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Chapter. 7. The Normal Probability Distribution. Section. 7.4. The Normal Approximation to the Binomial Probability Distribution. Objectives Approximate binomial probabilities using the normal distribution. Objective 1. Approximate Binomial Probabilities Using the Normal Distribution.

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  1. Chapter 7 The Normal Probability Distribution

  2. Section 7.4 The Normal Approximation to the Binomial Probability Distribution

  3. Objectives • Approximate binomial probabilities using the normal distribution

  4. Objective 1 • Approximate Binomial Probabilities Using the Normal Distribution

  5. Criteria for a Binomial Probability Experiment An experiment is said to be a binomial experiment if all of the following are true: 1. The experiment is performed n independent times. Each repetition of the experiment is called a trial. Independence means that the outcome of one trial will not affect the outcome of the other trials. 2. For each trial, there are two mutually exclusive outcomes - success or failure. 3. The probability of success, p, is the same for each trial of the experiment.

  6. For a fixed p, as the number of trials n in a binomial experiment increases, the probability distribution of the random variable X becomes more nearly symmetric and bell-shaped. As a rule of thumb, if np(1 – p) > 10, the probability distribution will be approximately symmetric and bell-shaped.

  7. The Normal Approximation to the Binomial Probability Distribution If np(1 – p) ≥ 10, the binomial random variable X is approximately normally distributed, with mean μX = np and standard deviation

  8. P(X = 18) ≈ P(17.5 <X< 18.5)

  9. P(X< 18) ≈ P(X< 18.5)

  10. Exact Probability Using Binomial: P(a) Approximate ProbabilityUsing Normal: P(a – 0.5 ≤ X ≤ a + 0.5) Graphical Depiction

  11. Exact Probability Using Binomial: P(X ≤ a) Approximate ProbabilityUsing Normal: P(X ≤ a + 0.5) Graphical Depiction

  12. Exact Probability Using Binomial: P(X ≥ a) Approximate ProbabilityUsing Normal: P(X ≥ a – 0.5) Graphical Depiction

  13. Exact Probability Using Binomial: P(a ≤ X ≤ b) Approximate ProbabilityUsing Normal: P(X ≥ a – 0.5) Graphical Depiction

  14. EXAMPLE Using the Binomial Probability Distribution Function • According to the Experian Automotive, 35% of all car-owning households have three or more cars. • In a random sample of 400 car-owning households, what is the probability that fewer than 150 have three or more cars?

  15. EXAMPLE Using the Binomial Probability Distribution Function According to the Experian Automotive, 35% of all car-owning households 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?

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