Section 7.5 The Normal Approximation to the Binomial Distribution 325 Since the sample proportion is approximately normally distributed whenever np ≥ 10 and n(1 − p) ≥ 10, the number of successes is also approximately normally distributed under these conditions. Therefore, the normal curve can also be used to compute approximate probabilities for the binomial distribution. We begin by reviewing the conditions under which a random variable has a binomial distribution. Conditions for the Binomial Distribution 1. A fixed number of trials are conducted. 2. There are two possible outcomes for each trial. One is labeled ‘‘success’’ and the other is labeled ‘‘failure.’’ 3. The probability of success is the same on each trial. 4. The trials are independent. This means that the outcome of one trial does not affect the outcomes of the other trials. 5. The random variable X represents the number of successes that occur. Notation: The following notation is commonly used: ∙ The number of trials is denoted by n. ∙ The probability of success is denoted by p, and the probability of failure is 1 − p. Mean, Variance, and Standard Deviation of a Binomial Random Variable Let X be a binomial random variable with n trials and success probability p. Then the mean of X is ��X = np The variance of X is ��2X = np(1 − p) The standard deviation of X is ��X = √ np(1 − p) Explain It Again Calculating binomial probabilities: Binomial probabilities can be computed exactly by using the methods described in Section 6.2. Using these methods by hand is extremely difficult. The normal approximation provides an easier way to approximate these probabilities when computing by hand. Binomial probabilities can be very difficult to compute exactly by hand, because many terms have to be calculated and added together. For example, imagine trying to compute the probability that the number of heads is between 75 and 125 when a coin is tossed 200 times. To do this, one would need to compute the following sum: P(X = 75) + P(X = 76) + · · · + P(X = 124) + P(X = 125) This is nearly impossible to do without technology. Fortunately, probabilities like this can be approximated very closely by using the normal curve. In the days before cheap computing became available, use of the normal curve was the only feasible method for doing these calculations. The normal approximation is somewhat less important now, but is still useful for quick ‘‘back of the envelope’’ calculations. Recall from Section 7.4 that a sample proportion ̂p is approximately normally distributed whenever np ≥ 10 and n(1 − p) ≥ 10. Now if X is a binomial random variable representing the number of successes in n trials, the sample proportion is given by ̂p = X∕n. Since ̂p is obtained simply by dividing X by the number of trials, it is reasonable to expect
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