So, what we're going to do is we're gonna look up this
z score on this table, and the way that you do
it, this first column, each rows tells us our z score
up until the tenths place and then each of these columns after that tell us which hundredths we're in. So, 0.57, the tenths
place is right over here, so we're going to be in this row. And then our hundredths
place is this seven, so we'll look right over here. So, 0.57, this tells us the proportion that is lower than 0.57 standard
deviations above the mean, and so it is 0.7157 or
another way to think about is if the heights are
truly normally distributed, 71.57% of the students would have a height less than Darnell's. But the answer to this question, what proportion of students' heights are lower than Darnell's height? Well, that would be 0.7157, and they want our answer
to four decimal places, which is exactly what we have done.

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