We’re now on problem number 4

from the Normal Distribution chapter from ck12.org’s

FlexBook on AP Statistics. You can go to their

site to download it. It’s all for free. So problem number 4, and

it’s, at least in my mind, pretty good practice. For a normal, or a standard

normal distribution, place the following in order

from smallest to largest. So let’s see, percentage of

data below 1, negative 1. OK. Let’s draw our standard

normal distribution. So a standard

normal distribution is one where the

mean is– sorry, I drew the standard

deviation– is one where the mean, mu for

mean, is where the mean is equal to 0, and the standard

deviation is equal to 1. So let me draw that standard

normal distribution. Let’s see, so let me draw

the axis right like that. Let me see if I can draw

a nice-looking bell curve. So there’s the bell

curve right there. You get the idea. And this is a standard normal

distribution, so the mean, or you can kind view the

center point right here. It’s not skewed. This says the mean is

going to be 0 right there, and the standard deviation is 1. So if we go 1 standard

deviation to the right, that is going to be 1. If you go 2 standard

deviations, it’s going to be 2, 3 standard

deviations, 3, just like that. 1 standard deviation to the

left is going to be minus 1. 2 standard deviations

to the left will be minus 2, and

so on, and so forth. Minus 3 will be 3 standard

deviations to the left because the standard

deviation is 1. So let’s see if we can

answer this question. So what’s the percentage

of data below 1? Part a, that’s this

stuff right here. So everything below 1, so

it’s all of– well, not just that little center portion. It keeps going. Everything below 1,

percentage of data below 1. So this is another

situation where we should use the

empirical rule. Never hurts to

get more practice. Empirical rule, or

maybe the better way to remember the empirical rule

is just the 68, 95, 99.7 rule. And I call that a

better way because it essentially gives you the rule. These are just the

numbers that you have to essentially memorize. And if you have a calculator

or a normal distribution table, you don’t have to do this. But sometimes in

class, or people want you to estimate

percentages, and so you can impress people

if you can do this in your head. So let’s see if we can use

the empirical rule to answer this question, the area under

the bell curve all the way up to 1, or everything

to the left of 1. So the empirical rule tells

us that this middle area between 1 standard

deviation to the left and 1 standard deviation to the

right, that right there is 68%. We saw that in the

previous video as well. That’s what the

empirical rule tells us. Now, if that’s 68%, we

saw in the last video that everything else

combined, it all has to add up to 1 or to 100%,

that this left-hand tail– let me draw it a little

bit– this part right here plus this part

right here has to add up, when you add it to 68, has

to add up to 1 or to 100%. So those two combined are 32%. 32 plus 68 is 100. Now, this is symmetrical. These two things

are the exact same. So if they add up to 32,

this right here is 16%, and this right here is 16%. Now, the question,

they want us to know the area of everything–

let me do it in a new color–

everything less than 1, the percentage of data

below 1, so everything to the left of this point. So it’s the 68%. It’s right there,

so it’s 68%, which is this middle area within

1 standard deviation, plus this left

branch right there. So 68 plus 16%, which is what? That’s equal to 84%. So part a is 84%. They’re going to want us to

put this in order eventually, but it’s good to just

solve because that’s really the hard part. Once we know the numbers,

ordering is pretty easy. Part b, the percentage

of data below minus 1. So minus 1 is right there. So they really just want

us to figure out this area right here, the percentage

of data below minus 1. Well, that’s going to be 16%. We just figured that out. And you could have already

known just without even knowing the empirical, just looking

at a normal distribution, that this entire area, that

part b is a subset of part a, so it’s going to be

a smaller number. So if you just have

to order things, you could have made

that intuition, but it’s good to do practice

with the empirical rule. Now part c, they want to

know, what’s the mean? Well, that’s the easiest thing. The mean of a standard

normal distribution, by definition, is 0. So number c is 0. d, the standard deviation. Well, by definition,

