1. Copyright © Cengage Learning. All rights reserved. 8 PROBABILITY DISTRIBUTIONS AND STATISTICS

2. Copyright © Cengage Learning. All rights reserved. 8.6 Applications of the Normal Distribution

3. 3 Approximating Binomial Distributions

4. 4 One important application of the normal distribution is that it provides us with an accurate approximation of other continuous probability distributions. Here, we show how a binomial distribution may be approximated by a suitable normal distribution. This technique leads to a convenient and simple solution to certain problems involving binomial probabilities.

5. 5 Approximating Binomial Distributions We know that a binomial distribution is a probability distribution of the form P(X = x) = C(n, x)p x q n–x For small values of n, the arithmetic computations of the binomial probabilities may be done with relative ease. However, if n is large, then the work involved becomes prodigious, even when tables of P(X = x) are available. x = 0, 1, 2,..., n (19)

6. 6 Approximating Binomial Distributions For example, if n = 50, p =.3, and q =.7, then the probability of ten or more successes is given by

7. 7 Approximating Binomial Distributions To see how the normal distribution helps us in such situations, let’s consider a coin-tossing experiment. Suppose a fair coin is tossed 20 times and we wish to compute the probability of obtaining 10 or more heads. The solution to this problem may be obtained, of course, by computing P(X  10) = P(X = 10) + P(X = 11) + · · · + P(X = 20)

8. 8 Approximating Binomial Distributions The inconvenience of this approach for solving the problem at hand has already been pointed out. As an alternative solution, let’s begin by interpreting the solution in terms of finding the area of suitable rectangles of the histogram for the distribution associated with the problem. We may use Equation (19) to compute the probability of obtaining exactly x heads in 20 coin tosses.

9. 9 Approximating Binomial Distributions The results lead to the binomial distribution displayed in Table 13. Table 13

10. 10 Approximating Binomial Distributions Using the data from the table, we next construct the histogram for the distribution (Figure 30). Figure 30 Histogram showing the probability of obtaining x heads in 20 coin tosses

11. 11 Approximating Binomial Distributions The probability of obtaining 10 or more heads in 20 coin tosses is equal to the sum of the areas of the shaded rectangles of the histogram of the binomial distribution shown in Figure 31. Figure 31 The shaded area gives the probability of obtaining 10 or more heads in 20 coin tosses.

12. 12 Approximating Binomial Distributions Next, observe that the shape of the histogram suggests that the binomial distribution under consideration may be approximated by a suitable normal distribution. Since the mean and standard deviation of the binomial distribution are given by

13. 13 Approximating Binomial Distributions respectively, the natural choice of a normal curve for this purpose is one with a mean of 10 and standard deviation of 2.24.

14. 14 Approximating Binomial Distributions Figure 32 shows such a normal curve superimposed on the histogram of the binomial distribution. The good fit suggests that the sum of the areas of the rectangles representing P(X  10), the probability of obtaining 10 or more heads in 20 coin tosses, may be approximated by the area of an appropriate region under the normal curve. Figure 32 Normal curve superimposed on the histogram for a binomial distribution

15. 15 Approximating Binomial Distributions To determine this region, let’s note that the base of the portion of the histogram representing the required probability extends from x = 9.5 on, since the base of the leftmost rectangle in the shaded region is centered at x = 10 and the base of each rectangle has length 1 (Figure 33). Figure 33 P(X  10) is approximated by the area under the normal curve.

16. 16 Approximating Binomial Distributions Therefore, the required region under the normal curve should also have x  9.5. Letting Y denote the continuous normal variable, we obtain P(X  10)  P(Y  9.5) = P(Y > 9.5)

17. 17 Approximating Binomial Distributions  P(Z > –0.22) = P(Z < 0.22) =.5871 The exact value of P(X  10) may be found by computing P(X = 10) + P(X = 11) + · · · + P(X = 20) in the usual fashion and is equal to.5881. Use the table of values of Z.

18. 18 Approximating Binomial Distributions Thus, the normal distribution with suitably chosen mean and standard deviation does provide us with a good approximation of the binomial distribution. In the general case, the following result, which is a special case of the central limit theorem, guarantees the accuracy of the approximation of a binomial distribution by a normal distribution under certain conditions.

19. 19 Approximating Binomial Distributions Note: It can be shown that if both np and nq are greater than 5, then the error resulting from this approximation is negligible.

20. 20 Applications Involving Binomial Random Variables

21. 21 Applied Example 4 – Quality Control An automobile manufacturer receives the microprocessors that are used to regulate fuel consumption in its automobiles in shipments of 1000 each from a certain supplier. It has been estimated that, on the average, 1% of the microprocessors manufactured by the supplier are defective. Determine the probability that more than 20 of the microprocessors in a single shipment are defective.

22. 22 Applied Example 4 – Solution Let X denote the number of defective microprocessors in a single shipment. Then X has a binomial distribution with n = 1000, p =.01, and q =.99, so

23. 23 Applied Example 4 – Solution Approximating the binomial distribution by a normal distribution with a mean of 10 and a standard deviation of 3.15, we find that the probability that more than 20 microprocessors in a shipment are defective is given by cont’d Where Y denotes the normal random variable

24. 24 Applied Example 4 – Solution  P(Z > 3.33)  P(Z < –3.33) =.0004 In other words, approximately 0.04% of the shipments containing 1000 microprocessors each will contain more than 20 defective units. cont’d

25. 25 Practice p. 489 Self-Check Exercises #2