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all right this video is going to be a
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very quick review of the normal model
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I'm not gonna lie to you the normal
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model is probably one of the most used
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models in all of statistics it could be
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used to do with distributions of data it
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can be used in probability it could be
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used in sampling it could literally be
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used in any situation that is normal and
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we see this normal model popping up
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across many different topics and all
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statistics in fact I often tell students
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that if you simply understand the normal
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model I guess I can't guarantee that
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you're gonna get it through on the AP
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stats test but you're gonna get a lot of
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questions right if you just understand
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the normal model alright so what do you
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need to know well all you need is the
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mean and standard deviation and there's
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nothing you cannot do all you need when
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you're working with the normal model is
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the mean and standard deviation in fact
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the normal model is defined by nothing
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more than the mean and the standard
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deviation of the data now to help you
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with the normal model you're going to
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need two functions on your calculator
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normal CDF on your calculator to find
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the area above below or in between and
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invert norm if you know the area above
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or below and you want to find the value
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now if you have been somewhat familiar
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with invert Norman normal CDF you should
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be ok with this now some teachers may
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have you use normal model formulas
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sheets or normal model calculation
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sheets where you have to look up all
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this information I don't do that in my
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classroom I let my kids use their ti-84
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normal CDF and invert norm hopefully
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your teacher does the same if not pay
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attention this video and I'll go over at
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all alright remember the normal model is
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this beautiful wonder normal model that
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is bell-shaped right the normal model
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says well smack dab in the middle is
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what you expect that's the mean you
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always expect the mean to happen but the
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further you move above or below the mean
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the less likely those types of values
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occur so we go one standard deviation
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above to send deviations above three
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cent deviations above one below one to
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below and then three below now
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traditionally we know that 68% of data
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is within one standard deviation that is
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why if you go all the way back to the
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very beginning of ap statistics we said
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that most data
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within one standard deviation of the
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mean and that's true 60% is most of the
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data that is within one standard
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deviation plus or minus 95 percent of
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data is within two standard deviations
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and then ninety-nine point seven percent
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of data is within three standard
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deviations so very little data is
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outside of three standard deviations
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from the mean as long as we follow a
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normal model all right let's just jump
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right into some problems because working
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with the normal model is quite simple
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and the problems are very easy all right
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so the mean height of teenage girls is
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64 inches with a standard deviation of
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two point six five inches so the mean
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height is 64 the standard deviation is
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two point six five and it does say the
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distribution Falls normal model which
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means we're free to use all of our
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normal model calculations now the only
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thing I'm going to say before we begin
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is the normal model is universal between
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many many many different scenarios so to
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use the normal model you need z-scores
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z-scores remember what's the z-score how
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many standard deviations you are above
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or below the mean that's exactly what a
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normal model shows so you always need
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z-scores all right so what percent of
