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T-statistic confidence interval | Inferential statistics | Probability and Statistics | Khan Academy - YouTube
Channel: Khan Academy
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This is the same problem that
we had in the last video.
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But instead of trying to figure
out whether the data
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supplies sufficient evidence to
conclude that the engines
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meet the actual emissions
requirement, and all of the
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hypothesis testing, I thought I
would also use the same data
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that we had in the last video to
actually come up with a 95%
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confidence interval.
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So you could ignore the
question right here.
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You can ignore all of this.
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I'm just using that same data
to come up with a 95%
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confidence interval for the
actual mean emission for this
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new engine design.
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So we want to find a 95%
confidence interval.
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And as you could imagine,
because we only have 10
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samples right here, we're
going to want to use a
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T-distribution.
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And right down here
I have a T-table.
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And we want a 95% confidence
interval.
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So we want to think about the
range of T-values that 95-- or
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the range that 95% of T-values
will fall under.
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So let's think about this way.
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So let me draw a
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T-distribution right over here.
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So a T-distribution looks
very similar to a normal
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distribution but it
has fatter tails.
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This end and this end will be
fatter than in a normal
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distribution.
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And then we want to find an
interval, so if this is a
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normalized T-distribution the
mean is going to be 0.
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And we want to find interval
of T-values between some
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negative value here and some
positive value here that
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contains 95% of the
probability.
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So this right here
has to be 95%.
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And to figure what these
critical T-values are at this
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end and this end, we can
just use a T-table.
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And we're going to use the
two-sided version of this
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because we're symmetric
around the center.
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So you look at the two-sided,
we want a 95% confidence
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interval, so we're going to
look right over here, 95%
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confidence interval.
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We have 10 data points,
which means we have
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9 degrees of freedom.
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So 9 degrees of freedom for
our 10 data points.
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We just took 10 minus 1.
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So if we look over here, so for
a T-distribution with 9
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degrees of freedom, you're
going to have 95% of the
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probability is going to be
contained within a T-value
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of-- so the T-value is going
to be between negative, so
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this value right here is 2.262,
and this value right
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here is negative 2.262.
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That's what this right
here tells us.
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That if you contain all the
values that are less than
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2.262 away from the center of
your T-distribution, you will
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contain 95% of the
probability.
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So that is our T-distribution
right there.
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Let me make it very clear.
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This is our T-distribution.
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So if you randomly pick
a T-value from this
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T-distribution, it has a 95%
chance of being within this
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far from the mean.
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Or maybe we should
write this way.
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If I pick a random T-value, if
I take a random T-statistic--
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let me write it this way--
there's a 95% chance that a
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random T-statistic is going
to be less than 2.262, and
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greater than negative 2.262.
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95% percent chance.
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Now when we took this sample, we
could also derive a random
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T-statistic from this.
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We have our sample mean and our
sample standard deviation,
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our sample mean here is
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17.17-- figured that out in the
last video, just add these
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up, divide by 10-- and
our sample standard
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deviation here is 2.98.
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So the T-statistic that we can
derive from this information
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right over here-- so let me
write it over here-- the
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T-statistic that we could derive
from this, and you can
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view this T-statistic as being
a random sample from a
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T-distribution.
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A T-distribution with 9
degrees of freedom.
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So the T-statistic that we
could derive from that is
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going to be our mean, 17.17
minus the true mean of our
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population.
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Or actually you would say the
true mean of our sampling
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distribution, which is also
going to be the same as the
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true mean of our population,
because that's our population
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mean over there, divided by s,
which is 2.98 over the square
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root of our number of samples.
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We've seen this multiple
times.
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This right here is
the T-statistic.
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So by taking this sample you
can say that we've randomly
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sampled a T-statistic from
this 9 degree of freedom
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T-distribution.
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So there's a 95% chance that
this thing right over here is
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going to be between-- is going
to be less than 2.262 and
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greater than negative 2.262.
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So the 95% probability still
applies to this right here.
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Now we just have to do some
math, calculate these things.
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So let me get my
calculator out.
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And so let me just
calculate this
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denominator right over here.
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So we have 2.98 divided by
the square root of 10.
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So that's 0.9423.
