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Statistics Notes: 7.6 Using the Normal Distribution to Approximate a Binomial Distribution. 7.6 The Normal Approximation to the Binomial Probability Distribution According to Nielson, 75% of all US households have cable TV. In a random sample of 60 households, find the probability that at least 58 have cable TV. e? p sha tion ribu dist dict Pre Skewed Left? Set up the NCTM core math tools simulator to run in the background. Binomial Probability Distribution: A probability distribution in which there are exactly two mutually exclusive outcomes. Criteria for a binomial probability experiment: 1. The experiment is performed a fixed number (n) of times (trials). 2. The trials are independent. 3. For each trial, there are 2 mutually exclusive outcomes: • Success P(success) = p • Failure P(failure) = 1 p 4. The P(success) is fixed for each trial of the experiment. Binomial Probability Distribution Function: The probability of obtaining x successes in n independent trials of a binomial experiment is: According to Nielson, 75% of all US households have cable TV. In a random sample of 60 households, find the probability that at least 58 have cable TV. What does the simulation show as the shape of the binomial distribution? after 34,808 samples of size 60. As the number of trials n (sample size) in a binomial experiment increases, the probability distribution becomes more nearly symmetric and normal. The Normal Approximation to the Binomial Probability Distribution If , then the binomial random variable X is approximately normally distributed with a mean and standard deviation 1 Statistics Notes: 7.6 Using the Normal Distribution to Approximate a Binomial Distribution. We have to approximate the area under the normal curve because we are using a continuous function to find a discrete probability. Exact Probability Using Binomial Approximate Probability Using Normal Curve Graphic Representation 2 Statistics Notes: 7.6 Using the Normal Distribution to Approximate a Binomial Distribution. A. The probability that at least 60 households have a gas stove. B. The probability that exactly 12 defective parts are in a shipment. C. The probability that fewer than 40 families have a pet D. The probability that there are between 30 and 40 tornadoes. Do you need to include the bars? Example Describe the area under the curve that should be found. According to the Information Please almanac, 80% of adult smokers started smoking before they were 18 years old. A researcher randomly selects 100 smokers. A. Can a normal approximation be used? B. Find the probability that exactly 80 individuals started smoking before they were 18. C. Find the probability that at least 80 of them started smoking before they were 18. D. Find the probability that fewer than 70 of them started before 18. E. Find the probability that between 70 and 90 individuals started smoking before they turned 18. 3 Statistics Notes: 7.6 Using the Normal Distribution to Approximate a Binomial Distribution. According to the Information Please almanac, 48% of women between the ages of 18 and 24 live with their parents. A researcher gives a survey to 200 randomly selected female students at a local community college. 120 of the women reported that they live with their parents. Is this unusual? Why? sample from a community college? HW: p442 C&V #1 3; p442 #1 25 odd. 4
