Normal Approximation to the Binomial

# Normal Approximation to the Binomial - P(X>4.5 since the...

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Normal Approximation to the Binomial Recall the binomial distribution where n = number of trials r = number of successes p = Probability (success) q = 1-p = Probability (failure) Thm : If np>5, and nq>5, then the binomial distribution can be approximated by a normal distribution with mean = u = np, and std.deviation sigma = (npq)^1/2, provided we apply a continuity correction. The conditions np>5 and nq>5 will guarantee that the distribution is bell shaped, unimodal, symmetric. In general , as n gets larger, the approximation gets better. Continuity Correction : when we use a continuous curve(normal) to approximate a discrete distribution (binomial), we must spread out the value of the discrete random variable over the continuous scale. Thus the whole number 2 corresponds to [1.5 , 2.5) on the continous scale. If we wish to say P(r greater than or equal to 5), this would correspond to

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Unformatted text preview: P(X>4.5), since the bar for 5 actually stretches from 4.5 to 5.5, and we need to include the entire bar of 5 and move right. If we wished to say P(r less than or equal to 5), this would correspond to P(X<5.5) , since again the bar ends at 5.5, and we need to include the whole bar for 5 and move left. Example : n = 25, p = .3, find P(r>11) . Use the normal approx to the binomial . Example : The probability is .65 that an animal survives migration to its winter location. For a random sample of 500 animals, use the normal approximation to the binomial to find the probability that between 310 and 340 animals (inclusive) survive migration. Example : The probability that a driver regularly wears a seatbelt is known to be . 75. Find the probability that in a random check of 100 cars, 70 or fewer drivers were wearing their seatbelts. Use the normal approximation....
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