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girls are above sixty six inches tall
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well if I'm thinking about that normal
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model there it is right smack dab in the
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middle of 64 up 2.5 I'm sorry two point
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six five up another up another down down
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down all right where does 66 fall well I
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can't answer what percent of girls are
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above sixty-six until I find were 66
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Falls so I need the z-score 66 minus the
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mean divided by the standard deviation
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right sorry for my very crappy writing
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but you'll get used to it
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sixty-six minus 64 divided by two point
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six five is 0.75 five okay which means
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that if you are 66 inches tall you fall
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somewhere right around here point seven
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five not even a full standard deviation
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above the mean now how do I find the
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percent above it well you could get out
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what's called a Z
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or z-table I'm not going to make anybody
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do that I'm just gonna ask that you use
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your calculator so you're gonna go to
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normal CDF we want to look above this so
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I'm gonna start at my z-score 0.75 5
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remembered normal CDF only speaks the
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language of z-scores now I need to go
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all the way up forever well that would
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be infinity and I don't have an infinity
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but on my calculator so I'm just gonna
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put a bunch of nines there to represent
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that I'm going way up leave your mean
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and standard deviation at 0 and 1 that's
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because those are the z-scores for a
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normal model right and normal model in
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terms of z-scores has a mean of 0
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because it is 0 standard deviations from
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itself and it has a standard deviation
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of 1 and hit enter couple times and boom
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twenty two point five one percent so
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twenty two point five one percent of
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girls are above six inches 66 inches
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tall alright what percent of girls are
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below 55 inches well once again I gotta
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find where 55 inches falls on the model
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so I'm going to find my z-score for 55
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inches by subtracting the mean and
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dividing by the standard deviation
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so in 55 minus 64 divided by my own TV
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ation is negative three point three nine
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six negative three point three nine six
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now if you understand the role model
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this is very unlikely very rare for a
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girl to be below 55 inches not one not
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two not three but somewhere right around
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here negative three point three nine six
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that's a very very low so to find that
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percent of girls that are below that I'm
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going to go and grab normal CDF now I
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want to look below so remember how
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normal CDF works that works from a lower
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value to an upper value so I want to
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look below so I'm gonna start way down
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at negative infinity way below well
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that's gonna be negative nine nine nine
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nine nine right because there is no
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infinity but under calculator and I'm
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gonna stop at my z-score of negative
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three point three nine six so what I'm
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doing is I'm asking the calculator to
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look at the normal model and
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me the percentage of data less than my
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z-score of negative three point three
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nine six hit enter enter enter enter and
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I get a very low number notice the e to
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the negative four that means I have to
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move the decimal four times to the left
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which will produce point zero zero zero
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three four two now it I'm sorry that
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should be up here now as a percent I got
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to move the decimal twice that would be
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point O 3 4 2 percent yeah
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that's pretty unlikely very very
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unexpected for a girl to be under 55
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inches tall a teenage girl under 55
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inches tall very very unlikely the
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probability or that I'm sorry the
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percentage of girls that are at that
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height or lower is under 1% very very
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low easy right that's how I use the
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normal model all right what percent what
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percentile is a girl at 69 inches well
