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Finding the Percentage under the Normal Distribution - YouTube
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okay so I had a few questions about
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finding the percentage under a normal
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distribution so this new video is gonna
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sort of address that and the problem
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that students were having was on this
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live stock company reports that the mean
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weight of a group of young steers is
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1099 pounds with a standard deviation of
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96 pounds and we want to know what the
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percentage of steers are gonna weigh in
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the following ranges and so the first
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one is over 1250 pounds right so um what
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I always do is I always start with a
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sketch of the normal curve oh my lord
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okay hold on a second okay let's try
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this again
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all right I'm just gonna tight
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so the first thing you want is a sketch
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of the normal distribution now I hope
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that this actually turns out okay I'm
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gonna try to draw the sketch and so I
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wanna horizontal line right here and
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then I want to draw that nice normal
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curve that's uni-modal and comes down
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and gets close to the axis there and
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then of course right in the middle is
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the mean and that's the 1099 pounds so
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right here I'm gonna have 1099 pounds
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okay and then here where I've got this
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point of inflection that's gonna be my
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first standard deviation right and so to
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do that I need my handy dandy calculator
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all right so I'm gonna take the 1099 and
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add to it one standard deviation which
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is this 1296 over here oops
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blossom on calculator all right so I'm
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gonna add 96 pounds to it and hit enter
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and so I get 1095 so I'm gonna go over
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here and do the 1195 and then I'm gonna
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go out one more standard deviation right
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because I would like to actually get
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past 1250 so I'm gonna go out one more
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standard deviation and so that's 1099
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plus and now I'm at the second standard
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deviation right so I'm going out two
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times ninety-six and hit enter and
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that's 1291 so that's far enough so here
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we go 1291
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now have to go the other direction right
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so here's my point of inflection so I'm
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going to go back this way right and so
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now I am at negative one standard
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deviation so I actually need to subtract
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the 96 pounds now when I go 1099 and
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then minus 96 pounds and that's gonna
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give me 1,000 three pounds
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and there's like a right there's my need
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one standard deviation two standard
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deviations and now I have to go back one
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more standard deviation so I'm gonna be
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back here and that's a negative 270
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asian so I'm gonna take my mean and I'm
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gonna subtract from it two standard
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deviations to get that other value which
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is 907 and so this is my normal
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distribution for the steers okay now
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once I sketch my normal distribution uh
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what I want to do is oh no what I want
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to do is find my z-score
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so wanna find my z-score for the cutoff
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value all right so what does that mean
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so what that means is I'm going to take
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this twelve hundred and fifty pounds
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right here and that's basically my
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cutoff value and I want to find the
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z-score for that right so let's do that
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so my z-score for 1250 is gonna be equal
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to 1250 minus 1000 99 pounds all divided
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by 96 right and that's gonna give me a
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value over here I'm gonna plug that into
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my handy-dandy calculator using the
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appropriate set of parentheses 1250 -
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1099 divided by 96 and I get one point
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five seven which hopefully makes sense
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because here's one standard deviation
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right here at eleven ninety five and
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here's two standard deviations at 1291
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and so basically I'm right here
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I'm right here with a 1250 that's equal
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to one point five seven standard
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deviations above right and now I'm
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actually doing over 1250 so what I'm
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gonna do is I'm gonna shade two over is
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above so I'm going to shade in this
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direction so I'm headed out here toward
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that 100 standard deviation cutoff that
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we're sort of making to the far right
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okay and that's gonna make sense here in
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a moment all right so now I'm gonna
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raise this up just a little bit so that
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we can see I'm just a tiny bit better to
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use the normal CDF remember the normal
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CDF function requires a left z-score and
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then a comma and then you're right
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z-score okay
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and that is gonna give you the decimal
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area under the curve okay
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and so that makes us super happy so
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let's go ahead and see how we do that in
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the calculator all right so I'm gonna
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have my normal CDF all right and that
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left z-score is the 1.57 that we got
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from over here right from our calculator
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all right and that's under the
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distribution button second distribution
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and the normal CDF is number two all
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right so I'm going to type that in and
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so now it's asking for a lower bound
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okay and so that's going to be the one
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point five seven all right
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so I'm going to type in 1.57 and hit
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enter and then the upper is this 100 to
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the far-right right I'm just gonna go
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that far over to 100 okay
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and so I'm gonna type that in and that's
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gonna give me a decimal answer now
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remember the MU is going to be 0 right
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because 0 is in the middle of my normal
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distribution and each standard deviation
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is a 1 so there's 1 2 3 right and so
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these guys are gonna stay the same so I
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hit enter enter and paste it in and hit
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enter again and basically just hit enter
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until you get this nice decimal right
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here point zero five eight okay and so
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I'm gonna put in point zero five eight
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okay and if I convert that to a
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percentage I'm gonna move the decimal
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over two places and that's going to give
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me five point eight percent all right so
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the conclusion is if I want to scroll up
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just a little bit so I can see the
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conclusion my conclusion is I predict I
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expect predict I should find that five
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point eight percent of steers way over
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1,250 pounds okay so that's what I'm
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predicting there right now this is the
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the first one okay which was over 1,250
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now what if I wanted to know how many of
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my steers were gonna be underweight
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right so let's say I go over here and
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let's say don't go down here thank you
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for B I want to figure out a steer
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that's maybe under 1,100 pounds okay
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well the the first thing that I'm gonna
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do with this under 1,100 pounds so I'm
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gonna just sketch this again okay oh
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look it's all going crazy I'm gonna
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sketch this again and the reason I'm
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going to sketch this again is because of
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this shading I don't want it to mess me
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up all right especially all those boxes
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ah look at all the boxes all right so
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what I want to do is sketch the curve
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I'm going to do that real quick here
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and again the 1099 is getting me in the
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center I've got my above I've got my
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below over here and of course I can just
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use the 907 1003 907 1003 and then of
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course the 11 95 and 1291 11 95 1291
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okay and then of course I want to cut
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off at 1100 pounds so 1100 pounds Wow
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1100 pound is he's actually right over
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here it's 1 pound more than this so I'm
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I'm like right here right at my 1100
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pounds and it's under that right so I'm
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gonna go this direction and it looks
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like it's gonna be a lot it looks like
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there's gonna be a lot in that direction
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okay and remember if I go out this way
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to super skinny remember this is
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negative 100 standard deviation shaded
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out to this tail the other direction
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it's positive 100 standard deviations
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this way is umm negative 100 right and
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so now what I want to do is find my
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z-score right so here be find z-score
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of 1,100 and that's going to be equal to
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1,100 - 1099 all over 96 and that's
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gonna give you 1 over 96 which is
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decimally approximately point zero one
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okay and so now I want to put that in
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the normal CDF right now here my lower
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score is the negative 100 and my upper
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score now is this point zero one right
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that it relates to right here at the
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upper end and when I do that second
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distribution number two right and that's
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gonna be from negative 100 to point zero
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one zero and then I'm just hitting enter
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until I get that number and so it's
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point five zero four one five zero four
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which makes sense right because I'm just
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past all right which makes sense cuz I'm
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just past halfway right so point five
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zero four which if I convert it is gonna
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be fifty point four percent and that
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makes us happy cuz that's about what I
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expect it anyway
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when this was just barely past halfway
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and halfway being 50% I knew I was just
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gonna be a little over 50% and so that
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really makes me happy okay so that's how
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you find the left end or a right end and
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that's how you sketch it and set it up
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I hope you enjoyed the video if you did
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please subscribe to my channel and that
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makes me happy
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