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So what I'm going to do is I'm
going to multiply both sides
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of this equation by this
expression right over here.
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So if I do that-- so let me just
do that right over-- so
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if I multiply this entire-- this
is really two equations
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or two inequalities
I should say.
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That this quantity is greater
than this quantity and that
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this quantity's greater
than that quantity.
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But we can operate on all of
them at the same time, this
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entire inequality.
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So what we want to do is
multiply this entire
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inequality by this value
right over here.
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And we just calculated it at
that value-- let me write it
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over here-- that 2.98-- I'll
write it right over here--
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2.98 over the square root
of 10 is equal to 0.942.
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So if I multiplied this entire
inequality by 0.942 I get, on
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this left-hand side over here
I have negative 2.262 times
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0.942-- and it's a positive
number that we're multiplying
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the whole inequality by, so the
inequality signs are still
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going to be in the same
direction-- is less than--
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we're multiplying this whole
expression by the same
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expression in the denominator
so it'll cancel out.
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So we're just going to be less
than 17.17 minus our
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population mean, which is going
to be less than 2.262
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times, once again, 0.942.
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Let me scroll over to the
right a little bit.
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0.942.
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Just be clear, I'm just
multiplying all three sides of
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this inequality by this number
right over here.
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In the middle this
cancels out.
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So if I multiply-- I'll just
write it over here-- 0.942,
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0.942, 0.942.
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This and this is the same number
so that's why those
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cancel out.
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And now let's get the calculator
to figure out what
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these numbers are.
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So if we have the 0.942
times 2.262.
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So we're going to say
times 2.262 is 2.13.
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So this number right
over here on the
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right-hand side is 2.13.
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This number on the left is just
the negative of that.
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So it's negative 2.13.
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And then we still have our
inequalities-- is going to be
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less than 17.17 minus the mean,
which is less than 2.13.
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Now what I want to do is
I actually want to
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solve for this mean.
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And I don't like that negative
sign in the mean.
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I'd rather have this
swapped around.
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I'd rather have the
mean minus 17.17.
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So what I'm going to do is
multiply this entire
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inequality by negative 1.
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If you do that, if you multiply
the entire thing
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times negative 1, this quantity
right here, this
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negative 2.13 will become
a positive 2.13.
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But since we are multiplying
an inequality by a negative
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number you have to swap
the inequality sign.
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So this less than will become
a greater than.
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This negative mu will become
a positive mu.
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This positive 17.17 will become
a negative 17.17.
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We're going to have to swap this
inequality sign as well,
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and this positive 2.13 will
become a negative 2.13.
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And we're almost there.
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We just want to solve for mu.
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Have this inequality expressed
in terms of mu.
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So what we can do is now just
add 17.17 to all three sides
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of this inequality, and we are
left with 2.13 plus 17.17 is
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greater than mu minus 17.17 plus
17.17 is just going to be
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mu, which is greater than-- so
this is greater than mu, which
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is greater than negative
2.13 plus 17.17.
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Or a more natural way to write
it since we actually have a
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bunch of greater than signs,
that this is actually the
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largest number and this-- oh
sorry, this is actually the
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smallest number and this over
here is actually the largest
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number, is actually flipped--
you can just re-write this
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inequality the other way.
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So now we can write-- actually
let's just figure out what
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these values are.
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So we have 2.13 plus 17.17.
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So that is the high
end of our range.
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So that is 19.3.
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So this value right over here,
so this is 19-- let me do it
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in that same color-- this value
right here is 19.3 is
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going to be greater than mu,
which is going to be greater
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than-- and this is negative
2.13 plus 17.17.
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Or we could have 17.17 minus
2.13, which gives us 15.04.
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And remember, the whole thing,
all of this, we started with,
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there was a 95% chance that a
random T-statistic will fall
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in this interval.
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We had a random T-statistic,
and all we did
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is a bunch of math.
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So there's a 95% chance that any
of these steps are true.
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So there's a 95% chance
that this is true.
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There's a 95% chance that the
true population mean, which is
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the same thing as the mean of
the sampling distribution of
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the sample mean, there's a 95%
chance, or that we are
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confident that there's a 95%
chance, that it will fall in
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this interval.
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And we're done.
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