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well this question is technically no
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different in the previous one because
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all you have to remember is the
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definition of a percentile is the
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percent of data below so all I have to
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do is figure out what percent of girls
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are below 69 inches and I'll have my
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answer because that's the definition of
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a percentile the percentage below so
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once again I have to figure out where
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does 69 inches fall on my model well to
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do that I need a z-score 69-64 divided
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by two point six five then I get a
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z-score of one point eight eight seven
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all right now one point eight eight
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seven here's one one point eight eight
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so that would be somewhere right around
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here so remember the definition of a
00:08:32
percentile is the percentage below so
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now I got to do is find the percent
00:08:36
below using normalcdf once again please
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remember when you're looking below you
00:08:41
want to actually start at negative
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infinity that is way below and you want
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to stop at your z-score of one point
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eight eight seven
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and this should be fairly large because
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I'm looking way below and yep it's at
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the 97th percentile so the 97th
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percentile so if you are a girl who's
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six nine inches tall
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consider yourself tall because only 3%
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of girls are taller than you and 97% of
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girls are shorter than you so you are
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fairly tall all right let's do one more
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question dealing with the height of
00:09:17
girls so if a girl is at the fourth
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percentile how tall is she so this is
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basically a problem I want to work
00:09:23
backwards a girl is at the 4th
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percentile I won't know how tall she is
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so this means that 40% of girls are
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shorter than her so 40% below well this
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is where I'm going to use invert normal
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invert norm is a very cool feature of
00:09:41
your calculator where you tell it the
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area below when it says area
00:09:47
you got to type in the area below well
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that's exactly what a percentile is so
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I'm going to type in point 400 there
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because I'm trying to find the percent
00:09:56
below so I know 40% below now when I do
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this what the calculator is going to
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give me is the z-score remember z-scores
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are the universal language of statistics
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so when I get enter here in a moment
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it's going to give me the z-score that
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has 40% below it all you have to do is
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type in 40% point 4 and it'll tell you
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the z-score so easy score 0.25 three-
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point two five three that is the z-score
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that represents 40% below I just have to
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forgot what height that is while I'm
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still work backwards the form of the for
00:10:35
a z-score is a height minus the mean
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divided by the standard deviation I
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don't know what that height is but I
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could solve for it so I know the z-score
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I'm trying to solve for that x-value all
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I got to do is multiply the 2.65 over
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and then add the 64 so I'm going to take
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the negative 0.25 three look I'm gonna
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clear this out sorry let's see here
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negative
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two five three I'm going to multiply the
00:11:06
standard deviation and then I'm going to
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add the 64 and I get that sixty three
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point three three inches so if you are
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sixty three point three three inches
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tall that will produce a z-score of
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negative 0.25 three which in turn puts
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you at the fortieth percentile so if you
00:11:28
a girl who is around sixty three point
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three three inches tall about forty
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percent of girls are shorter than you
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hence you fall at the fortieth
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percentile pretty easy now I know
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through I went through all that quick
00:11:39
guys but we've learned this all before
00:11:40
hopefully this is nothing more than a
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quick review for you alright
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now the cool thing about normal models
00:11:47
it could be used in lots of applications
00:11:49
so here's another application it can be
00:11:51
used with random variables remember
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random variable is a number that you
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don't know right for example how long is
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the song on the radio I don't know how
00:12:01
long a song on the radio is there's
00:12:04
probably a mean but they probably could
00:12:07
deviate right oh this is cool I'm giving
