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Unformatted text preview: = 0.1, with δ = 0.02
ˆ
and n = 1000.
(b) If p = 0.5 and n = 1000, ﬁnd δ so P {p−p ≤ δ } ≈ 0.99. Equivalently, P {p ∈ [ˆ−δ, p+δ ]} ≈ 0.99.
ˆ
p
ˆ
Note that p is not random, but the conﬁdence interval [ˆ − δ, p + δ ] is random. So we want to ﬁnd
p
ˆ
the halfwidth δ of the interval so we have 99% conﬁdence that the interval will contain the true
value of p.
(c) Repeat part (b), but for p = 0.1.
(d) However, the campus network engineer doesn’t know p to begin with, so she can’t select the
halfwidth δ of the conﬁdence interval as a function of p. A reasonable approach is to select δ so
that, the Gaussian approximation to P {p ∈ [ˆ − δ, p + δ ]} is greater than or equal to 0.99 for any
p
ˆ
value of p. Find such a δ for n = 1000.
(e) Using the same approach as in part (d), what n is needed (not depending on p) so that the
random conﬁdence interval [ˆ − 0.01, p + 0.01] contains p with probability at least 0.99 (according
p
ˆ
to the Gaussian approximation of the binomial)? 3.7. ML PARAMETER ESTIMATION FOR CONTIN...
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 Spring '08
 Zahrn
 The Land

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