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Bernoulli Trials Two Possible Outcomes –Success, with probability p –Failure, with probability q = 1 p Trials are independent.

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Binomial Distribution For n Bernoulli trials, the number of successes X is a binomial random variable. The probability of k successes is given by the binomial probability formula: As k varies with fixed n and p, the binomial probabilities define a binomial probability distribution over {0, 1, 2, …, n}.

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Sampling Distribution of the Count in a SRS When the population is much larger than the sample, the count of X successes in a SRS of size n has approximately the Binomial(n,p) distribution (given that the true proportion of successes in the population is p). As a rule of thumb, we use the binomial sampling distribution for counts when the population is at least 10 times as large as the sample.

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Mean and Standard Deviation of a Binomial RV X (i.e., of a sample count) Mean and Standard Deviation of a sample proportion,

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Law of Large Numbers Informal: If n is large, the proportion of successes in n Bernoulli trials will be very close to p. Formal: For Bernoulli trials with n and p, as n , for all > 0, where is the sample proportion.

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Normal Approximation Draw a SRS of size n from a large population having proportion p of successes. Let X be the count of successes in the sample and = X/n the sample proportion. When n is large, the sampling distributions of the two statistics are approximately normal: As a rule of thumb, we use the approximation for values of n and p such that np 10 and n(1 p) 10.

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Example 1 – Normal Approximation of Counts Suppose you flip a balanced coin 1000 times. What is the probability of getting between 480 and 532 heads?

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Example 2 – Normal Approximation of Proportions A corporation receives 100 applications for a position from recent college graduates in business. Assuming that these applications constitute a random sample of graduates in business, what is the probability that between 25% and 35% of the applicants are women if 30% of all recent college graduates in business are women?

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