00:12:10
you everything you need so the mean
00:12:13
length of a song on the radio I'll use
00:12:15
an S for song is three point four
00:12:18
minutes but guess what
00:12:19
songs deviate so the standard deviation
00:12:21
for a song is 0.8 and as long as the
00:12:26
problem says it follows a normal model
00:12:28
which I hopefully it will then I could
00:12:30
figure this out right because a random
00:12:32
variable is hey I wonder how long that
00:12:34
song is I don't know that's four in a
00:12:37
variable right it's I don't know the
00:12:38
answer that's the whole point of it
00:12:39
being a random variable but if it
00:12:42
follows a normal model then I do know a
00:12:44
lot for example I know that three point
00:12:47
four falls right in the center because
00:12:48
the average length of a song is three
00:12:50
point four but I could go up point eight
00:12:54
and that would take me to four point two
00:12:57
I can go up another point eight and that
00:13:00
would take me to five I can go up
00:13:02
another four point eight another point
00:13:04
eight and that would take me to five
00:13:06
point eight I could also go down that
00:13:10
would take me to two point six I can go
00:13:13
down another point eight that would take
00:13:15
me to one point eight
00:13:17
and I could go down another point eight
00:13:19
and that would take me to one so what
00:13:21
this tells me is it would be very very
00:13:24
weird for a song to be shorter than one
00:13:25
minute don't think there's many songs
00:13:27
out there Shore than one minute and
00:13:28
there's very few songs that are over
00:13:31
five point eight minutes
00:13:32
all right but now that I have the normal
00:13:34
model I can answer probability questions
00:13:36
with it right because that's what where
00:13:38
the variables are all about what's the
00:13:39
probability that a variable is this or
00:13:41
that well here we go what is the
00:13:43
probability that the next song I hear on
00:13:45
the radio is over four minutes okay well
00:13:48
the first thing I got to figure out is
00:13:49
where does four minutes follow my model
00:13:51
over the z score so four minus three
00:13:54
point four divided by point eight four
00:13:58
minus three point four divided by 0.8 is
00:14:02
0.75 so a song that's four minutes long
00:14:06
will fall right about here not even once
00:14:09
an intubation above mean so how do I
00:14:12
find the probability that song is over
00:14:14
four minutes oh I just need the normal
00:14:17
CDF so once again just remember that
00:14:20
normal CDF only deals with z-scores so
00:14:22
I'm gonna start at my z-score 0.75 I'm
00:14:25
gonna go all the way up towards infinity
00:14:27
which is basically a bunch of nines and
00:14:30
I'm gonna hone hit pace point two two
00:14:32
six six so point two two six six or
00:14:36
twenty two point six seven percent is
00:14:38
the probability that the next song on
00:14:40
the radio is over four minutes guys the
00:14:43
normal model is awesome it can be used
00:14:45
in so many situations all right now
00:14:48
let's answer this question because this
00:14:50
is a good one this is actually going to
00:14:52
incorporate a couple different ideas
00:14:53
that I'm trying to review with you what
00:14:54
is the probability that twenty songs
00:14:57
will fit into six minutes okay well we
00:15:00
know that one song is supposed to be
00:15:02
three point four and deviate by 0.8 well
00:15:06
I'm not talking about one song anymore
00:15:08
I'm talking about twenty songs so what's
00:15:12
the average for twenty songs
00:15:14
well three point four for each song
00:15:18
would be three point four times twenty
00:15:21
68 total minutes so probably going to be
00:15:25
kind of on the low side that I can fit
00:15:28
20
00:15:28
songs in 60 minutes but that's what I'm
00:15:31
trying to find out what's the
00:15:32
probability that it does happen
00:15:33
all right so 20 songs are supposed to
00:15:35
take 68 minutes all right what about the
00:15:38
standard deviation for 20 songs well hmm
00:15:42
I'm not allowed to just combine standard
00:15:46
deviations so I can't just times it by
00:15:48
20 because you know remember what
00:15:50
timesing by 20 is right I'm doing three
00:15:51
point four for their first song three
00:15:53
point four for the second Psalm the
00:15:55
three point four for the third song
00:15:56
yadda yadda yadda but I'm just speeding
00:15:58
that up by doing three point four times
00:16:00
20 I'm not allowed to do that with
00:16:02
standard deviation cuz you're not
00:16:03
allowed to repeatedly add standard
00:16:05
deviations together but but but remember
00:16:07
what you are allowed to do you are
00:16:09
allowed to work with variance the
00:16:11
variance for one song is point eight
00:16:13
squared
00:16:13
variance is just standard deviation
00:16:15
squared multiply that by 24 24 variance
00:16:21
but then don't forget to square root all
00:16:24
of that to get back to a standard
00:16:26
deviation so let's see what that would
00:16:29
be so the square root would be 0.8
00:16:32
squared times 20 so my standard
00:16:36
deviation would be three point five
00:16:39
seven eight all right so now that I
00:16:45
understand what 20 songs look like 20
00:16:47
songs should be about 68 minutes logged
00:16:50
but it could deviate by three point five
00:16:52
seven eight so now I'm trying to find
00:16:54
the probability that I fit into 60
00:16:56
minutes well as long as I'm less than 60
00:16:59
I'm gonna fit so you know I could draw a
00:17:01
normal model if I really wanted to
00:17:02
showing my 68 minutes in the middle up
00:17:05
up short short short down down down
00:17:08
right by my standard deviation but you
00:17:11
know essentially all I got to do is
00:17:12
figure out where does 60 minutes fall so
00:17:15
I'm gonna find the z-score for 60
00:17:16
minutes now remember 20 songs is
00:17:18
supposed to be 68 minutes standard
00:17:20
deviation for 20 songs is three point
00:17:22
five seven eightt so let's see what this
00:17:25
would be this would be negative two
00:17:29
point two three six negative two point
00:17:33
two three six alright so that is my
00:17:36
z-score so 60 minutes would fall
00:17:38
somewhere down here so if I need my
00:17:41
twenty saw
00:17:42
to fit in 60 minutes it's got to be
00:17:44
something less than 60 because anything
00:17:46
more than 60 it's not gonna fit right so
00:17:50
all I'm gonna do now is go and grab my
00:17:52
normal CDF I need to look less than my
00:17:57
z-score so I'm gonna start way down at
00:17:59
negative 9 9 9 9 9 I'm gonna stop at my
00:18:02
z-score of negative 2 point 2 3 6 and
00:18:07
what this is going to do is this is
00:18:09
going to calculate the probability or
00:18:11
the percentage of data where this song
00:18:13
is the total for 20 songs is under 60
00:18:17
minutes all right so I get whoa very
00:18:20
unlikely one point two seven percent one
00:18:24
point two seven percent so if I own a
00:18:28
radio station and I need to play 20
00:18:30
songs in a 60 minute window mmm probably
00:18:34
not gonna happen I mean it could happen
00:18:36
it's not impossible but the probability
00:18:39
is very low so guys hopefully this was a
00:18:42
very quick video on how to use in the
00:18:43
normal model to be honest the normal
00:18:46
model comes up all over the AP test I
00:18:49
cannot tell you how many multiple-choice
00:18:50
questions are gonna somehow relate to
00:18:53
the normal model you need to understand
00:18:55
how the normal model works you need to
00:18:56
understand that it incorporates z-scores
00:18:58
it deals with percentiles and if you're
00:19:02
going to use your calculator make sure
00:19:03
you know how to use invert norm and
00:19:05
normal CDF those are the two keys to
00:19:08
being able to do any calculations
00:19:09
resulting in the normal model but you
00:19:11
need to know the mean you need to know
00:19:13
the standard deviation and then the
00:19:15
normal model becomes pretty easy to use
00:19:17
the normal model is gonna pop up in some
00:19:19
of these other review videos because it
00:19:21
really is something extremely important
00:19:23
that you need to know for the AP stats
00:19:25
tests
