3.2 THE QUANTILE TEST AND ESTIMATION OF XP 135 ROBLEMS The continuity correction. It is obvious that if Y has a binomial distribution, then 1. P(Y 4) P(Ys 4.1) = = P(Ys 4.999) because Y takes on 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 trying to use a 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 only integer values, such as 4 or 5, but no values between integers. P(Y 4) PZ40.5-np 12 Vngp 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. 0.1, find the exact value of P(Ys 1) from Table A3. Use the normal 20, p (a) For n = 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 n1 and pi and n2 and p2, respectively. 2. (a) Show that Y1/n1- Y2/n2 has mean pi - P2 (b) Show that Y1/n1 Y2/n2 has variance p(1 - p1)/n1 + p2(1 - pa)/n2. Y2)/n2 as an estimate of the variance of fE (c) Justify the use of Y1(n, Y1)/n3 +Y2(n2 Y1/n1 Y2/n2. (d) If Y1/n1 Y2/n2 is approximately normal, show how an approximate 1 - a confidence interval for (p1 - p2) is given by Y1 Y2-Z1-a/2S
3.2 THE QUANTILE TEST AND ESTIMATION OF XP 135 ROBLEMS The continuity correction. It is obvious that if Y has a binomial distribution, then 1. P(Y 4) P(Ys 4.1) = = P(Ys 4.999) because Y takes on 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 trying to use a 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 only integer values, such as 4 or 5, but no values between integers. P(Y 4) PZ40.5-np 12 Vngp 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. 0.1, find the exact value of P(Ys 1) from Table A3. Use the normal 20, p (a) For n = 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 n1 and pi and n2 and p2, respectively. 2. (a) Show that Y1/n1- Y2/n2 has mean pi - P2 (b) Show that Y1/n1 Y2/n2 has variance p(1 - p1)/n1 + p2(1 - pa)/n2. Y2)/n2 as an estimate of the variance of fE (c) Justify the use of Y1(n, Y1)/n3 +Y2(n2 Y1/n1 Y2/n2. (d) If Y1/n1 Y2/n2 is approximately normal, show how an approximate 1 - a confidence interval for (p1 - p2) is given by Y1 Y2-Z1-a/2S
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Transcribed Image Text:3.2 THE QUANTILE TEST AND ESTIMATION OF XP
135
ROBLEMS
The continuity correction. It is obvious that if Y has a binomial distribution, then
1.
P(Y
4) P(Ys 4.1) = = P(Ys 4.999)
because Y takes on
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 trying to use a
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
only integer values, such as 4 or 5, but no values between integers.
P(Y 4) PZ40.5-np
12
Vngp
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.
0.1, find the exact value of P(Ys 1) from Table A3. Use the normal
20, p
(a) For n =
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 n1 and pi and
n2 and p2, respectively.
2.
(a) Show that Y1/n1- Y2/n2 has mean pi - P2
(b) Show that Y1/n1 Y2/n2 has variance p(1 - p1)/n1 + p2(1 - pa)/n2.
Y2)/n2 as an estimate of the variance of
fE
(c) Justify the use of Y1(n, Y1)/n3 +Y2(n2
Y1/n1 Y2/n2.
(d) If Y1/n1 Y2/n2 is approximately normal, show how an approximate 1 - a confidence
interval for (p1 - p2) is given by
Y1 Y2-Z1-a/2S <p1-P2 -2+21-a/25
n1 n2
n1 n2
where
s VY1(n- Y/ni + Y2(n2- Y2) / n
and where z1-a/2 is obtained from Table A1.
THE QUANTILE TEST AND
MATION OF Xp
The binomial test may be used to test hypotheses concerning the quantiles of a
call it the quantile test. For example, we may
TAIE
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