the standard deviation for the standard normal

distribution is 1. So this is 1 right here. This is easier than I

thought it would be. Part e, the percentage

of data above 2. So they want the

percentage of data above 2. So we know from the

68, 95, 99.7 rule that if we want to

know how much data is within 2 standard

deviations– so let me do it in a new color. Let me do it in a more

vibrant color, green. If we’re looking from

this point to this point– so it’s within 2 standard

deviations, right, the standard

deviation here is 1– if we’re looking within

2 standard deviations, that whole area right there,

by the empirical rule, is 95%, within 2 standard deviations. This is 95%. Which tells us that

everything else combined– so if you take

this yellow portion right here and

this yellow portion right here, so everything

beyond 2 standard deviations in either direction– that

has to be the remainder. So you know everything

in the middle was 95 within 2

standard deviations. So that has to be 5%, if you

add that tail and that tail together, everything to

the left and right of 2 standard deviations. Well, I’ve made the

argument before, everything is symmetrical. This and this are equal. So this right here

is 2 and 1/2%, and this right here

is also 2 and 1/2%. So they’re asking

us the percentage of data above 2, that’s this

tail, just this tail right here, the percentage of data

more than 2 standard deviations away from the mean. So that’s 2 and 1/2%. Let me do it in a darker

color– 2 and 1/2%. Now, they’re asking

us, let’s see, place the following in order

from smallest to largest. So there’s a little

bit of ambiguity here. Because if they’re saying the

percentage of data below 1, do they want us to

say, well, it’s 84%. So should we consider

the answer to part a, 84? Or should we consider– if

they said the fraction of data below 1, I would say 0.84. So it depends on how they

want to interpret it. Same way here. The percentage of data below

minus 1, I could say the answer is 16. 16 is the percentage

below minus 1. But the actual number, if

I said the fraction of data below minus 1, I would say 0.16. So this actually would be very

different in how we order it. Similarly, if someone

me asked me the percent, I’d say, oh, that’s 2.5. But the actual number is 0.025. That’s the actual fraction

or the actual decimal. So I mean, this is

just ordering numbers, so I shouldn’t fixate

on this too much. But let’s just say that

they’re going with the decimal. So if we wanted

to do it that way, they want to do it from smallest

to largest, the smallest number we have here is c, right? That’s 0. And then the next smallest

is e, which is 0.025. Then the next smallest

is b, which is 0.16. And then the next one after

that is a, which is 0.84. And then the largest would

be the standard deviation, d. So the answer is c, e, b, a , d. And obviously, the

order would be different if the answer to this,

instead of saying it’s 0.84, if you said it was 84 because

it’s asking for the percentage. So a little bit of ambiguity. If we had a question

like this on the exam, I would clarify that

with the teacher. But hopefully you

found this useful.

It's kind of hard when numbers aren't whole numbers. My mean came out to 2.92 so it's very hard for me to graph it 🙁 But I can kind of understand how to do this better. My professor barely went over this so I'm pretty confused. This helps a lot 🙂

Brilliant!

Had to re-watch this video atleast 9 times, to get it down my brain but holy crap, was it worth it.

Thanks.

Thanx for your helping us brother.

1. what is the probability that a standard normal variate Z will be(a) greater than 1.09; (b) less than -1.65 (c) lying between -1.00 and 1.96; (d) lying between 1.25 and 2.75? please give solution

i love colorful graph..lol

I love you SAL! thankssss

thanks sal.. you are a good teacher

greatly helpful, thanks

Thank you so much, you explained this so well!!

Ye number C yeah lol

if the standard deviation is 1 shouldnt d be a subset of a??

Thank you so much. The teacher for statistics we should've gotten got sick at the start of the year. So we got another teacher, but unfortunately he understood everything, but wasn't able to explain it to us, litteraly everyone of our class didn't understand what he was talking about. And with these episodes I still maneged to get a 17/20

I feel like a>d, If question d is interpreted as the area 1 sd from the mean, which is 34%.

thank you

the standard deviation is one? how is that easy? what is the interval that you place it in?

This question isn't the best

isnt the standard deviation in data from 1 to 0.1? making it the third largest