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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