INTRODUCTION TO STATISTICS AND ECONOMETRICS
NORMAL APPROXIMATION OF BINOMIAL
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Here we shall consider in detail the normal approximation of a binomial variable as an application of the Lindeberg-Levy CLT (Theorem 6.2.2). In Definition 5.1.1 we defined a binomial variable X as a sum of i. i.d. Bernoulli variables (T,): that is, X = Z”=1F,-, where Yt = 1 with probability p and = 0 with probability q = 1 — p. Since (F,) satisfy the conditions of the Lindeberg-Levy CLT, with EYt = p and VY, = pq, we can conclude
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FIGURE 6.1 Normal approximation of B(b,0.5) |
As we stated in the last paragraph of Section 6.2, we may replace A
A A
above by ~. Or we may state alternatively that X/n ~ N(p, pq/n) or that
д
X ~ (np, npq). We shall consider three examples of a normal approximation of a binomial.
EXAMPLE 6.3.1 Let X be as defined in Example 5.1.1. Since EX = 2.5 and VX = 1.25 in this case, we shall approximate binomial X by normal X* ~ N{2.5, 1.25). The density function f(x) of iV(2.5, 1.25) is, after some rounding off,
(6.3.2) f(x) = g-g exp [-(ж - 2.5)2/2.5].
Using (5.1.12) and (6.3.2), we draw the probability step function of binomial X and the density function of normal X* in Figure 6.1. The figure suggests that P(X = 1) should be approximated by P(0.5 < X* < 1.5), P(X = 2) by P(1.5 < X* < 2.5), and so on. As for P(X = 0), it may be approximated either by P(X* < 0.5) or P(—0.5 < X* < 0.5). The same is true of P(X = 5). The former seems preferable, however, because it makes the sum of the approximate probabilities equal to unity. The true probabilities and their approximations are given in Table 6.1.
EXAMPLE 6.3.2 Change the above example to p = 0.2. Then EX = 1 and VX = 0.8. The results are summarized in Table 6.2 and Figure 6.2.
example 6.3.3 If 5% of the labor force is unemployed, what is the probability that one finds three or more unemployed workers among
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table 6.1 Normal approximation of В(5, 0.5)
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table 6.2 Normal approximation of B(5, 0.2) |
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k |
P(X = k) |
Approximation |
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0 |
0.3277 |
0.2877 |
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1 |
0.4096 |
0.4246 |
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2 |
0.2048 |
0.2412 |
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3 |
0.0512 |
0.0439 |
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4 |
0.0064] |
0.0026 |
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5 |
0.0003J |
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FIGURE 6.2 Normal approximation of B(5,0.2) |
twelve randomly chosen workers? What if 50% of the labor force is unemployed?
Let X be the number of unemployed workers among twelve workers. Then X ~ B(12, p), where we first assume p = 0.05. We first calculate the exact probability:
(6.3.3)
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P(X > 3) = 1 - P(X = 0) - P(X = 1) - P(X = 2)
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= 0.02.
P(Z > 2.52) = 0.0059,
where Z is N(0, 1). This is a poor approximation.
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Next, put p = 0.5. Then the exact probability is given by
and the approximation using X* ~ N(6, 3) yields
(6.3.6) P(X > 3) = P(X* > 2.5) = P(Z > -2.02) = 0.9783,
which is a good approximation.
