a. To determine Find the approximation value of P ( X ≤ 70 ) . Explanation Given info: The random variable X is follows a binomial distribution with n = 200 and p = 0.4 . Calculation: The variable X is a binomial random variable with parameter n and p , Z = X − n p n p ( 1 − p ) is approximately a standard normal random variable. The formula for mean in binomial distribution is, μ = n p The formula for standard deviation is, σ = n p ( 1 − p ) The value of mean is, μ = 200 × 0.4 = 80 The value of standard deviation is, σ = 200 × 0.4 ( 1 − 0.4 ) = 200 × ( 0.4 × 0.6 ) = 48 = 6.93 The requirements to check whether the normal distribution can be used for approximate the distribution of x , the binomial random variable are n p ≥ 5 and n ( 1 − p ) ≥ 5 . That is, Condition 1: n p ≥ 5 Condition 2: n ( 1 − p ) ≥ 5 Condition 1: n p ≥ 5 Substitute 200 for n and 0.4 for p in the np . n p = 200 × 0.4 = 80 ≥ 5 Thus, the given requirement n p ( = 80 ) ≥ 5 is satisfied b. To determine Find the approximation value of P ( 70 < X < 90 ) . c. To determine Find the approximation value of P ( X = 80 ) .

Applied Statistics and Probability for Engineers

6th Edition

ISBN: 9781118539712

Author: Douglas C. Montgomery

Publisher: WILEY

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Chapter 4.7, Problem 95E

To determine

Find the approximation value of P(X≤70) .

To determine

Find the approximation value of P(70<X<90) .

To determine

Find the approximation value of P(X=80) .

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Chapter 4.6, Problem 94E

Chapter 4.7, Problem 96E

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Applied Statistics and Probability for Engineers

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