3.2 THE QUANTILE TEST AND ESTIMATION OF Xp 135 PROBLEMS 1. The continuity correction. It is obvious that if Y has a binomial distribution, then P(Y 4) P(Y 4.1) = .. = P(Y 4.999) because Y takes on only integer values, such as 4 or 5, but no values between integers. Therefore, which number should be used in the normal approximation to the binomial distribution: 4, or 4.1, or what? The continuity correction (because we are continuous distribution such as the normal to approximate a discrete distribution such as the binomial) says to use the number midway between two adjacent values in the discrete distribution. That is, in the binomial distribution estimate P(Y 4), with trying to use a P(Y4) PZ 0.5-np Vngp 4 dbu where Z has a normal distribution, because 4.5 is halfway between 4 and 5. Usually the continuity correction works well when using the normal distribution to approximate binomial probabilities. (a) For n 20, p = 0.1, find the exact value of P(Y 1) from Table A3. Use the normal approximation to estimate P(Y 1), first without the continuity correction and then with the continuity correction. Which estimate is closer? (b) Repeat part a, but change from p = 0.1 to p = 0.3. Now which estimate is closer? Let Y1 and Y2 be independent binomial random variables with parameters n and p, and n2 and p2, respectively 2. (a) Show that Y1/n1-Y2/n2 has mean p1 - P2 (b) Show that Y1/n1 Y2/n2 has variance p,(1 - pi)/n1 + p2(1 - p2)/n2. (c) Justify the use of Y1(n, - Y)/n, + Y2(nz Y2)/n2 as an estimate of the variance of Y1/n1 Y2/n2. 3 (d) If Y1/n1 Y2/n2 is approximately normal, show how an approximate 1 - a confidence interval for (pi - P2) is given by Y1 Y2 Y2 Y1 +Z1-a/25 П2 21-a/25
3.2 THE QUANTILE TEST AND ESTIMATION OF Xp 135 PROBLEMS 1. The continuity correction. It is obvious that if Y has a binomial distribution, then P(Y 4) P(Y 4.1) = .. = P(Y 4.999) because Y takes on only integer values, such as 4 or 5, but no values between integers. Therefore, which number should be used in the normal approximation to the binomial distribution: 4, or 4.1, or what? The continuity correction (because we are continuous distribution such as the normal to approximate a discrete distribution such as the binomial) says to use the number midway between two adjacent values in the discrete distribution. That is, in the binomial distribution estimate P(Y 4), with trying to use a P(Y4) PZ 0.5-np Vngp 4 dbu where Z has a normal distribution, because 4.5 is halfway between 4 and 5. Usually the continuity correction works well when using the normal distribution to approximate binomial probabilities. (a) For n 20, p = 0.1, find the exact value of P(Y 1) from Table A3. Use the normal approximation to estimate P(Y 1), first without the continuity correction and then with the continuity correction. Which estimate is closer? (b) Repeat part a, but change from p = 0.1 to p = 0.3. Now which estimate is closer? Let Y1 and Y2 be independent binomial random variables with parameters n and p, and n2 and p2, respectively 2. (a) Show that Y1/n1-Y2/n2 has mean p1 - P2 (b) Show that Y1/n1 Y2/n2 has variance p,(1 - pi)/n1 + p2(1 - p2)/n2. (c) Justify the use of Y1(n, - Y)/n, + Y2(nz Y2)/n2 as an estimate of the variance of Y1/n1 Y2/n2. 3 (d) If Y1/n1 Y2/n2 is approximately normal, show how an approximate 1 - a confidence interval for (pi - P2) is given by Y1 Y2 Y2 Y1 +Z1-a/25 П2 21-a/25
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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2.d in pictures
Transcribed Image Text:3.2 THE QUANTILE TEST AND ESTIMATION OF Xp
135
PROBLEMS
1.
The continuity correction. It is obvious that if Y has a binomial distribution, then
P(Y 4) P(Y 4.1) = ..
= P(Y 4.999)
because Y takes on only integer values, such as 4 or 5, but no values between integers.
Therefore, which number should be used in the normal approximation to the binomial
distribution: 4, or 4.1, or what? The continuity correction (because we are
continuous distribution such as the normal to approximate a discrete distribution such as
the binomial) says to use the number midway between two adjacent values in the discrete
distribution. That is, in the binomial distribution estimate P(Y 4), with
trying to use a
P(Y4) PZ 0.5-np
Vngp
4
dbu
where Z has a normal distribution, because 4.5 is halfway between 4 and 5.
Usually the continuity correction works well when using the normal distribution to
approximate binomial probabilities.
(a) For n
20, p = 0.1, find the exact value of P(Y
1) from Table A3. Use the normal
approximation to estimate P(Y 1), first without the continuity correction and then with
the continuity correction. Which estimate is closer?
(b) Repeat part a, but change from p = 0.1 to p = 0.3. Now which estimate is closer?
Let Y1 and Y2 be independent binomial random variables with parameters n and p, and
n2 and p2, respectively
2.
(a) Show that Y1/n1-Y2/n2 has mean p1 - P2
(b) Show that Y1/n1 Y2/n2 has variance p,(1 - pi)/n1 + p2(1 - p2)/n2.
(c) Justify the use of Y1(n, - Y)/n, + Y2(nz Y2)/n2 as an estimate of the variance of
Y1/n1 Y2/n2.
3
(d) If Y1/n1 Y2/n2 is approximately normal, show how an approximate 1 - a confidence
interval for (pi - P2) is given by
Y1 Y2
Y2
Y1
+Z1-a/25
П2
21-a/25 <p1-P2
n2
n1
n1
where
+Y2(n2- Y2)/ni
S=VY1(n1- Y)/ng
and where z1-a/2 is obtained from Table A1
THE QUANTILE TEST AND
MATION OF Xp
ihogo goncerning the auantiles of a
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