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AP Statistics Notes 6.4 Binomial Setting – 4 conditions MUST be satisfied: 1. Binary – each observation falls into one of two categories: “success” or “failure” 2. Independent – knowing the result of one trial does not have any effect on the result of any other trial 3. Number – must be fixed in advance 4. Success – probability , p, of success is the same for each trial Binomial Distribution: probability distribution of X with parameters n and p abbreviation: B(n, p) Is this a binomial setting? 1. A balanced die is tossed 5 times; X is the number of times a 6 is rolled. 2. An SRS of 100 disposable razors; X is the number of defective razors. On average, 500 out of every 100,000 razors are defective. 3. An engineer chooses an SRS of 10 switches from a shipment of 10,000 switches. Suppose that (unknown to the engineer) 10% of the switches in the shipment are bad. The engineer counts the number X of bad switches in the sample. 4. Blood type is inherited. If both parents carry genes for the O and A blood types, then each child has probability 0.25 of getting two O genes and so of having blood type O. Different children inherit independently of each other. The number of O blood types among 5 children of these parents is the count X of successes. 5. Deal 10 cards from a shuffled deck and count the number X of red cards. Binomial Probability Distribution Function OR Binompdf Examples: 1. Fifty-seven percent of companies in the U.S. use networking to recruit workers. What is the probability that in a survey of ten companies exactly half of them use networking to recruit workers? 2. If X represents the number of girls in families having four children, then X is a binomial random variable with n = 4 and p = .5. What is the probability that a family has 2 girls? Binomial Cumulative Distribution Function OR Binomcdf for every possible value of k 3. Sixty percent of teenagers who drink alcohol do so because of peer pressure. In a sample of 15 teenagers who drink, find a. P(X = 5) b. probability that five or fewer do so because of peer pressure. c. probability that five or more do so because of peer pressure. d. probability that more than five do so because of peer pressure. 4. Approximately 10% of students are left-handed. If we select a random sample of 15 students, find: a. the probability that exactly 3 students in the sample are left-handed. b. the probability that more than 3 students in the sample are left-handed. c. the probability that at most 3 students in the sample are left-handed. d. the probability that at least 3 students in the sample are left-handed. Mean and standard deviation of a Binomial Random Variable: B np B np (1 p ) Examples: 5. If X represents the number of girls in families having four children, then X is a binomial random variable with n = 4 and p = .5. What is the mean and standard deviation? Normal Approximation to Binomial Distributions When n increases, the formula becomes awkward to use. That leaves us with two options: 1. use software or statistical calculator OR 2. as the number of trials n gets larger, the binomial distribution gets close to a normal distribution As n gets larger B(n,p) N B , B ***As a rule of thumb, we only use the normal approximation when n and p satisfy np 10 AND n(1 p) 10 Examples. 6. Sixty percent of teenagers who drink alcohol do so because of peer pressure. Find the normal approximation probability that in a sample of 25 teenagers who drink, a. 20 or fewer do so because of peer pressure. b. at least 20 do so because of peer pressure.