WEBVTT
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We now discuss how to come up
with confidence intervals when
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we try to estimate the unknown
mean of some random variable,
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or of some distribution,
using the
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sample mean as our estimator.
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So here X1 up to Xn are
independent, identically
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distributed random variables
that are drawn from a
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distribution that has a certain
mean theta, the
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quantity that we want to
estimate, and some variance
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sigma squared.
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Let us say that we want
to construct a
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95% confidence interval.
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Our starting point will be the
fact that the sample mean,
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according to the central limit
theorem, can be described
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approximately using normal
distributions.
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And we look up at the normal
table, and we observe the
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following--
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that if we take a standard
normal random variable, then
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there is probability, 97.5% of
falling below this number,
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1.96, which means that there
is probability 2 1/2% of
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falling above that number.
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And by symmetry, the probability
of falling below
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minus 1.96 is also 2 1/2%.
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This means that this middle
interval here has probability
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95%, and we exploit this
fact as follows.
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If we take the sample mean,
subtract the true mean, and
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then divide by the standard
deviation of the sample mean,
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then we obtain a random
variable, which is
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approximately a standard
normal.
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Therefore, what we have here
is the probability of an
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approximately standard normal
random variable.
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Or actually, its absolute value
falling below 1.96.
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This is just the event that
our standard normal falls
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inside this middle interval
here, according to this entry
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from the normal tables and the
previous discussion, this
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probability is going to
be approximately 95%.
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And now we take this statement,
send this term to
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the other side of the
inequality, and then interpret
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what it means for an
absolute value to
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be less than something.
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And we obtain an equivalent
statement.
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This event here is algebraically
identical to the
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event that we have up there, and
this provides us with the
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desired confidence interval.
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We think of this quantity here
as the lower end of the
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confidence interval.
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This quantity here is
the upper end of
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the confidence interval.
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And this statement tells us
that there is probability
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approximately equal to 95% that
the confidence interval
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constructed this way contains
the true value
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of the unknown parameter.
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So this is how we obtain a
95% confidence interval.
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If instead we wanted a 90%
confidence interval, we would
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proceed in more or less
in the same way.
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Here, we would want to
have the number 0.95.
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Why is that?
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We want this middle interval to
have probability 90%, which
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means that we want to have
probability 5% at
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each one of the tails.
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And then we look up at the
normal tables, and we find
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that the entry that gives us
probability 95% of being below
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that value is 1.645.
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So if we use 1.645 in place
of 1.96, we obtain a 90%
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confidence interval, and
similarly for other choices.
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For example, if we want a
99% confidence interval.
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There's only one issue that's
left to discuss, and this is
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the following.
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In order to obtain numerical
values for the endpoints of
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the confidence interval, we
need to know sigma, the
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standard deviation
of the random
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variables that we are observing.
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But if we do not know the value
of sigma, then we may
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have to do some additional
work.