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Sta220 Section 4.7- Approximating a Binomial Distribution with a Normal Distribution As a rule of thumb, if the interval ________________________ should lie within the range of the binomial random variable x (i.e., from 0 to n), the probability distribution will be approximately symmetric and bell-shaped The Normal Approximation to the Binomial Probability Distribution If the interval 𝜇 ± 3𝜎 lies in the range from 0 to n, the binomial random variable X is approximately normally distributed, with mean μ = np and standard deviation 𝜎 = √𝑛𝑝𝑞. Exact Probability Using Binomial 𝑃(𝑎) 𝑃(𝑋 ≤ 𝑎) 𝑃(𝑋 ≥ 𝑎) 𝑃(𝑎 ≤ 𝑋 ≤ 𝑏) Approximate Probability Using Normal Graphical Depiction EXAMPLE According to the Experian Automotive, 35% of all car-owning households have three or more cars. a. In a random sample of 400 car-owning households, what is the probability that fewer than 150 have three or more cars? b. In a random sample of 400 car-owning households, what is the probability that at least 160 have three or more cars? EXAMPLE Suppose a manufacturer of calculators chooses 200 stamped circuits from the day’s production and determines x, the number of defective circuits in the sample. Suppose that up to a 6% rate of defectives is considered acceptable for the process. a. Find the mean and standard deviation of x, assuming that the rate of defectives is 6%. b. Use the normal approximation to determine the probability that 20 or more defectives are observed in the sample of 200 circuits.
