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Lesson 7.8 Normal Approximation and the Binomial Distribution Notes Statistics Page 1 of 3 Normal Approximation to the Binomial Distributions Consider a binomial distribution where o n = the number of trials o r = number of successes o p = probability of success on a single trial o q = 1 – p = probability of failure on a single trial If np > 5 and nq > 5, then r has a binomial distribution that is approximated by a normal distribution with µ = np and σ = npq . o As n increases, the approximation becomes better. Example 1: The owner of a new apartment building must install 25 water heaters. From past experience in other apartment buildings, she knows that Quick Hot is a good brand. A Quick Hot heater is guaranteed for only 5 years, but from the owner’s past experience, she knows that the probability that it will last 10 years is 0.25. a. What is the probability that 8 or more of the 25 water heaters will last at least 10 years? Define success to mean a water heater lasts at least 10 years. b. How does this result compare with the result we can obtain by using the formula for binomial probability distribution with n = 25 and p = 0.25? c. How do the results of parts (a) and (b) compare? Lesson 7.8 Normal Approximation and the Binomial Distribution Notes Statistics Page 2 of 3 Continuity correction Needs to be made when approximating a binomial distribution with a normal distribution o If the discrete variable is a left point of an interval, subtract 0.5 to obtain the corresponding normal variable. o If the discrete variable is a right point of an interval, add 0.5 to obtain the corresponding normal variable. Example 2: For many years of observation, a biologists knows that the probability is only 0.65 that any given Artic tern will survive the migration from its summer nesting area to its winter feeding grounds. A random sample of 500 Arctic terns were banded at their summer nesting area. Use the normal approximation to the binomial and the following to find the probability that between 310 and 340 of the banded Arctic terns will survive the migration. Let r be the number of surviving terns. a. To approximate P(310 ≤ r ≤340), we use the normal curve with µ = _________ and σ = _________. b. P(310 ≤ r ≤340), is approximately equal to P( _____ ≤ x ≤ ____ ), where x is a variable from the normal distribution described in part (a). c. Convert the condition P( _____ ≤ x ≤ ____ ) to a condition in standard units. (Find the z scores.) d. Find P( _____ ≤ r ≤ ____ ). e. Will the normal distribution make a good approximation to the binomial for this problem? Explain your answer. Lesson 7.8 Normal Approximation and the Binomial Distribution Notes Statistics Page 3 of 3 Example 3: Throw a fair coin for 200 times and let r be the number of Tail occurring in the 200 trials. Since the coin is fair, p = 0.5 and q = 0.5. a. To approximate P( 90 ≤ r ≤ 120 ), we use the normal curve with µ = _________ and σ = _________. b. P( 90 ≤ r ≤ 120 ) is approximately equal to P( _____ ≤ x ≤ ____ ), where x is a variable from the normal distribution described in part (a). c. Convert the probability about x in part (b) into a probability about the standard normal distribution z: P( _____ ≤ z ≤ ____ ). d. Find the approximate value using the result of part (c). e. Will the normal distribution make a good approximation to the binomial for this problem? Explain your answer. Assignment: p. 294 # 1, 5, 11; p. 298 # 5, 8, 10, 17, 19