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- [Instructor] Miriam was
testing her null hypothesis
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that the population mean of some data set
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is equal to 18 versus her
alternative hypothesis
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is that the mean is less than 18
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with a sample of seven observations.
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Her test statistic, I
can never say that right,
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was t is equal to negative 1.9.
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Assume that the conditions
for inference were met.
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What is the approximate p
value for Miriam's test?
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So, pause this video and see if you
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can figure this out on your own.
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Alright, what I always like
to remind ourselves what's
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going on here before I go ahead
and calculate the p value.
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There's some data set,
some population here
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and the null hypothesis is
that the true mean is 18,
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the alternative is that it's less than 18.
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To test that null hypothesis,
Miriam takes a sample,
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sample size is equal to seven.
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From that, she would
calculate her sample mean
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and her sample standard deviation,
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and from that, she would
calculate this t statistic.
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The way she would do that or if they
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didn't tell us ahead
of time what that was.
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We would say the t statistic
is equal to her sample mean,
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minus the assumed mean
from the null hypothesis,
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that's what we have over here, divided by
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and this is a mouthful, our approximation
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of the standard error of the mean.
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The way we get that approximation,
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we take our sample standard deviation
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and divide it by the square
root of our sample size.
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Well, they've calculated
this ahead of time for us.
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This is equal to negative 1.9.
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So, if we think about a t distribution,
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I'll try to hand draw a rough
t distribution really fast,
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and if this is the mean
of the t distribution,
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what we are curious about,
because our alternative
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hypothesis is that the
mean is less than 18.
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What we care about is,
what is the probability
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of getting a t value that
is more than 1.9 below
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the mean so this right
over here, negative 1.9.
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It's this area, right there.
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I'm gonna do this with a TI-84,
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at least an emulator of a TI-84.
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All we have to do is, we
would go to 2nd distribution
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and then I would use the t cumulative
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distribution function so let's go there,
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that's the number six
right there, click enter.
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My lower bound...
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Yeah, I essentially wanted
it to be negative infinity
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and we can just call
that negative infinity.
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It's an approximation of negative
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infinity, very, very low number.
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Our upper bound would be
negative 1.9, negative 1.9.
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And then our degrees of freedom,
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that's our sample size minus one.
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Our sample size is seven so our
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degrees of freedom would be six.
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There we have it.
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This would be, our p value
would be approximately 0.053.
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Our p value would be approximately 0.053.
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Then what Miriam would do is,
would compare this p value
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to her preset significance
level, to alpha.
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If this is below alpha,
then she would reject
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her null hypothesis, which
would suggest the alternative.
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If this is above alpha, then she would
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fail to reject her null hypothesis.