Example 3.6.10 A campus network engineer would like to estimate the fraction p of p going over the fiber optic link from the campus network to Bardeen Hall that are digital vide (DVD) packets. The engineer writes a script to examine n packets, counts the number X th DVD packets, and uses p = to estimate p. The inspected packets are separated by hundr other packets, so it is reasonable to assume that each packet is a DVD packet with probabi independently of the other packets. X n (a) Using the Gaussian approximation to the binomial distribution, find an approximation t P{p - p ≤ 8} as a function of p, n, and 8. Evaluate it for p = 0.5 and for p = 0.1, with 8 and n = 1000. 1060 DELO
Example 3.6.10 A campus network engineer would like to estimate the fraction p of p going over the fiber optic link from the campus network to Bardeen Hall that are digital vide (DVD) packets. The engineer writes a script to examine n packets, counts the number X th DVD packets, and uses p = to estimate p. The inspected packets are separated by hundr other packets, so it is reasonable to assume that each packet is a DVD packet with probabi independently of the other packets. X n (a) Using the Gaussian approximation to the binomial distribution, find an approximation t P{p - p ≤ 8} as a function of p, n, and 8. Evaluate it for p = 0.5 and for p = 0.1, with 8 and n = 1000. 1060 DELO
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![Example 3.6.10 A campus network engineer would like to estimate the fraction p of packets
going over the fiber optic link from the campus network to Bardeen Hall that are digital video disk
(DVD) packets. The engineer writes a script to examine n packets, counts the number X that are
DVD packets, and uses p= to estimate p. The inspected packets are separated by hundreds of
other packets, so it is reasonable to assume that each packet is a DVD packet with probability p,
independently of the other packets.
(a) Using the Gaussian approximation to the binomial distribution, find an approximation to
P{p-p ≤ 8} as a function of p, n, and 8. Evaluate it for p = 0.5 and for p = 0.1, with 8 = 0.02
and n 1000.
(b) If p = 0.5 and n = 1000, find 8 so P{p-p ≤ 8} ≈ 0.99. Equivalently, P{p € [p-8,p+6]} ≈ 0.99.
Note that p is not random, but the confidence interval [p-8,p+8] is random. So we want to find
the half-width 8 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 of the confidence interval as a function of p. A reasonable approach is to select & so
that, the Gaussian approximation to P{p € [p − 6, p + 6]} is greater than or equal to 0.99 for any
value of p. Find such a 6 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 [p-0.01, p +0.01] contains p with probability at least 0.99 (according
to the Gaussian approximation of the binomial)?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fe27963c7-4ee2-4128-b995-9ca1637a6da7%2Fb37fa48b-5ecc-4b6c-ad10-640a9bdc6ddf%2Fdpu5jva_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Example 3.6.10 A campus network engineer would like to estimate the fraction p of packets
going over the fiber optic link from the campus network to Bardeen Hall that are digital video disk
(DVD) packets. The engineer writes a script to examine n packets, counts the number X that are
DVD packets, and uses p= to estimate p. The inspected packets are separated by hundreds of
other packets, so it is reasonable to assume that each packet is a DVD packet with probability p,
independently of the other packets.
(a) Using the Gaussian approximation to the binomial distribution, find an approximation to
P{p-p ≤ 8} as a function of p, n, and 8. Evaluate it for p = 0.5 and for p = 0.1, with 8 = 0.02
and n 1000.
(b) If p = 0.5 and n = 1000, find 8 so P{p-p ≤ 8} ≈ 0.99. Equivalently, P{p € [p-8,p+6]} ≈ 0.99.
Note that p is not random, but the confidence interval [p-8,p+8] is random. So we want to find
the half-width 8 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 of the confidence interval as a function of p. A reasonable approach is to select & so
that, the Gaussian approximation to P{p € [p − 6, p + 6]} is greater than or equal to 0.99 for any
value of p. Find such a 6 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 [p-0.01, p +0.01] contains p with probability at least 0.99 (according
to the Gaussian approximation of the binomial)?
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