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Normal Approximation Notes Spring

# Normal Approximation Notes Spring - 2 of 6 1.2 The...

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2 of 6 1.2 The approximation 0 5 10 15 20 25 30 0.0 0.1 0.2 0.3 0.4 P binom ( x , n = 20 , p = 0.05 29 x P binom ( x 29 0 5 10 15 20 25 30 P binom ( x , n = 20 , p = 0.1 29 x 0 5 10 15 20 25 30 P binom ( x , n = 20 , p = 0.5 29 x 0 5 10 15 20 25 30 P binom ( x , n = 20 , p = 0.95 29 x Sometimes the binomial has the same shape as the normal. 1.2 The approximation Normal distribution approximation of the binomial distribution. Definition 1.1 A binomial distribution can be approximated as a normal distribution when: np 5 and nq 5 (1) recall that a binomial random variable has: μ = np (2) σ = npq (3) Illustration of normal approximation Given a binomial distribution with n = 20 and p = 0 . 5 then np = nq = 10 5, therefore the approximation should be valid. Anthony Tanbakuchi MAT167

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3 of 6 0 5 10 15 20 25 30 0.00 0.05 0.10 0.15 Normal Approx to Binom: n=20, p=0.5 x binomial dist P(x) normal approx f(x) Thus, if np 5 and nq 5 we can use the normal distribution to approxi- mately describe a binomial random variable. Question 1 . If we use the normal distribution to approximate the binomial, can we ﬁnd P ( x = 10) with the normal distribution? Area under the normal distribution Given a binomial distribution with n = 20 and p = 0 . 5, ﬁnd P ( x = 10) using the normal approximation. Anthony Tanbakuchi
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Normal Approximation Notes Spring - 2 of 6 1.2 The...

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