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well look at that we're already to
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chapter 12
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and we're only doing one section so this
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is actually the last section
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for the whole semester so this is on
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anova which stands for analysis
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of variance and remember variance is
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just the standard deviation squared
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so in the past we were able to use the t
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test to taste the claim about two means
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but what if it wasn't just two means
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what if you had three or more groups
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and you wanted to see if the mean was
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equal for those three groups
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so the claim could look something like
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this mean one equals mean
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two equals mean three and you could even
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have four
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five six seven you could do a whole
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bunch of groups all at the same time
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with one simple test
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so for each group you have to find the
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mean and standard deviation
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and then we're going to see how much
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variation there is in the three means
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in other words if the three groups are
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the same
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an average of five an average of five an
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average of five point one
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they're all pretty much the same then
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the variance of those means is going to
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be very very small
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so we're going to use what's called the
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f test and the critical value is going
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to be in table a5
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now for this f test and for this anova
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test you can actually use the calculator
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but i'm going to do a simple example
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using the formula to show you the idea
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of where it came from
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and then after that i'll show you how to
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use the calculator
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all right here we go so use anova to
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test the claim of equal means
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here's these three groups so perhaps
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this is
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three different classes trigonometry
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statistics and calculus something like
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that and then these are the ages of the
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students
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so just by looking at them does it look
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like there's a difference
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well to me these people look pretty
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young
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these people are in their early 20s and
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then these people
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are in their late 20s to early 30s so
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group three definitely looks
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older so i think that at the end i'm
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gonna get reject
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they are not the same okay so
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first find the mean and standard
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deviation each group has five people
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sure enough for these people they're
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younger 19.8
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for these people they're older 29.2 and
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we also need the standard deviation for
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each group
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okay so that's pretty much what i just
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said
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so here's the same data and now find the
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overall mean
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so what is the average of the three
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averages so just add up these three
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averages divide by three and the overall
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average is 23.867
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so how much do the means vary from
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23.867
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so like i said we can use the calculator
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i'll show you
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very easy very fast using the calculator
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but this is what the formula is
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basically saying
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how much does this group vary from the
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average how much does this group vary
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from the average and how much does this
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group vary from the average
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since i'm not taking the square root of
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it this is the variance called the
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variance of the means
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so that's a 23.29
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now also we need to pay attention to how
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much each individual group
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varies or how much variance there is for
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each one
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so variance just means s squared so you
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square each one
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add them up and divide by 3.
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now the formula for the test statistic
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f is going to be n the sample size
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times the variance of the means that's
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this one
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divided by which called this is called
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the pooled variance which is this 4.897
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or the mean of the variances
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so just reiterating the top one is going
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to be 23.92
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and the bottom one is going to be 4.897
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now when you divide those the test
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statistic turns out to be 23.78
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and then use table a5 it's a right till
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only test
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and degrees of freedom number one is
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the number of categories so there is
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three groups so k equals three
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so categories minus one that's degrees
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of freedom number one
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degrees of freedom number two will be
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that three times n minus one
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so this was five people in each group so
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this would be three
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times five minus one
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so there are the two degrees of freedom
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a two and a twelve
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so on table 85 i go to 2 for the top one
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12 for the side one so it should be this
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one right here
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let's see if the magical rectangle
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agrees ah yes
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so the critical value is 3.8853
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so i'm using 95 percent confidence
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that's why this
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right here is a 0.05 or right here it
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says .05 in the right tail because it's
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a right till only
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and we're using 95 confidence
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so then you draw the little picture and
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the critical value is 3.8853
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the test statistic was 23.78
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which goes way past three point eight
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eight five three the critical value that
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means reject the claim
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they are not the same so
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in intuitively
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now just by looking at the numbers at
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the beginning i was trying to get all
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fancy sorry about that
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um from the beginning i could see group
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one was younger
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group three was older now i have actual
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proof
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the means are not equal because group
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one is younger
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group three is older okay now how to use
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the calculator
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that's right you don't have to use the
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formula
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so put the data into list one list two
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list three
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and of course if you have more than
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three you can go ahead and use list four
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list five list six
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so there are my numbers i typed them
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into the good old ti 84
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list one list two list three then you go
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to stat
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tests and go down to the bottom
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so the one we've used so much
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stat and then tests
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and you can scroll down to the bottom
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but because it's the bottom you could
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also scroll
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up and then it says anova right there
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i don't know if you can see it but i'll
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show you in just a second
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so it will say anova and then you just
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have to put the list make sure you put a
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comma
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in between so it has to be list one
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comma list two comma list three
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and then hit enter
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now the test statistic right here
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i think this is even a little more
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accurate than i had because mine had a
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little bit of rounding error
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because i was working with three
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decimals and this is working with even
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more decimals so this
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is not only faster easier funner
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but it's more accurate as well so that's
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the test statistic
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f and then for the degrees of freedom
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you don't even have to do the formula
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for that
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because this is degrees of freedom
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number one
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and then there's this little arrow
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saying scroll down so if you hit the
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down button
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it will then tell you degrees of freedom
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number one
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is a two and the second degrees of
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freedom is a twelve
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you can just ignore these words factor
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and error
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and ignore the ss and the ms all you
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need is degrees of freedom number one
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and degrees of freedom number two and
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that's it
