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Section 5.5

Section 5.5. Normal Approximations to Binomial Distributions. RECALL:. For a binomial distribution: n = the number of independent trials p = the probability of success q = the probability of failure µ = np σ = √ npq. approximated as normal:. TWO conditions: np > 5 and nq > 5

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Section 5.5

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  1. Section 5.5 Normal Approximations to Binomial Distributions

  2. RECALL: • For a binomial distribution: • n = the number of independent trials • p = the probability of success • q = the probability of failure • µ = np σ = √npq

  3. approximated as normal: TWO conditions: np> 5 and nq> 5 If conditions are met, then the random variable x is normally distributed.

  4. Can the following be approximated by the normal distribution?

  5. Continuity Correction • Binomial distributions are DISCRETE, but the normal distribution is CONTINUOUS. • The binomial probability formulas from CH 4 are for exact probabilities. i.e., P(X = 4) • To adjust for continuity, move 0.5 units to the left and right of the midpoint. This allows you to include all x-values in the interval. i.e., P(3.5 < X < 4.5)

  6. Convert each BINOMIAL interval to a NORMAL interval • 1. The probability if getting between 39 and 77 successes, inclusive. • 2. The probability of getting at least 80 successes. • 3. The probability of getting fewer than 50 successes.

  7. To Approx. Binomial Probabilities • Find n, p, and q • Is np > 5? Is nq > 5? • Find µ and σ • Correct for Continuity (+ 0.5) • Find z • Use standard normal table to finish

  8. 24. A survey of US adults ages 50-64 found that 70% use the Internet. You randomly select 80 adults ages 50-64 and ask them if they use the Internet. • A. Find the prob that at least 70 people say they use the Internet. • B. Find the prob that exactly 50 people say they use the internet. • C. Find the prob that more than 60 people say they use the internet.

  9. 25. About 34% of workers in the US are college graduates. You randomly select 50 workers and ask them if they are a college graduate. • A. Find the prob that exactly 12 workers are college graduates. • B. Find the prob that more than 23 workers are college graduates. • C. Find the prob that at most 18 workers are college graduates.

  10. #26, continued • D. A committee is looking for 30 working college graduates to volunteer at a career fair. The committee randomly selects 125 workers. What is the probability that there will not be enough college graduates?

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